{"id":3309,"date":"2024-06-18T19:57:52","date_gmt":"2024-06-18T19:57:52","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus1\/?post_type=chapter&#038;p=3309"},"modified":"2024-08-04T10:40:11","modified_gmt":"2024-08-04T10:40:11","slug":"basic-functions-and-graphs-cheat-sheet","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus1\/chapter\/basic-functions-and-graphs-cheat-sheet\/","title":{"raw":"Basic Functions and Graphs: Cheat Sheet","rendered":"Basic Functions and Graphs: Cheat Sheet"},"content":{"raw":"<p style=\"text-align: right;\"><a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Calculus+1+2024+Build\/Cheat+Sheets\/Calculus+1+Cheat+Sheet_+Basic+Functions+and+Graphs.pdf\" target=\"_blank\" rel=\"noopener\"><span style=\"font-size: 14pt;\">Download a PDF of this page here.<\/span><\/a><\/p>\r\n<p style=\"text-align: right;\"><span style=\"font-size: 14pt;\">Download the Spanish version here.<\/span><\/p>\r\n<h2>Essential Concepts<\/h2>\r\n<p><strong>Review of Functions<\/strong><\/p>\r\n<ul id=\"fs-id1170572176922\">\r\n\t<li>A function is a mapping from a set of inputs to a set of outputs with exactly one output for each input.<\/li>\r\n\t<li>If no domain is stated for a function [latex]y=f(x)[\/latex], the domain is considered to be the set of all real numbers [latex]x[\/latex] for which the function is defined.<\/li>\r\n\t<li>When sketching the graph of a function [latex]f[\/latex], each vertical line may intersect the graph, at most, once.<\/li>\r\n\t<li>A function may have any number of zeros, but it has, at most, one [latex]y[\/latex]-intercept.<\/li>\r\n\t<li>To define the composition [latex]g\\circ f[\/latex], the range of [latex]f[\/latex] must be contained in the domain of [latex]g[\/latex].<\/li>\r\n\t<li>Even functions are symmetric about the [latex]y[\/latex]-axis whereas odd functions are symmetric about the origin.<\/li>\r\n<\/ul>\r\n<p><strong>Basic Classes of Functions<\/strong><\/p>\r\n<ul id=\"fs-id1170573581308\">\r\n\t<li>The power function [latex]f(x)=x^n[\/latex] is an even function if [latex]n[\/latex] is even and [latex]n \\ne 0[\/latex], and it is an odd function if [latex]n[\/latex] is odd.<\/li>\r\n\t<li>The root function [latex]f(x)=x^{1\/n}[\/latex] has the domain [latex][0,\\infty )[\/latex] if [latex]n[\/latex] is even and the domain [latex](-\\infty,\\infty )[\/latex] if [latex]n[\/latex] is odd. If [latex]n[\/latex] is odd, then [latex]f(x)=x^{1\/n}[\/latex] is an odd function.<\/li>\r\n\t<li>The domain of the rational function [latex]f(x)=p(x)\/q(x)[\/latex], where [latex]p(x)[\/latex] and [latex]q(x)[\/latex] are polynomial functions, is the set of [latex]x[\/latex] such that [latex]q(x) \\ne 0[\/latex].<\/li>\r\n\t<li>Functions that involve the basic operations of addition, subtraction, multiplication, division, and powers are algebraic functions. All other functions are transcendental. Trigonometric, exponential, and logarithmic functions are examples of transcendental functions.<\/li>\r\n\t<li>A polynomial function [latex]f[\/latex] with degree [latex]n \\ge 1[\/latex] satisfies [latex]f(x) \\to \\pm \\infty [\/latex] as [latex]x \\to \\pm \\infty[\/latex]. The sign of the output as [latex]x \\to \\infty [\/latex] depends on the sign of the leading coefficient only and on whether [latex]n[\/latex] is even or odd.<\/li>\r\n\t<li>Vertical and horizontal shifts, vertical and horizontal scalings, and reflections about the [latex]x[\/latex]- and [latex]y[\/latex]-axes are examples of transformations of functions.<\/li>\r\n<\/ul>\r\n<h2>Key Equations<\/h2>\r\n<ul id=\"fs-id1170572554776\">\r\n\t<li><strong>Composition of two functions<\/strong><br \/>\r\n[latex](g\\circ f)(x)=g(f(x))[\/latex]<\/li>\r\n\t<li><strong>Absolute value function<\/strong><br \/>\r\n[latex]f(x) = |x| = \\begin{cases} x, &amp; x \\ge 0 \\\\ -x, &amp; x &lt; 0 \\end{cases}[\/latex]<\/li>\r\n\t<li><strong>Point-slope equation of a line<\/strong><br \/>\r\n[latex]y-y_1=m(x-x_1)[\/latex]<\/li>\r\n\t<li><strong>Slope-intercept form of a line<\/strong><br \/>\r\n[latex]y=mx+b[\/latex]<\/li>\r\n\t<li><strong>Standard form of a line<\/strong><br \/>\r\n[latex]ax+by=c[\/latex]<\/li>\r\n\t<li><strong>Polynomial function<\/strong><br \/>\r\n[latex]f(x)=a_nx^n+a_{n-1}x^{n-1}+ \\ldots +a_1x+a_0[\/latex]<\/li>\r\n<\/ul>\r\n<h2>Glossary<\/h2>\r\n<dl id=\"fs-id1170572443492\" class=\"definition\">\r\n<dt>absolute value function<\/dt>\r\n<dd id=\"fs-id1170572443498\">[latex]f(x) = |x| = \\begin{cases} x, &amp; x \\ge 0 \\\\ -x, &amp; x &lt; 0 \\end{cases}[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170572443550\" class=\"definition\">\r\n<dt>\r\n<dl id=\"fs-id1170573596356\" class=\"definition\">\r\n<dt>algebraic function<\/dt>\r\n<dd id=\"fs-id1170573596362\">a function involving any combination of only the basic operations of addition, subtraction, multiplication, division, powers, and roots applied to an input variable [latex]x[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170573596721\" class=\"definition\">\r\n<dt>composite function<\/dt>\r\n<dd id=\"fs-id1170572443555\">given two functions [latex]f[\/latex] and [latex]g[\/latex], a new function, denoted [latex]g\\circ f[\/latex], such that [latex](g\\circ f)(x)=g(f(x))[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170572548156\" class=\"definition\">\r\n<dt>\r\n<dl id=\"fs-id1170573596371\" class=\"definition\">\r\n<dt>cubic function<\/dt>\r\n<dd id=\"fs-id1170573596376\">a polynomial of degree [latex]3[\/latex]; that is, a function of the form [latex]f(x)=ax^3+bx^2+cx+d[\/latex], where [latex]a \\ne 