{"id":931,"date":"2024-04-09T16:26:49","date_gmt":"2024-04-09T16:26:49","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus1\/?post_type=chapter&#038;p=931"},"modified":"2025-08-17T15:52:23","modified_gmt":"2025-08-17T15:52:23","slug":"basic-classes-of-functions-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus1\/chapter\/basic-classes-of-functions-fresh-take\/","title":{"raw":"Basic Classes of Functions: Fresh Take","rendered":"Basic Classes of Functions: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li>Recognize the degree of a polynomial, find the roots of quadratic polynomials, and describe the graphs of basic odd and even polynomial functions<\/li>\r\n\t<li>Graph a piecewise-defined function<\/li>\r\n\t<li>Explain the difference between algebraic and transcendental functions<\/li>\r\n\t<li>Sketch the graph of a function that has been shifted, stretched, or reflected from its initial graph position<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Polynomial Functions<\/h2>\r\n<div class=\"textbox shaded\">\r\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\r\n<ul>\r\n\t<li class=\"whitespace-normal break-words\">A polynomial function is of the form [latex]f(x) = a\u2081x\u207f + a\u2082x\u207f\u207b\u00b9 + ... + a\u2099\u208b\u2081x + a\u2099, [\/latex] where [latex]n[\/latex] is a non-negative integer.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Key Terms:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Degree: The highest power of [latex]x[\/latex] in the polynomial<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Leading Term: The term with the highest degree<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Leading Coefficient: The coefficient of the leading term<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Types of Polynomials:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Linear: Degree [latex]1[\/latex] (e.g., [latex]f(x) = mx + b[\/latex])<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Quadratic: Degree [latex]2[\/latex] (e.g., [latex]f(x) = ax\u00b2 + bx + c[\/latex])<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Cubic: Degree [latex]3[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Higher degrees: Quartic ([latex]4[\/latex]), Quintic ([latex]5[\/latex]), etc.<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">The zero function [latex]f(x) = 0[\/latex] is considered a polynomial of degree [latex]0[\/latex].<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n<p class=\"whitespace-pre-wrap break-words\">Identify the degree, leading term, and leading coefficient for each of the following functions:<\/p>\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n\t<li class=\"whitespace-pre-wrap break-words\">[latex]h(y) = 2y\u2074 - 3y\u00b2 + 5y - 7[\/latex]<\/li>\r\n\t<li class=\"whitespace-pre-wrap break-words\">[latex]k(s) = -0.5s\u00b3 + 2s - 1[\/latex]<\/li>\r\n\t<li class=\"whitespace-pre-wrap break-words\">[latex]m(w) = w\u2075 + 3w\u2074 - 2w\u00b2 + w[\/latex]<\/li>\r\n<\/ol>\r\n<p><br \/>\r\n[reveal-answer q=\"101020\"]Show Answer[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"101020\"]<\/p>\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n\t<li class=\"whitespace-pre-wrap break-words\">\u00a0[latex]h(y) = 2y\u2074 - 3y\u00b2 + 5y - 7[\/latex]\r\n\r\n<ul>\r\n\t<li class=\"whitespace-pre-wrap break-words\">Degree: [latex]4[\/latex]<\/li>\r\n\t<li class=\"whitespace-pre-wrap break-words\">Leading Term: [latex]2y\u2074[\/latex]<\/li>\r\n\t<li class=\"whitespace-pre-wrap break-words\">Leading Coefficient: [latex]2[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li>[latex]k(s) = -0.5s\u00b3 + 2s - 1[\/latex]\r\n\r\n<ul>\r\n\t<li>Degree: [latex]3[\/latex]<\/li>\r\n\t<li>Leading Term: [latex]-0.5s\u00b3[\/latex]<\/li>\r\n\t<li>Leading Coefficient: [latex]-0.5[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li>[latex]m(w) = w\u2075 + 3w\u2074 - 2w\u00b2 + w[\/latex]\r\n\r\n<ul>\r\n\t<li>Degree: [latex]5[\/latex]<\/li>\r\n\t<li>Leading Term: [latex]w\u2075[\/latex]<\/li>\r\n\t<li>Leading Coefficient: [latex]1[\/latex] (remember, when not written, the coefficient is [latex]1[\/latex])<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<h2>Zeros of Polynomial Functions<\/h2>\r\n<div class=\"textbox shaded\">\r\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\r\n<ul>\r\n\t<li class=\"whitespace-normal break-words\">For quadratic functions [latex]f(x) = ax\u00b2 + bx + c[\/latex], zeros can be found using:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Factoring (if possible)<\/li>\r\n\t<li class=\"whitespace-normal break-words\">The quadratic formula: [latex]x = \\dfrac{-b \u00b1 \\sqrt{(b\u00b2 - 4ac)}}{ (2a)}[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">The discriminant [latex](b\u00b2 - 4ac)[\/latex] determines the nature of quadratic solutions:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Positive: Two distinct real roots<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Zero: One real root (double root)<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Negative: No real roots<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Higher-degree polynomials may have more complex methods for finding zeros.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n<p>Find the zeros of the quadratic function [latex]g(x) = 2x\u00b2 + 5x - 3[\/latex].