0[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170573596434\" class=\"definition\">\r\n<dt><\/dt>\r\n<\/dl>\r\n<dl id=\"fs-id1170573596721\" class=\"definition\">\r\n<dt>decreasing on the interval [latex]I[\/latex]<\/dt>\r\n<dd id=\"fs-id1170572548165\">a function decreasing on the interval [latex]I[\/latex] if, for all [latex]x_1, \\, x_2\\in I, \\, f(x_1)\\ge f(x_2)[\/latex] if [latex]x_1&lt;x_2[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170573520938\" class=\"definition\">\r\n<dt>\r\n<dl id=\"fs-id1170573596434\" class=\"definition\">\r\n<dt>degree<\/dt>\r\n<dd id=\"fs-id1170573596439\">for a polynomial function, the value of the largest exponent of any term<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170573596721\" class=\"definition\">\r\n<dt>dependent variable<\/dt>\r\n<dd id=\"fs-id1170572548252\">the output variable for a function<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170572548257\" class=\"definition\">\r\n<dt>domain<\/dt>\r\n<dd id=\"fs-id1170572548262\">the set of inputs for a function<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170572548267\" class=\"definition\">\r\n<dt>even function<\/dt>\r\n<dd id=\"fs-id1170572548272\">a function is even if [latex]f(\u2212x)=f(x)[\/latex] for all [latex]x[\/latex] in the domain of [latex]f[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170572548320\" class=\"definition\">\r\n<dt>function<\/dt>\r\n<dd id=\"fs-id1170572549920\">a set of inputs, a set of outputs, and a rule for mapping each input to exactly one output<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170572549925\" class=\"definition\">\r\n<dt>graph of a function<\/dt>\r\n<dd id=\"fs-id1170572549930\">the set of points [latex](x,y)[\/latex] such that [latex]x[\/latex] is in the domain of [latex]f[\/latex] and [latex]y=f(x)[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170572550070\" class=\"definition\">\r\n<dt>independent variable<\/dt>\r\n<dd id=\"fs-id1170572550076\">the input variable for a function<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170572550080\" class=\"definition\">\r\n<dt>\r\n<dl id=\"fs-id1170573596443\" class=\"definition\">\r\n<dt>linear function<\/dt>\r\n<dd id=\"fs-id1170573596448\">a function that can be written in the form [latex]f(x)=mx+b[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170573596475\" class=\"definition\">\r\n<dt>logarithmic function<\/dt>\r\n<dd id=\"fs-id1170573596481\">a function of the form [latex]f(x)=\\log_b(x)[\/latex] for some base [latex]b&gt;0, \\, b \\ne 1[\/latex] such that [latex]y=\\log_b(x)[\/latex] if and only if [latex]b^y=x[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170573596572\" class=\"definition\">\r\n<dt>mathematical model<\/dt>\r\n<dd id=\"fs-id1170573596577\">A method of simulating real-life situations with mathematical equations<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170573596721\" class=\"definition\">\r\n<dt>odd function<\/dt>\r\n<dd id=\"fs-id1170572550086\">a function is odd if [latex]f(\u2212x)=\u2212f(x)[\/latex] for all [latex]x[\/latex] in the domain of [latex]f[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170572550135\" class=\"definition\">\r\n<dt>\r\n<dl id=\"fs-id1170573596582\" class=\"definition\">\r\n<dt>piecewise-defined function<\/dt>\r\n<dd id=\"fs-id1170573596587\">a function that is defined differently on different parts of its domain<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170573596592\" class=\"definition\">\r\n<dt>point-slope equation<\/dt>\r\n<dd id=\"fs-id1170573596597\">equation of a linear function indicating its slope and a point on the graph of the function<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170573596602\" class=\"definition\">\r\n<dt>polynomial function<\/dt>\r\n<dd id=\"fs-id1170573596608\">a function of the form [latex]f(x)=a_nx^n+a_{n-1}x^{n-1}+ \\cdots +a_1x+a_0[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170573596682\" class=\"definition\">\r\n<dt>power function<\/dt>\r\n<dd id=\"fs-id1170573596687\">a function of the form [latex]f(x)=x^n[\/latex] for any positive integer [latex]n \\ge 1[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170573596721\" class=\"definition\">\r\n<dt>quadratic function<\/dt>\r\n<dd id=\"fs-id1170573596726\">a polynomial of degree 2; that is, a function of the form [latex]f(x)=ax^2+bx+c[\/latex] where [latex]a \\ne 0[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170573596721\" class=\"definition\">\r\n<dt>range<\/dt>\r\n<dd id=\"fs-id1170572235100\">the set of outputs for a function<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170572235105\" class=\"definition\">\r\n<dt>\r\n<dl id=\"fs-id1170573596772\" class=\"definition\">\r\n<dt>rational function<\/dt>\r\n<dd id=\"fs-id1170573596778\">a function of the form [latex]f(x)=p(x)\/q(x)[\/latex], where [latex]p(x)[\/latex] and [latex]q(x)[\/latex] are polynomials<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170573596850\" class=\"definition\">\r\n<dt>root function<\/dt>\r\n<dd id=\"fs-id1170573596855\">a function of the form [latex]f(x)=x^{1\/n}[\/latex] for any integer [latex]n \\ge 2[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170573596895\" class=\"definition\">\r\n<dt>slope<\/dt>\r\n<dd id=\"fs-id1170573596900\">the change in [latex]y[\/latex] for each unit change in [latex]x[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170573596915\" class=\"definition\">\r\n<dt>slope-intercept form<\/dt>\r\n<dd id=\"fs-id1170573596921\">equation of a linear function indicating its slope and [latex]y[\/latex]-intercept<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-8773456\" class=\"definition\">\r\n<dt>standard form<\/dt>\r\n<dd id=\"fs-8773456\">equation of a linear function with both variable terms set equal to a constant,\u00a0[latex]ax+by=c[\/latex].