<\/p>\r\n<p><br \/>\r\n[reveal-answer q=\"169751\"]Show Answer[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"169751\"]<\/p>\r\n<p>Set the function equal to zero:<\/p>\r\n<p style=\"text-align: center;\">[latex]2x\u00b2 + 5x - 3 = 0[\/latex]<\/p>\r\n<p>We'll use the quadratic formula since this doesn't factor easily. Here [latex]a = 2, b = 5[\/latex], and [latex]c = -3[\/latex]<\/p>\r\n<p>Apply the quadratic formula:<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{align*}<br \/>\r\nx &amp;= \\frac{-b \\pm \\sqrt{b^2 - 4ac}}{2a} \\\\<br \/>\r\nx &amp;= \\frac{-5 \\pm \\sqrt{5^2 - 4(2)(-3)}}{2(2)} \\\\<br \/>\r\nx &amp;= \\frac{-5 \\pm \\sqrt{25 + 24}}{4} \\\\<br \/>\r\nx &amp;= \\frac{-5 \\pm \\sqrt{49}}{4} \\\\<br \/>\r\nx &amp;= \\frac{-5 \\pm 7}{4}<br \/>\r\n\\end{align*}[\/latex]<\/p>\r\n<p>Simplify:<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{align*} x &amp;= \\frac{(-5 + 7)}{4} \\text{ or } \\frac{(-5 - 7)}{4} \\\\[6pt] x &amp;= \\frac{2}{4} \\text{ or } \\frac{-12}{4} \\\\[6pt] x &amp;= \\frac{1}{2} \\text{ or } -3 \\end{align*}[\/latex]<\/p>\r\n<p>We can check our solutions by plugging them back in for [latex]x[\/latex] and solving to see if the answer is [latex]0[\/latex]:<\/p>\r\n<p style=\"text-align: center;\">[latex]g(\\frac{1}{2}) = 2(\\frac{1}{2})\u00b2 + 5(\\frac{1}{2}) - 3 = \\frac{1}{2} + \\frac{5}{2} - 3 = 0[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]g(-3) = 2(-3)\u00b2 + 5(-3) - 3 = 18 - 15 - 3 = 0[\/latex]<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">Therefore, the zeros of [latex]g(x)[\/latex] are [latex]x = \\frac{1}{2}[\/latex] and [latex]x = -3[\/latex].<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<h2>Graphs of Polynomial Functions Basics<\/h2>\r\n<div class=\"textbox shaded\">\r\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\r\n<ul>\r\n\t<li class=\"whitespace-normal break-words\">The degree of a polynomial greatly influences its graph's shape.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Even-degree polynomials:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Both ends of the graph point in the same direction (up or down)<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Similar to a parabola, but may be flatter near the origin<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Odd-degree polynomials:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Ends of the graph point in opposite directions<\/li>\r\n\t<li class=\"whitespace-normal break-words\">One end goes up, the other goes down<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">The sign of the leading coefficient determines the direction of the graph's ends:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Positive leading coefficient: right end goes up<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Negative leading coefficient: right end goes down<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">As the degree increases, graphs tend to flatten near the origin and become steeper away from it.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n<p class=\"whitespace-pre-wrap break-words\">Describe the end behavior and basic shape of the graph for the polynomial function:<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex]f(x) = -2x\u2075 + 3x\u2074 - x\u00b2 + 7[\/latex]<\/p>\r\n<p><br \/>\r\n[reveal-answer q=\"399650\"]Show Answer[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"399650\"]<\/p>\r\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Identify key features:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Degree: [latex]5[\/latex] (odd)<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Leading coefficient: [latex]-2[\/latex] (negative)<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Describe end behavior:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">As [latex]x[\/latex] approaches positive infinity, [latex]f(x)[\/latex] approaches negative infinity<\/li>\r\n\t<li class=\"whitespace-normal break-words\">As [latex]x[\/latex] approaches negative infinity, [latex]f(x)[\/latex] approaches negative infinity<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Basic shape:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">The graph will have one arm pointing down on the right side<\/li>\r\n\t<li class=\"whitespace-normal break-words\">The other arm will point up on the left side<\/li>\r\n\t<li class=\"whitespace-normal break-words\">There may be turns or oscillations between these extremes<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\n\r\n[caption id=\"attachment_3625\" align=\"aligncenter\" width=\"500\"]<img class=\"wp-image-3625\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/34\/2024\/04\/01181605\/Screenshot-2024-07-01-141554-300x274.png\" alt=\"A graph of the function f(x) = -2x\u2075 + 3x\u2074 - x\u00b2 + 7 on a coordinate plane. The red curve starts in the upper left quadrant, descends steeply near the y-axis with a small bump around y=7, then continues downward crossing into the lower right quadrant. \" width=\"500\" height=\"457\" \/> Graph of the polynomial function f(x) = -2x\u2075 + 3x\u2074 - x\u00b2 + 7[\/caption]\r\n\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<h2>Piecewise-Defined Functions<\/h2>\r\n<div class=\"textbox shaded\">\r\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\r\n<ul>\r\n\t<li class=\"whitespace-normal break-words\">A piecewise-defined function uses different formulas for different parts of its domain.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Each piece of the function has its own subdomain, which together form the function's complete domain.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Piecewise functions may be continuous or discontinuous, depending on how the pieces connect.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Each piece is graphed separately within its specified domain, paying special attention to the transition points.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\">\r\n<p id=\"fs-id1170573569350\">Sketch a graph of the function<\/p>\r\n<p>[latex]f(x)=\\begin{cases} 2-x, &amp; x \\le 2 \\\\ x+2, &amp; x&gt;2 \\end{cases}[\/latex]<\/p>\r\n<p>[reveal-answer q=\"705322\"]Hint[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"705322\"]<\/p>\r\n<p>Graph one linear function for [latex]x \\le 2[\/latex] and then graph a different linear function for [latex]x&gt;2[\/latex].<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<p>[reveal-answer q=\"fs-id1170573351776\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1170573351776\"]<\/p>\r\n\r\n[caption id=\"\" align=\"alignnone\" width=\"462\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202257\/CNX_Calc_Figure_01_02_012.jpg\" alt=\"&quot;An\" width=\"462\" height=\"384\" \/> Graph of f(x)[\/caption]\r\n\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox example\">\r\n<p id=\"fs-id1170573582214\">The cost of mailing a letter is a function of the weight of the letter. Suppose the cost of mailing a letter is [latex]49\\text{\u00a2}[\/latex] for the first ounce and [latex]21\\text{\u00a2}[\/latex] for each additional ounce. Write a piecewise-defined function describing the cost [latex]C[\/latex] as a function of the weight [latex]x[\/latex] for [latex]0 &lt; x \\le 3[\/latex], where [latex]C[\/latex] is measured in cents and [latex]x[\/latex] is measured in ounces.<\/p>\r\n<p>[reveal-answer q=\"981114\"]Hint[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"981114\"]<\/p>\r\n<p>The piecewise-defined function is constant on the intervals [latex](0,1], \\, (1,2], \\, \\cdots[\/latex]<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<p>[reveal-answer q=\"fs-id1170573359603\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1170573359603\"]<\/p>\r\n<p id=\"fs-id1170573359603\">[latex]C(x)=\\begin{cases} 49, &amp; 0 &lt; x \\le 1 \\\\ 70, &amp; 1 &lt; x \\le 2 \\\\ 91, &amp; 2 &lt; x \\le 3 \\end{cases}[\/latex]<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<h2>Algebraic Functions and Transcendental Functions<\/h2>\r\n<div class=\"textbox shaded\">\r\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\r\n<ul>\r\n\t<li class=\"whitespace-normal break-words\">Algebraic Functions:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Involve addition, subtraction, multiplication, division, rational powers, and roots<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Include polynomial, rational, and root functions<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Examples: [latex]f(x) = x\u00b3 + 2x - 1, g(x) = \\frac{(x\u00b2 + 1) }{ (x - 2)}, h(x) = \\sqrt{(x + 3)}[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Transcendental Functions:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Cannot be expressed using a finite number of algebraic operations<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Main types: trigonometric, exponential, and logarithmic functions<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Examples: [latex]sin(x), e^x, log\u2082(x)[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\">\r\n<p id=\"fs-id1170573577836\">Is [latex]f(x)=\\dfrac{x}{2}[\/latex] an algebraic or a transcendental function?<\/p>\r\n<p>[reveal-answer q=\"fs-id1170573577863\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1170573577863\"]<\/p>\r\n<p id=\"fs-id1170573577863\">Algebraic<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n<p class=\"whitespace-pre-wrap break-words\">Classify the following functions as algebraic or transcendental, and identify their specific type if possible:<\/p>\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n\t<li>[latex]f(x) = \\frac{(x\u2074 + 2x\u00b2 - 5)}{ (x - 1)}[\/latex]<\/li>\r\n\t<li>[latex]g(x) = sin(x\u00b2) + cos(x)[\/latex]<\/li>\r\n\t<li>[latex]h(x) = \\sqrt[3]{(x\u00b2 + 1)} + log\u2083(x)[\/latex]<\/li>\r\n\t<li>[latex]k(x) = \\frac{(2^x + 3^x)}{x}[\/latex]<\/li>\r\n<\/ol>\r\n<p><br \/>\r\n[reveal-answer q=\"715773\"]Show Answer[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"715773\"]<\/p>\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n\t<li class=\"whitespace-pre-wrap break-words\">\u00a0Algebraic; It's a ratio of polynomials.<\/li>\r\n\t<li class=\"whitespace-pre-wrap break-words\">Transcendental;\u00a0 Involves sine and cosine, which are transcendental.<\/li>\r\n\t<li class=\"whitespace-pre-wrap break-words\">Mixed - Both algebraic and transcendental; [latex] \\sqrt[3]{(x\u00b2 + 1)}[\/latex] is algebraic (root function), but [latex]log\u2083(x)[\/latex] is transcendental.<\/li>\r\n\t<li class=\"whitespace-pre-wrap break-words\">Transcendental;\u00a0 [latex]2^x[\/latex] and [latex]3^x[\/latex] are exponential (transcendental), and the overall function can't be simplified to an algebraic form.