<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170573596721\" class=\"definition\">\r\n<dt>symmetry about the origin<\/dt>\r\n<dd id=\"fs-id1170572235110\">the graph of a function [latex]f[\/latex] is symmetric about the origin if [latex](\u2212x,\u2212y)[\/latex] is on the graph of [latex]f[\/latex] whenever [latex](x,y)[\/latex] is on the graph<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170572235164\" class=\"definition\">\r\n<dt>symmetry about the [latex]y[\/latex]-axis<\/dt>\r\n<dd id=\"fs-id1170572235175\">the graph of a function [latex]f[\/latex] is symmetric about the [latex]y[\/latex]-axis if [latex](\u2212x,y)[\/latex] is on the graph of [latex]f[\/latex] whenever [latex](x,y)[\/latex] is on the graph<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170572235230\" class=\"definition\">\r\n<dt>table of values<\/dt>\r\n<dd id=\"fs-id1170572235236\">a table containing a list of inputs and their corresponding outputs<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170572235240\" class=\"definition\">\r\n<dt>\r\n<dl id=\"fs-446634\" class=\"definition\">\r\n<dt>transcendental function<\/dt>\r\n<dd id=\"fs-446634\"><span style=\"font-size: 1em;\">a function that cannot be expressed by a combination of basic arithmetic operations<\/span><\/dd>\r\n<\/dl>\r\n<dl id=\"fs-3772119\" class=\"definition\">\r\n<dt>transformation of a function<\/dt>\r\n<dd id=\"fs-3772119\">a shift, scaling, or reflection of a function<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170573596721\" class=\"definition\">\r\n<dt>vertical line test<\/dt>\r\n<dd id=\"fs-id1170572235245\">given the graph of a function, every vertical line intersects the graph, at most, once<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170572235250\" class=\"definition\">\r\n<dt>zeros of a function<\/dt>\r\n<dd id=\"fs-id1170572235255\">when a real number [latex]x[\/latex] is a zero of a function [latex]f[\/latex], [latex]f(x)=0[\/latex]<\/dd>\r\n<\/dl>\r\n<h2 class=\"font-bold\"><strong>Study Tips<\/strong><\/h2>\r\n<p><strong>Functions<\/strong><\/p>\r\n<ul class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Practice evaluating functions by substituting given values for [latex]x[\/latex] in the function's formula.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">When working with tables, pay attention to how the output changes as the input changes.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">When looking at graphs, observe the overall behavior of the function, including any increases, decreases, or constant sections.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">For algebraic formulas, practice creating tables of values to help visualize the function.<\/li>\r\n<\/ul>\r\n<p><strong>The Domain and Range of a Function<\/strong><\/p>\r\n<ul>\r\n\t<li class=\"whitespace-normal break-words\">When determining the domain, look for potential mathematical restrictions like division by zero or even roots of negative numbers.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">For the range, consider the nature of the function (linear, quadratic, etc.) and any limitations on the output values.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Practice using both set notation and interval notation to express domains and ranges.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Remember that infinity ([latex]\\infty[\/latex]) is not a real number but a concept used to describe unbounded sets.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">When working with rational functions, pay special attention to values that make the denominator zero.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">For functions involving square roots, ensure the expression under the root is non-negative.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Visualize the function graph to help identify the domain and range, especially for common function types.<\/li>\r\n<\/ul>\r\n<p><strong>Intercepts of a Function<\/strong><\/p>\r\n<ul>\r\n\t<li>To find [latex]x[\/latex]-intercepts, set the function equal to zero and solve for [latex]x[\/latex].<\/li>\r\n\t<li>To find the [latex]y[\/latex]-intercept, evaluate the function at [latex]x = 0[\/latex].<\/li>\r\n\t<li>Remember that [latex]x[\/latex]-intercepts are in the form ([latex]x, 0[\/latex]), while the [latex]y[\/latex]-intercept is in the form ([latex]0, y[\/latex]).<\/li>\r\n\t<li>Remember that not all functions have x-intercepts, but most will have a [latex]y[\/latex]-intercept.<\/li>\r\n<\/ul>\r\n<p><strong>Symmetry of Functions\u00a0<\/strong><\/p>\r\n<ul>\r\n\t<li>To check for symmetry algebraically:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Even functions: Replace [latex]x[\/latex] with [latex]-x[\/latex] and see if the result is identical<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Odd functions: Replace [latex]x[\/latex] with [latex]-x[\/latex] and see if the result is the negative of the original function<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Visualize symmetry:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Even functions: Mirror image across [latex]y[\/latex]-axis<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Odd functions: [latex]180\u00b0[\/latex] rotation around the origin<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">For absolute value 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\">Remember they create a \"V\" shape on the graph<\/li>\r\n\t<li class=\"whitespace-normal break-words\">The output is always non-negative<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Transformations can shift or stretch the basic [latex]|x|[\/latex] function<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Practice identifying symmetry in various function types (polynomials, rationals, etc.)