<\/li>\r\n<\/ol>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<h2>Transformations of Functions\u00a0<\/h2>\r\n<div class=\"textbox shaded\">\r\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\r\n<ul>\r\n\t<li class=\"whitespace-normal break-words\">Vertical Shift: [latex]f(x) \u00b1 c[\/latex]\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">[latex]+c [\/latex] shifts up, [latex]-c[\/latex] shifts down<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Does not change shape of graph<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Horizontal Shift: [latex]f(x \u00b1 c)[\/latex]\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">[latex]-c[\/latex] shifts right, [latex]+c[\/latex] shifts left<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Does not change shape of graph<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Vertical Scaling: [latex]cf(x)[\/latex]\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">[latex]|c| &gt; 1[\/latex] stretches vertically, [latex]0 &lt; |c| &lt; 1[\/latex] compresses vertically<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Changes height of graph<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Horizontal Scaling: [latex]f(cx)[\/latex]\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">[latex]0 &lt; |c| &lt; 1 [\/latex]stretches horizontally, [latex]|c| &gt; 1[\/latex] compresses horizontally<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Changes width of graph<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Reflections:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">[latex]-f(x)[\/latex] reflects over [latex]x[\/latex]-axis<\/li>\r\n\t<li class=\"whitespace-normal break-words\">[latex]f(-x)[\/latex] reflects over [latex]y[\/latex]-axis<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Multiple Transformations: [latex]y = cf(a(x + b)) + d[\/latex]\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Apply in order: horizontal shift, horizontal scaling, vertical scaling, vertical shift<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n<p>Transform the following function which has the base tool-kit function [latex]f(x) = \\sqrt{x}[\/latex]:<\/p>\r\n<p style=\"text-align: center;\">[latex]g(x) = 2\\sqrt{(x - 1)} + 3[\/latex]<\/p>\r\n<p><br \/>\r\n[reveal-answer q=\"635560\"]Show Answer[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"635560\"]<\/p>\r\n<ul>\r\n\t<li class=\"whitespace-normal break-words\">Identify the base function: [latex]f(x) = \\sqrt{x}[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Break down the transformations:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Inside the square root:[latex] (x - 1)[\/latex] indicates a horizontal shift<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Outside the square root: [latex]2[\/latex] indicates vertical scaling<\/li>\r\n\t<li class=\"whitespace-normal break-words\">[latex]+3[\/latex] at the end indicates a vertical shift<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Apply transformations in order:\r\n\r\n<ul>\r\n\t<li class=\"whitespace-normal break-words\">Horizontal shift: [latex]\\sqrt{(x - 1)}[\/latex] shifts the graph [latex]1[\/latex] unit right<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Vertical scaling: [latex]2\\sqrt{(x - 1)}[\/latex] stretches the graph vertically by a factor of [latex]2[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Vertical shift: [latex]2\\sqrt{(x - 1)} + 3[\/latex] shifts the graph up [latex]3[\/latex] units<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Describe the final graph:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">The graph of [latex]g(x)[\/latex] is the square root function shifted [latex]1[\/latex] unit right, stretched vertically by a factor of [latex]2[\/latex], and shifted up [latex]3[\/latex] units.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">The domain is [latex][1, \u221e)[\/latex] because [latex]x - 1[\/latex] must be non-negative under the square root.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">The range is [latex][3, \u221e)[\/latex] due to the vertical stretch and shift.<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<p>We can check this by plugging both equations into a graphing software.<\/p>\r\n\r\n[caption id=\"attachment_3627\" align=\"aligncenter\" width=\"500\"]<img class=\"wp-image-3627\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/34\/2024\/04\/01183152\/Screenshot-2024-07-01-143131.png\" alt=\"A coordinate plane showing two graphs. The red curve represents f(x) = \u221ax, starting at the origin and gradually increasing with a decreasing slope. The blue curve represents g(x) = 2\u221a(x-1) + 3, beginning at (1, 3) and also increasing with a decreasing slope, but steeper and higher than the red curve. \" width=\"500\" height=\"500\" \/> Comparison of square root functions: f(x) = \u221ax (red) and g(x) = 2\u221a(x-1) + 3 (blue)[\/caption]\r\n\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Recognize the degree of a polynomial, find the roots of quadratic polynomials, and describe the graphs of basic odd and even polynomial functions<\/li>\n<li>Graph a piecewise-defined function<\/li>\n<li>Explain the difference between algebraic and transcendental functions<\/li>\n<li>Sketch the graph of a function that has been shifted, stretched, or reflected from its initial graph position<\/li>\n<\/ul>\n<\/section>\n<h2>Polynomial Functions<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">A polynomial function is of the form [latex]f(x) = a\u2081x\u207f + a\u2082x\u207f\u207b\u00b9 + ... + a\u2099\u208b\u2081x + a\u2099,[\/latex] where [latex]n[\/latex] is a non-negative integer.<\/li>\n<li class=\"whitespace-normal break-words\">Key Terms:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Degree: The highest power of [latex]x[\/latex] in the polynomial<\/li>\n<li class=\"whitespace-normal break-words\">Leading Term: The term with the highest degree<\/li>\n<li class=\"whitespace-normal break-words\">Leading Coefficient: The coefficient of the leading term<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Types of Polynomials:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Linear: Degree [latex]1[\/latex] (e.g., [latex]f(x) = mx + b[\/latex])<\/li>\n<li class=\"whitespace-normal break-words\">Quadratic: Degree [latex]2[\/latex] (e.g., [latex]f(x) = ax\u00b2 + bx + c[\/latex])<\/li>\n<li class=\"whitespace-normal break-words\">Cubic: Degree [latex]3[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Higher degrees: Quartic ([latex]4[\/latex]), Quintic ([latex]5[\/latex]), etc.<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">The zero function [latex]f(x) = 0[\/latex] is considered a polynomial of degree [latex]0[\/latex].<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p class=\"whitespace-pre-wrap break-words\">Identify the degree, leading term, and leading coefficient for each of the following functions:<\/p>\n<ol style=\"list-style-type: lower-alpha;\">\n<li class=\"whitespace-pre-wrap break-words\">[latex]h(y) = 2y\u2074 - 3y\u00b2 + 5y - 7[\/latex]<\/li>\n<li class=\"whitespace-pre-wrap break-words\">[latex]k(s) = -0.5s\u00b3 + 2s - 1[\/latex]<\/li>\n<li class=\"whitespace-pre-wrap break-words\">[latex]m(w) = w\u2075 + 3w\u2074 - 2w\u00b2 + w[\/latex]<\/li>\n<\/ol>\n<div class=\"wp-nocaption \"><\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q101020\">Show Answer<\/button><\/p>\n<div id=\"q101020\" class=\"hidden-answer\" style=\"display: none\">\n<ol style=\"list-style-type: lower-alpha;\">\n<li class=\"whitespace-pre-wrap break-words\">\u00a0[latex]h(y) = 2y\u2074 - 3y\u00b2 + 5y - 7[\/latex]\n<ul>\n<li class=\"whitespace-pre-wrap break-words\">Degree: [latex]4[\/latex]<\/li>\n<li class=\"whitespace-pre-wrap break-words\">Leading Term: [latex]2y\u2074[\/latex]<\/li>\n<li class=\"whitespace-pre-wrap break-words\">Leading Coefficient: [latex]2[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li>[latex]k(s) = -0.5s\u00b3 + 2s - 1[\/latex]\n<ul>\n<li>Degree: [latex]3[\/latex]<\/li>\n<li>Leading Term: [latex]-0.5s\u00b3[\/latex]<\/li>\n<li>Leading Coefficient: [latex]-0.5[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li>[latex]m(w) = w\u2075 + 3w\u2074 - 2w\u00b2 + w[\/latex]\n<ul>\n<li>Degree: [latex]5[\/latex]<\/li>\n<li>Leading Term: [latex]w\u2075[\/latex]<\/li>\n<li>Leading Coefficient: [latex]1[\/latex] (remember, when not written, the coefficient is [latex]1[\/latex])<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/section>\n<h2>Zeros of Polynomial Functions<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">For quadratic functions [latex]f(x) = ax\u00b2 + bx + c[\/latex], zeros can be found using:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Factoring (if possible)<\/li>\n<li class=\"whitespace-normal break-words\">The quadratic formula: [latex]x = \\dfrac{-b \u00b1 \\sqrt{(b\u00b2 - 4ac)}}{ (2a)}[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">The discriminant [latex](b\u00b2 - 4ac)[\/latex] determines the nature of quadratic solutions:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Positive: Two distinct real roots<\/li>\n<li class=\"whitespace-normal break-words\">Zero: One real root (double root)<\/li>\n<li class=\"whitespace-normal break-words\">Negative: No real roots<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Higher-degree polynomials may have more complex methods for finding zeros.<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p>Find the zeros of the quadratic function [latex]g(x) = 2x\u00b2 + 5x - 3[\/latex].<\/p>\n<div class=\"wp-nocaption \"><\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q169751\">Show Answer<\/button><\/p>\n<div id=\"q169751\" class=\"hidden-answer\" style=\"display: none\">\n<p>Set the function equal to zero:<\/p>\n<p style=\"text-align: center;\">[latex]2x\u00b2 + 5x - 3 = 0[\/latex]<\/p>\n<p>We&#8217;ll use the quadratic formula since this doesn&#8217;t factor easily. Here [latex]a = 2, b = 5[\/latex], and [latex]c = -3[\/latex]<\/p>\n<p>Apply the quadratic formula:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align*}<br \/>  x &= \\frac{-b \\pm \\sqrt{b^2 - 4ac}}{2a} \\\\<br \/>  x &= \\frac{-5 \\pm \\sqrt{5^2 - 4(2)(-3)}}{2(2)} \\\\<br \/>  x &= \\frac{-5 \\pm \\sqrt{25 + 24}}{4} \\\\<br \/>  x &= \\frac{-5 \\pm \\sqrt{49}}{4} \\\\<br \/>  x &= \\frac{-5 \\pm 7}{4}<br \/>  \\end{align*}[\/latex]<\/p>\n<p>Simplify:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align*} x &= \\frac{(-5 + 7)}{4} \\text{ or } \\frac{(-5 - 7)}{4} \\\\[6pt] x &= \\frac{2}{4} \\text{ or } \\frac{-12}{4} \\\\[6pt] x &= \\frac{1}{2} \\text{ or } -3 \\end{align*}[\/latex]<\/p>\n<p>We can check our solutions by plugging them back in for [latex]x[\/latex] and solving to see if the answer is [latex]0[\/latex]:<\/p>\n<p style=\"text-align: center;\">[latex]g(\\frac{1}{2}) = 2(\\frac{1}{2})\u00b2 + 5(\\frac{1}{2}) - 3 = \\frac{1}{2} + \\frac{5}{2} - 3 = 0[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]g(-3) = 2(-3)\u00b2 + 5(-3) - 3 = 18 - 15 - 3 = 0[\/latex]<\/p>\n<p class=\"whitespace-pre-wrap break-words\">Therefore, the zeros of [latex]g(x)[\/latex] are [latex]x = \\frac{1}{2}[\/latex] and [latex]x = -3[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/section>\n<h2>Graphs of Polynomial Functions Basics<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">The degree of a polynomial greatly influences its graph&#8217;s shape.