<\/li>\r\n\t<li class=\"whitespace-normal break-words\">When solving absolute value equations, consider both positive and negative cases<\/li>\r\n<\/ul>\r\n<p><strong>Composing Functions\u00a0<\/strong><\/p>\r\n<ul>\r\n\t<li class=\"whitespace-normal break-words\">When combining functions, treat each function as a unit and apply the operation.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">For composite functions, work from the inside out: first apply the inner function, then the outer function.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Pay attention to the domains when combining or composing functions, especially for quotients and composites.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Practice identifying the difference between [latex](f \u2218 g)(x)[\/latex] and [latex](g \u2218 f)(x)[\/latex].<\/li>\r\n\t<li class=\"whitespace-normal break-words\">When working with composing 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\">Add\/subtract: Combine like terms<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Multiply: Use the FOIL method for binomials or distribute each term<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<p><strong>Polynomial Functions\u00a0<\/strong><\/p>\r\n<ul>\r\n\t<li class=\"whitespace-normal break-words\">Always arrange polynomials in descending order of degree for easy identification of the leading term and degree.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Remember that the degree is determined by the highest power of the variable, not the largest number in the function.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">The leading coefficient can be positive or negative, and it's always the number in front of the highest power term.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">When the leading coefficient is [latex]1[\/latex] or [latex]-1[\/latex], it's often not written explicitly. For example, [latex]x\u00b3 + 2x\u00b2[\/latex] has a leading coefficient of [latex]1[\/latex].<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Be careful with negative exponents or fractional exponents - these are not considered polynomials.<\/li>\r\n<\/ul>\r\n<p><strong>Zeros of Polynomial Functions<\/strong><\/p>\r\n<ul>\r\n\t<li class=\"whitespace-normal break-words\">For quadratic functions, try factoring first if the coefficients are simple.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Memorize the quadratic formula and practice using it.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Pay attention to the sign of the discriminant to predict the nature of roots before solving.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">When using the quadratic formula, simplify under the square root if possible before calculating further.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Double-check your solutions by plugging them back into the original equation.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Remember that the number of real zeros a polynomial can have is at most equal to its degree.<\/li>\r\n<\/ul>\r\n<p><strong>Graphs of Polynomial Functions Basics<\/strong><\/p>\r\n<ul>\r\n\t<li class=\"whitespace-normal break-words\">Always identify the degree and the sign of the leading coefficient first.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Remember: Even degree = same direction ends, Odd degree = opposite direction ends.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Practice sketching basic shapes for different degrees and leading coefficient signs.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">For odd-degree polynomials, a negative input will always yield a negative output (assuming positive leading coefficient).<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Look at the behavior of the graph as [latex]x[\/latex] approaches positive and negative infinity to determine the degree's parity and leading coefficient's sign.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Compare graphs of polynomials with the same degree but different leading coefficient signs to see the reflection effect.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">When in doubt, plot a few key points to confirm the graph's behavior.<\/li>\r\n<\/ul>\r\n<p><strong>Piecewise-Defined Functions<\/strong><\/p>\r\n<ul>\r\n\t<li class=\"whitespace-normal break-words\">Clearly identify the domain for each piece of the function.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">When graphing, use open or closed circles at endpoints to show inclusion or exclusion of boundary points.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Check for continuity at the points where the function definition changes.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">When solving equations involving piecewise functions, consider each piece separately.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Use technology to help visualize complex piecewise functions.<\/li>\r\n<\/ul>\r\n<p><strong>Algebraic Functions and Transcendental Functions<\/strong><\/p>\r\n<ul>\r\n\t<li class=\"whitespace-normal break-words\">To identify algebraic functions, look for operations limited to addition, subtraction, multiplication, division, and rational exponents.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Remember that any finite combination of algebraic functions is still algebraic.