<\/li>\n<li class=\"whitespace-normal break-words\">Even-degree polynomials:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Both ends of the graph point in the same direction (up or down)<\/li>\n<li class=\"whitespace-normal break-words\">Similar to a parabola, but may be flatter near the origin<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Odd-degree polynomials:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Ends of the graph point in opposite directions<\/li>\n<li class=\"whitespace-normal break-words\">One end goes up, the other goes down<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">The sign of the leading coefficient determines the direction of the graph&#8217;s ends:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Positive leading coefficient: right end goes up<\/li>\n<li class=\"whitespace-normal break-words\">Negative leading coefficient: right end goes down<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">As the degree increases, graphs tend to flatten near the origin and become steeper away from it.<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p class=\"whitespace-pre-wrap break-words\">Describe the end behavior and basic shape of the graph for the polynomial function:<\/p>\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex]f(x) = -2x\u2075 + 3x\u2074 - x\u00b2 + 7[\/latex]<\/p>\n<div class=\"wp-nocaption \"><\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q399650\">Show Answer<\/button><\/p>\n<div id=\"q399650\" class=\"hidden-answer\" style=\"display: none\">\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Identify key features:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Degree: [latex]5[\/latex] (odd)<\/li>\n<li class=\"whitespace-normal break-words\">Leading coefficient: [latex]-2[\/latex] (negative)<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Describe end behavior:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">As [latex]x[\/latex] approaches positive infinity, [latex]f(x)[\/latex] approaches negative infinity<\/li>\n<li class=\"whitespace-normal break-words\">As [latex]x[\/latex] approaches negative infinity, [latex]f(x)[\/latex] approaches negative infinity<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Basic shape:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">The graph will have one arm pointing down on the right side<\/li>\n<li class=\"whitespace-normal break-words\">The other arm will point up on the left side<\/li>\n<li class=\"whitespace-normal break-words\">There may be turns or oscillations between these extremes<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<figure id=\"attachment_3625\" aria-describedby=\"caption-attachment-3625\" style=\"width: 500px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-3625\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/34\/2024\/04\/01181605\/Screenshot-2024-07-01-141554-300x274.png\" alt=\"A graph of the function f(x) = -2x\u2075 + 3x\u2074 - x\u00b2 + 7 on a coordinate plane. The red curve starts in the upper left quadrant, descends steeply near the y-axis with a small bump around y=7, then continues downward crossing into the lower right quadrant.\" width=\"500\" height=\"457\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/34\/2024\/04\/01181605\/Screenshot-2024-07-01-141554-300x274.png 300w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/34\/2024\/04\/01181605\/Screenshot-2024-07-01-141554-65x59.png 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/34\/2024\/04\/01181605\/Screenshot-2024-07-01-141554-225x206.png 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/34\/2024\/04\/01181605\/Screenshot-2024-07-01-141554-350x320.png 350w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/34\/2024\/04\/01181605\/Screenshot-2024-07-01-141554.png 738w\" sizes=\"(max-width: 500px) 100vw, 500px\" \/><figcaption id=\"caption-attachment-3625\" class=\"wp-caption-text\">Graph of the polynomial function f(x) = -2x\u2075 + 3x\u2074 &#8211; x\u00b2 + 7<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/section>\n<h2>Piecewise-Defined Functions<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">A piecewise-defined function uses different formulas for different parts of its domain.<\/li>\n<li class=\"whitespace-normal break-words\">Each piece of the function has its own subdomain, which together form the function&#8217;s complete domain.<\/li>\n<li class=\"whitespace-normal break-words\">Piecewise functions may be continuous or discontinuous, depending on how the pieces connect.<\/li>\n<li class=\"whitespace-normal break-words\">Each piece is graphed separately within its specified domain, paying special attention to the transition points.<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\">\n<p id=\"fs-id1170573569350\">Sketch a graph of the function<\/p>\n<p>[latex]f(x)=\\begin{cases} 2-x, & x \\le 2 \\\\ x+2, & x>2 \\end{cases}[\/latex]<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q705322\">Hint<\/button><\/p>\n<div id=\"q705322\" class=\"hidden-answer\" style=\"display: none\">\n<p>Graph one linear function for [latex]x \\le 2[\/latex] and then graph a different linear function for [latex]x>2[\/latex].<\/p>\n<\/div>\n<\/div>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1170573351776\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1170573351776\" class=\"hidden-answer\" style=\"display: none\">\n<figure style=\"width: 462px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11202257\/CNX_Calc_Figure_01_02_012.jpg\" alt=\"&quot;An\" width=\"462\" height=\"384\" \/><figcaption class=\"wp-caption-text\">Graph of f(x)<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p id=\"fs-id1170573582214\">The cost of mailing a letter is a function of the weight of the letter. Suppose the cost of mailing a letter is [latex]49\\text{\u00a2}[\/latex] for the first ounce and [latex]21\\text{\u00a2}[\/latex] for each additional ounce. Write a piecewise-defined function describing the cost [latex]C[\/latex] as a function of the weight [latex]x[\/latex] for [latex]0 < x \\le 3[\/latex], where [latex]C[\/latex] is measured in cents and [latex]x[\/latex] is measured in ounces.<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q981114\">Hint<\/button><\/p>\n<div id=\"q981114\" class=\"hidden-answer\" style=\"display: none\">\n<p>The piecewise-defined function is constant on the intervals [latex](0,1], \\, (1,2], \\, \\cdots[\/latex]<\/p>\n<\/div>\n<\/div>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1170573359603\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1170573359603\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170573359603\">[latex]C(x)=\\begin{cases} 49, & 0 < x \\le 1 \\\\ 70, & 1 < x \\le 2 \\\\ 91, & 2 < x \\le 3 \\end{cases}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<h2>Algebraic Functions and Transcendental Functions<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Algebraic Functions:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Involve addition, subtraction, multiplication, division, rational powers, and roots<\/li>\n<li class=\"whitespace-normal break-words\">Include polynomial, rational, and root functions<\/li>\n<li class=\"whitespace-normal break-words\">Examples: [latex]f(x) = x\u00b3 + 2x - 1, g(x) = \\frac{(x\u00b2 + 1) }{ (x - 2)}, h(x) = \\sqrt{(x + 3)}[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Transcendental Functions:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Cannot be expressed using a finite number of algebraic operations<\/li>\n<li class=\"whitespace-normal break-words\">Main types: trigonometric, exponential, and logarithmic functions<\/li>\n<li class=\"whitespace-normal break-words\">Examples: [latex]sin(x), e^x, log\u2082(x)[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\">\n<p id=\"fs-id1170573577836\">Is [latex]f(x)=\\dfrac{x}{2}[\/latex] an algebraic or a transcendental function?<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1170573577863\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1170573577863\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170573577863\">Algebraic<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p class=\"whitespace-pre-wrap break-words\">Classify the following functions as algebraic or transcendental, and identify their specific type if possible:<\/p>\n<ol style=\"list-style-type: lower-alpha;\">\n<li>[latex]f(x) = \\frac{(x\u2074 + 2x\u00b2 - 5)}{ (x - 1)}[\/latex]<\/li>\n<li>[latex]g(x) = sin(x\u00b2) + cos(x)[\/latex]<\/li>\n<li>[latex]h(x) = \\sqrt[3]{(x\u00b2 + 1)} + log\u2083(x)[\/latex]<\/li>\n<li>[latex]k(x) = \\frac{(2^x + 3^x)}{x}[\/latex]<\/li>\n<\/ol>\n<div class=\"wp-nocaption \"><\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q715773\">Show Answer<\/button><\/p>\n<div id=\"q715773\" class=\"hidden-answer\" style=\"display: none\">\n<ol style=\"list-style-type: lower-alpha;\">\n<li class=\"whitespace-pre-wrap break-words\">\u00a0Algebraic; It&#8217;s a ratio of polynomials.<\/li>\n<li class=\"whitespace-pre-wrap break-words\">Transcendental;\u00a0 Involves sine and cosine, which are transcendental.<\/li>\n<li class=\"whitespace-pre-wrap break-words\">Mixed &#8211; Both algebraic and transcendental; [latex]\\sqrt[3]{(x\u00b2 + 1)}[\/latex] is algebraic (root function), but [latex]log\u2083(x)[\/latex] is transcendental.<\/li>\n<li class=\"whitespace-pre-wrap break-words\">Transcendental;\u00a0 [latex]2^x[\/latex] and [latex]3^x[\/latex] are exponential (transcendental), and the overall function can&#8217;t be simplified to an algebraic form.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/section>\n<h2>Transformations of Functions\u00a0<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Vertical Shift: [latex]f(x) \u00b1 c[\/latex]\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]+c[\/latex] shifts up, [latex]-c[\/latex] shifts down<\/li>\n<li class=\"whitespace-normal break-words\">Does not change shape of graph<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Horizontal Shift: [latex]f(x \u00b1 c)[\/latex]\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]-c[\/latex] shifts right, [latex]+c[\/latex] shifts left<\/li>\n<li class=\"whitespace-normal break-words\">Does not change shape of graph<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Vertical Scaling: [latex]cf(x)[\/latex]\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]|c| > 1[\/latex] stretches vertically, [latex]0 < |c| < 1[\/latex] compresses vertically<\/li>\n<li class=\"whitespace-normal break-words\">Changes height of graph<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Horizontal Scaling: [latex]f(cx)[\/latex]\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]0 < |c| < 