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Transcendental functions often involve trigonometric ratios, e, pi, or logarithms.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Practice identifying functions by their form:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Fraction of polynomials? Rational function.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Variable under a root? Root function.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Variable in the exponent? Likely exponential (transcendental).<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Be cautious with composite functions - they may combine algebraic and transcendental elements.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">When in doubt, ask yourself: \"Can this be broken down into basic algebraic operations?\"<\/li>\r\n<\/ul>\r\n<p><strong>Transformations of Functions<\/strong><\/p>\r\n<ul>\r\n\t<li class=\"whitespace-normal break-words\">Practice identifying transformations from equations.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Sketch basic function graphs and apply transformations step-by-step.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Remember that horizontal transformations work in the \"opposite\" direction of what you might expect.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Use the order of operations for multiple transformations: parentheses, exponents, multiplication\/division, addition\/subtraction.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Familiarize yourself with basic toolkit functions as starting points for transformations.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">When dealing with complex transformations, break them down into individual steps.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Use technology to verify your hand-drawn transformed graphs.<\/li>\r\n<\/ul>\r\n<\/dt>\r\n<\/dl>\r\n<\/dt>\r\n<\/dl>\r\n<\/dt>\r\n<\/dl>\r\n<\/dt>\r\n<\/dl>\r\n<\/dt>\r\n<\/dl>\r\n<\/dt>\r\n<\/dl>\r\n<\/dt>\r\n<\/dl>","rendered":"<p style=\"text-align: right;\"><a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Calculus+1+2024+Build\/Cheat+Sheets\/Calculus+1+Cheat+Sheet_+Basic+Functions+and+Graphs.pdf\" target=\"_blank\" rel=\"noopener\"><span style=\"font-size: 14pt;\">Download a PDF of this page here.<\/span><\/a><\/p>\n<p style=\"text-align: right;\"><span style=\"font-size: 14pt;\">Download the Spanish version here.<\/span><\/p>\n<h2>Essential Concepts<\/h2>\n<p><strong>Review of Functions<\/strong><\/p>\n<ul id=\"fs-id1170572176922\">\n<li>A function is a mapping from a set of inputs to a set of outputs with exactly one output for each input.<\/li>\n<li>If no domain is stated for a function [latex]y=f(x)[\/latex], the domain is considered to be the set of all real numbers [latex]x[\/latex] for which the function is defined.<\/li>\n<li>When sketching the graph of a function [latex]f[\/latex], each vertical line may intersect the graph, at most, once.<\/li>\n<li>A function may have any number of zeros, but it has, at most, one [latex]y[\/latex]-intercept.<\/li>\n<li>To define the composition [latex]g\\circ f[\/latex], the range of [latex]f[\/latex] must be contained in the domain of [latex]g[\/latex].<\/li>\n<li>Even functions are symmetric about the [latex]y[\/latex]-axis whereas odd functions are symmetric about the origin.<\/li>\n<\/ul>\n<p><strong>Basic Classes of Functions<\/strong><\/p>\n<ul id=\"fs-id1170573581308\">\n<li>The power function [latex]f(x)=x^n[\/latex] is an even function if [latex]n[\/latex] is even and [latex]n \\ne 0[\/latex], and it is an odd function if [latex]n[\/latex] is odd.<\/li>\n<li>The root function [latex]f(x)=x^{1\/n}[\/latex] has the domain [latex][0,\\infty )[\/latex] if [latex]n[\/latex] is even and the domain [latex](-\\infty,\\infty )[\/latex] if [latex]n[\/latex] is odd. If [latex]n[\/latex] is odd, then [latex]f(x)=x^{1\/n}[\/latex] is an odd function.<\/li>\n<li>The domain of the rational function [latex]f(x)=p(x)\/q(x)[\/latex], where [latex]p(x)[\/latex] and [latex]q(x)[\/latex] are polynomial functions, is the set of [latex]x[\/latex] such that [latex]q(x) \\ne 0[\/latex].<\/li>\n<li>Functions that involve the basic operations of addition, subtraction, multiplication, division, and powers are algebraic functions. All other functions are transcendental. Trigonometric, exponential, and logarithmic functions are examples of transcendental functions.<\/li>\n<li>A polynomial function [latex]f[\/latex] with degree [latex]n \\ge 1[\/latex] satisfies [latex]f(x) \\to \\pm \\infty[\/latex] as [latex]x \\to \\pm \\infty[\/latex]. The sign of the output as [latex]x \\to \\infty[\/latex] depends on the sign of the leading coefficient only and on whether [latex]n[\/latex] is even or odd.<\/li>\n<li>Vertical and horizontal shifts, vertical and horizontal scalings, and reflections about the [latex]x[\/latex]&#8211; and [latex]y[\/latex]-axes are examples of transformations of functions.<\/li>\n<\/ul>\n<h2>Key Equations<\/h2>\n<ul id=\"fs-id1170572554776\">\n<li><strong>Composition of two functions<\/strong><br \/>\n[latex](g\\circ f)(x)=g(f(x))[\/latex]<\/li>\n<li><strong>Absolute value function<\/strong><br \/>\n[latex]f(x) = |x| = \\begin{cases} x, & x \\ge 0 \\\\ -x, & x < 0 \\end{cases}[\/latex]<\/li>\n<li><strong>Point-slope equation of a line<\/strong><br \/>\n[latex]y-y_1=m(x-x_1)[\/latex]<\/li>\n<li><strong>Slope-intercept form of a line<\/strong><br \/>\n[latex]y=mx+b[\/latex]<\/li>\n<li><strong>Standard form of a line<\/strong><br \/>\n[latex]ax+by=c[\/latex]<\/li>\n<li><strong>Polynomial function<\/strong><br \/>\n[latex]f(x)=a_nx^n+a_{n-1}x^{n-1}+ \\ldots +a_1x+a_0[\/latex]<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1170572443492\" class=\"definition\">\n<dt>absolute value function<\/dt>\n<dd id=\"fs-id1170572443498\">[latex]f(x) = |x| = \\begin{cases} x, & x \\ge 0 \\\\ -x, & x < 0 \\end{cases}[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1170572443550\" class=\"definition\">\n<dt>\n<\/dt>\n<dt>algebraic function<\/dt>\n<dd id=\"fs-id1170573596362\">a function involving any combination of only the basic operations of addition, subtraction, multiplication, division, powers, and roots applied to an input variable [latex]x[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1170573596721\" class=\"definition\">\n<dt>composite function<\/dt>\n<dd id=\"fs-id1170572443555\">given two functions [latex]f[\/latex] and [latex]g[\/latex], a new function, denoted [latex]g\\circ f[\/latex], such that [latex](g\\circ f)(x)=g(f(x))[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1170572548156\" class=\"definition\">\n<dt>\n<\/dt>\n<dt>cubic function<\/dt>\n<dd id=\"fs-id1170573596376\">a polynomial of degree [latex]3[\/latex]; that is, a function of the form [latex]f(x)=ax^3+bx^2+cx+d[\/latex], where [latex]a \\ne 0[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1170573596434\" class=\"definition\">\n<dt><\/dt>\n<\/dl>\n<dl class=\"definition\">\n<dt>decreasing on the interval [latex]I[\/latex]<\/dt>\n<dd id=\"fs-id1170572548165\">a function decreasing on the interval [latex]I[\/latex] if, for all [latex]x_1, \\, x_2\\in I, \\, f(x_1)\\ge f(x_2)[\/latex] if [latex]x_1<x_2[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1170573520938\" class=\"definition\">\n<dt>\n<\/dt>\n<dt>degree<\/dt>\n<dd id=\"fs-id1170573596439\">for a polynomial function, the value of the largest exponent of any term<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>dependent variable<\/dt>\n<dd id=\"fs-id1170572548252\">the output variable for a function<\/dd>\n<\/dl>\n<dl id=\"fs-id1170572548257\" class=\"definition\">\n<dt>domain<\/dt>\n<dd id=\"fs-id1170572548262\">the set of inputs for a function<\/dd>\n<\/dl>\n<dl id=\"fs-id1170572548267\" class=\"definition\">\n<dt>even function<\/dt>\n<dd id=\"fs-id1170572548272\">a function is even if [latex]f(\u2212x)=f(x)[\/latex] for all [latex]x[\/latex] in the domain of [latex]f[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1170572548320\" class=\"definition\">\n<dt>function<\/dt>\n<dd id=\"fs-id1170572549920\">a set of inputs, a set of outputs, and a rule for mapping each input to exactly one output<\/dd>\n<\/dl>\n<dl id=\"fs-id1170572549925\" class=\"definition\">\n<dt>graph of a function<\/dt>\n<dd id=\"fs-id1170572549930\">the set of points [latex](x,y)[\/latex] such that [latex]x[\/latex] is in the domain of [latex]f[\/latex] and [latex]y=f(x)[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1170572550070\" class=\"definition\">\n<dt>independent variable<\/dt>\n<dd id=\"fs-id1170572550076\">the input variable for a function<\/dd>\n<\/dl>\n<dl id=\"fs-id1170572550080\" class=\"definition\">\n<dt>\n<\/dt>\n<dt>linear function<\/dt>\n<dd id=\"fs-id1170573596448\">a function that can be written in the form [latex]f(x)=mx+b[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1170573596475\" class=\"definition\">\n<dt>logarithmic function<\/dt>\n<dd id=\"fs-id1170573596481\">a function of the form [latex]f(x)=\\log_b(x)[\/latex] for some base [latex]b>0, \\, b \\ne 1[\/latex] such that [latex]y=\\log_b(x)[\/latex] if and only if [latex]b^y=x[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1170573596572\" class=\"definition\">\n<dt>mathematical model<\/dt>\n<dd id=\"fs-id1170573596577\">A method of simulating real-life situations with mathematical equations<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>odd function<\/dt>\n<dd id=\"fs-id1170572550086\">a function is odd if [latex]f(\u2212x)=\u2212f(x)[\/latex] for all [latex]x[\/latex] in the domain of [latex]f[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1170572550135\" class=\"definition\">\n<dt>\n<\/dt>\n<dt>piecewise-defined function<\/dt>\n<dd id=\"fs-id1170573596587\">a function that is defined differently on different parts of its domain<\/dd>\n<\/dl>\n<dl id=\"fs-id1170573596592\" class=\"definition\">\n<dt>point-slope equation<\/dt>\n<dd id=\"fs-id1170573596597\">equation of a linear function indicating its slope and a point on the graph of the function<\/dd>\n<\/dl>\n<dl id=\"fs-id1170573596602\" class=\"definition\">\n<dt>polynomial function<\/dt>\n<dd id=\"fs-id1170573596608\">a function of the form [latex]f(x)=a_nx^n+a_{n-1}x^{n-1}+ \\cdots +a_1x+a_0[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1170573596682\" class=\"definition\">\n<dt>power function<\/dt>\n<dd id=\"fs-id1170573596687\">a function of the form [latex]f(x)=x^n[\/latex] for any positive integer [latex]n \\ge 1[\/latex]<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>quadratic function<\/dt>\n<dd id=\"fs-id1170573596726\">a polynomial of degree 2; that is, a function of the form [latex]f(x)=ax^2+bx+c[\/latex] where [latex]a \\ne 0[\/latex]<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>range<\/dt>\n<dd id=\"fs-id1170572235100\">the set of outputs for a function<\/dd>\n<\/dl>\n<dl id=\"fs-id1170572235105\" class=\"definition\">\n<dt>\n<\/dt>\n<dt>rational function<\/dt>\n<dd id=\"fs-id1170573596778\">a function of the form [latex]f(x)=p(x)\/q(x)[\/latex], where [latex]p(x)[\/latex] and [latex]q(x)[\/latex] are polynomials<\/dd>\n<\/dl>\n<dl id=\"fs-id1170573596850\" class=\"definition\">\n<dt>root function<\/dt>\n<dd id=\"fs-id1170573596855\">a function of the form [latex]f(x)=x^{1\/n}[\/latex] for any integer [latex]n \\ge 2[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1170573596895\" class=\"definition\">\n<dt>slope<\/dt>\n<dd id=\"fs-id1170573596900\">the change in [latex]y[\/latex] for each unit change in [latex]x[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1170573596915\" class=\"definition\">\n<dt>slope-intercept form<\/dt>\n<dd id=\"fs-id1170573596921\">equation of a linear function indicating its slope and [latex]y[\/latex]-intercept<\/dd>\n<\/dl>\n<dl id=\"fs-8773456\" class=\"definition\">\n<dt>standard form<\/dt>\n<dd>equation of a linear function with both variable terms set equal to a constant,\u00a0[latex]ax+by=c[\/latex].