1[\/latex]stretches horizontally, [latex]|c| > 1[\/latex] compresses horizontally<\/li>\n<li class=\"whitespace-normal break-words\">Changes width of graph<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Reflections:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]-f(x)[\/latex] reflects over [latex]x[\/latex]-axis<\/li>\n<li class=\"whitespace-normal break-words\">[latex]f(-x)[\/latex] reflects over [latex]y[\/latex]-axis<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Multiple Transformations: [latex]y = cf(a(x + b)) + d[\/latex]\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Apply in order: horizontal shift, horizontal scaling, vertical scaling, vertical shift<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p>Transform the following function which has the base tool-kit function [latex]f(x) = \\sqrt{x}[\/latex]:<\/p>\n<p style=\"text-align: center;\">[latex]g(x) = 2\\sqrt{(x - 1)} + 3[\/latex]<\/p>\n<div class=\"wp-nocaption \"><\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q635560\">Show Answer<\/button><\/p>\n<div id=\"q635560\" class=\"hidden-answer\" style=\"display: none\">\n<ul>\n<li class=\"whitespace-normal break-words\">Identify the base function: [latex]f(x) = \\sqrt{x}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Break down the transformations:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Inside the square root:[latex](x - 1)[\/latex] indicates a horizontal shift<\/li>\n<li class=\"whitespace-normal break-words\">Outside the square root: [latex]2[\/latex] indicates vertical scaling<\/li>\n<li class=\"whitespace-normal break-words\">[latex]+3[\/latex] at the end indicates a vertical shift<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Apply transformations in order:\n<ul>\n<li class=\"whitespace-normal break-words\">Horizontal shift: [latex]\\sqrt{(x - 1)}[\/latex] shifts the graph [latex]1[\/latex] unit right<\/li>\n<li class=\"whitespace-normal break-words\">Vertical scaling: [latex]2\\sqrt{(x - 1)}[\/latex] stretches the graph vertically by a factor of [latex]2[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Vertical shift: [latex]2\\sqrt{(x - 1)} + 3[\/latex] shifts the graph up [latex]3[\/latex] units<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Describe the final graph:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">The graph of [latex]g(x)[\/latex] is the square root function shifted [latex]1[\/latex] unit right, stretched vertically by a factor of [latex]2[\/latex], and shifted up [latex]3[\/latex] units.<\/li>\n<li class=\"whitespace-normal break-words\">The domain is [latex][1, \u221e)[\/latex] because [latex]x - 1[\/latex] must be non-negative under the square root.<\/li>\n<li class=\"whitespace-normal break-words\">The range is [latex][3, \u221e)[\/latex] due to the vertical stretch and shift.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p>We can check this by plugging both equations into a graphing software.<\/p>\n<figure id=\"attachment_3627\" aria-describedby=\"caption-attachment-3627\" style=\"width: 500px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-3627\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/34\/2024\/04\/01183152\/Screenshot-2024-07-01-143131.png\" alt=\"A coordinate plane showing two graphs. The red curve represents f(x) = \u221ax, starting at the origin and gradually increasing with a decreasing slope. The blue curve represents g(x) = 2\u221a(x-1) + 3, beginning at (1, 3) and also increasing with a decreasing slope, but steeper and higher than the red curve.\" width=\"500\" height=\"500\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/34\/2024\/04\/01183152\/Screenshot-2024-07-01-143131.png 680w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/34\/2024\/04\/01183152\/Screenshot-2024-07-01-143131-300x300.png 300w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/34\/2024\/04\/01183152\/Screenshot-2024-07-01-143131-150x150.png 150w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/34\/2024\/04\/01183152\/Screenshot-2024-07-01-143131-65x65.png 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/34\/2024\/04\/01183152\/Screenshot-2024-07-01-143131-225x225.png 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/34\/2024\/04\/01183152\/Screenshot-2024-07-01-143131-350x350.png 350w\" sizes=\"(max-width: 500px) 100vw, 500px\" \/><figcaption id=\"caption-attachment-3627\" class=\"wp-caption-text\">Comparison of square root functions: f(x) = \u221ax (red) and g(x) = 2\u221a(x-1) + 3 (blue)<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/section>\n","protected":false},"author":15,"menu_order":18,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":143,"module-header":"fresh_take","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/931"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":14,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/931\/revisions"}],"predecessor-version":[{"id":4736,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/931\/revisions\/4736"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/parts\/143"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/931\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/media?parent=931"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapter-type?post=931"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/contributor?post=931"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/license?post=931"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}