<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>symmetry about the origin<\/dt>\n<dd id=\"fs-id1170572235110\">the graph of a function [latex]f[\/latex] is symmetric about the origin if [latex](\u2212x,\u2212y)[\/latex] is on the graph of [latex]f[\/latex] whenever [latex](x,y)[\/latex] is on the graph<\/dd>\n<\/dl>\n<dl id=\"fs-id1170572235164\" class=\"definition\">\n<dt>symmetry about the [latex]y[\/latex]-axis<\/dt>\n<dd id=\"fs-id1170572235175\">the graph of a function [latex]f[\/latex] is symmetric about the [latex]y[\/latex]-axis if [latex](\u2212x,y)[\/latex] is on the graph of [latex]f[\/latex] whenever [latex](x,y)[\/latex] is on the graph<\/dd>\n<\/dl>\n<dl id=\"fs-id1170572235230\" class=\"definition\">\n<dt>table of values<\/dt>\n<dd id=\"fs-id1170572235236\">a table containing a list of inputs and their corresponding outputs<\/dd>\n<\/dl>\n<dl id=\"fs-id1170572235240\" class=\"definition\">\n<dt>\n<\/dt>\n<dt>transcendental function<\/dt>\n<dd><span style=\"font-size: 1em;\">a function that cannot be expressed by a combination of basic arithmetic operations<\/span><\/dd>\n<\/dl>\n<dl id=\"fs-3772119\" class=\"definition\">\n<dt>transformation of a function<\/dt>\n<dd>a shift, scaling, or reflection of a function<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>vertical line test<\/dt>\n<dd id=\"fs-id1170572235245\">given the graph of a function, every vertical line intersects the graph, at most, once<\/dd>\n<\/dl>\n<dl id=\"fs-id1170572235250\" class=\"definition\">\n<dt>zeros of a function<\/dt>\n<dd id=\"fs-id1170572235255\">when a real number [latex]x[\/latex] is a zero of a function [latex]f[\/latex], [latex]f(x)=0[\/latex]<\/dd>\n<\/dl>\n<h2 class=\"font-bold\"><strong>Study Tips<\/strong><\/h2>\n<p><strong>Functions<\/strong><\/p>\n<ul class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Practice evaluating functions by substituting given values for [latex]x[\/latex] in the function&#8217;s formula.<\/li>\n<li class=\"whitespace-normal break-words\">When working with tables, pay attention to how the output changes as the input changes.<\/li>\n<li class=\"whitespace-normal break-words\">When looking at graphs, observe the overall behavior of the function, including any increases, decreases, or constant sections.<\/li>\n<li class=\"whitespace-normal break-words\">For algebraic formulas, practice creating tables of values to help visualize the function.<\/li>\n<\/ul>\n<p><strong>The Domain and Range of a Function<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">When determining the domain, look for potential mathematical restrictions like division by zero or even roots of negative numbers.<\/li>\n<li class=\"whitespace-normal break-words\">For the range, consider the nature of the function (linear, quadratic, etc.) and any limitations on the output values.<\/li>\n<li class=\"whitespace-normal break-words\">Practice using both set notation and interval notation to express domains and ranges.<\/li>\n<li class=\"whitespace-normal break-words\">Remember that infinity ([latex]\\infty[\/latex]) is not a real number but a concept used to describe unbounded sets.<\/li>\n<li class=\"whitespace-normal break-words\">When working with rational functions, pay special attention to values that make the denominator zero.<\/li>\n<li class=\"whitespace-normal break-words\">For functions involving square roots, ensure the expression under the root is non-negative.<\/li>\n<li class=\"whitespace-normal break-words\">Visualize the function graph to help identify the domain and range, especially for common function types.<\/li>\n<\/ul>\n<p><strong>Intercepts of a Function<\/strong><\/p>\n<ul>\n<li>To find [latex]x[\/latex]-intercepts, set the function equal to zero and solve for [latex]x[\/latex].<\/li>\n<li>To find the [latex]y[\/latex]-intercept, evaluate the function at [latex]x = 0[\/latex].<\/li>\n<li>Remember that [latex]x[\/latex]-intercepts are in the form ([latex]x, 0[\/latex]), while the [latex]y[\/latex]-intercept is in the form ([latex]0, y[\/latex]).<\/li>\n<li>Remember that not all functions have x-intercepts, but most will have a [latex]y[\/latex]-intercept.<\/li>\n<\/ul>\n<p><strong>Symmetry of Functions\u00a0<\/strong><\/p>\n<ul>\n<li>To check for symmetry algebraically:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Even functions: Replace [latex]x[\/latex] with [latex]-x[\/latex] and see if the result is identical<\/li>\n<li class=\"whitespace-normal break-words\">Odd functions: Replace [latex]x[\/latex] with [latex]-x[\/latex] and see if the result is the negative of the original function<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Visualize symmetry:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Even functions: Mirror image across [latex]y[\/latex]-axis<\/li>\n<li class=\"whitespace-normal break-words\">Odd functions: [latex]180\u00b0[\/latex] rotation around the origin<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">For absolute value functions:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Remember they create a &#8220;V&#8221; shape on the graph<\/li>\n<li class=\"whitespace-normal break-words\">The output is always non-negative<\/li>\n<li class=\"whitespace-normal break-words\">Transformations can shift or stretch the basic [latex]|x|[\/latex] function<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Practice identifying symmetry in various function types (polynomials, rationals, etc.)<\/li>\n<li class=\"whitespace-normal break-words\">When solving absolute value equations, consider both positive and negative cases<\/li>\n<\/ul>\n<p><strong>Composing Functions\u00a0<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">When combining functions, treat each function as a unit and apply the operation.<\/li>\n<li class=\"whitespace-normal break-words\">For composite functions, work from the inside out: first apply the inner function, then the outer function.<\/li>\n<li class=\"whitespace-normal break-words\">Pay attention to the domains when combining or composing functions, especially for quotients and composites.<\/li>\n<li class=\"whitespace-normal break-words\">Practice identifying the difference between [latex](f \u2218 g)(x)[\/latex] and [latex](g \u2218 f)(x)[\/latex].<\/li>\n<li class=\"whitespace-normal break-words\">When working with composing polynomials:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Add\/subtract: Combine like terms<\/li>\n<li class=\"whitespace-normal break-words\">Multiply: Use the FOIL method for binomials or distribute each term<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p><strong>Polynomial Functions\u00a0<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Always arrange polynomials in descending order of degree for easy identification of the leading term and degree.<\/li>\n<li class=\"whitespace-normal break-words\">Remember that the degree is determined by the highest power of the variable, not the largest number in the function.<\/li>\n<li class=\"whitespace-normal break-words\">The leading coefficient can be positive or negative, and it&#8217;s always the number in front of the highest power term.<\/li>\n<li class=\"whitespace-normal break-words\">When the leading coefficient is [latex]1[\/latex] or [latex]-1[\/latex], it&#8217;s often not written explicitly. For example, [latex]x\u00b3 + 2x\u00b2[\/latex] has a leading coefficient of [latex]1[\/latex].<\/li>\n<li class=\"whitespace-normal break-words\">Be careful with negative exponents or fractional exponents &#8211; these are not considered polynomials.<\/li>\n<\/ul>\n<p><strong>Zeros of Polynomial Functions<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">For quadratic functions, try factoring first if the coefficients are simple.<\/li>\n<li class=\"whitespace-normal break-words\">Memorize the quadratic formula and practice using it.<\/li>\n<li class=\"whitespace-normal break-words\">Pay attention to the sign of the discriminant to predict the nature of roots before solving.<\/li>\n<li class=\"whitespace-normal break-words\">When using the quadratic formula, simplify under the square root if possible before calculating further.<\/li>\n<li class=\"whitespace-normal break-words\">Double-check your solutions by plugging them back into the original equation.<\/li>\n<li class=\"whitespace-normal break-words\">Remember that the number of real zeros a polynomial can have is at most equal to its degree.<\/li>\n<\/ul>\n<p><strong>Graphs of Polynomial Functions Basics<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Always identify the degree and the sign of the leading coefficient first.<\/li>\n<li class=\"whitespace-normal break-words\">Remember: Even degree = same direction ends, Odd degree = opposite direction ends.<\/li>\n<li class=\"whitespace-normal break-words\">Practice sketching basic shapes for different degrees and leading coefficient signs.<\/li>\n<li class=\"whitespace-normal break-words\">For odd-degree polynomials, a negative input will always yield a negative output (assuming positive leading coefficient).<\/li>\n<li class=\"whitespace-normal break-words\">Look at the behavior of the graph as [latex]x[\/latex] approaches positive and negative infinity to determine the degree&#8217;s parity and leading coefficient&#8217;s sign.<\/li>\n<li class=\"whitespace-normal break-words\">Compare graphs of polynomials with the same degree but different leading coefficient signs to see the reflection effect.<\/li>\n<li class=\"whitespace-normal break-words\">When in doubt, plot a few key points to confirm the graph&#8217;s behavior.<\/li>\n<\/ul>\n<p><strong>Piecewise-Defined Functions<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Clearly identify the domain for each piece of the function.<\/li>\n<li class=\"whitespace-normal break-words\">When graphing, use open or closed circles at endpoints to show inclusion or exclusion of boundary points.<\/li>\n<li class=\"whitespace-normal break-words\">Check for continuity at the points where the function definition changes.<\/li>\n<li class=\"whitespace-normal break-words\">When solving equations involving piecewise functions, consider each piece separately.<\/li>\n<li class=\"whitespace-normal break-words\">Use technology to help visualize complex piecewise functions.<\/li>\n<\/ul>\n<p><strong>Algebraic Functions and Transcendental Functions<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">To identify algebraic functions, look for operations limited to addition, subtraction, multiplication, division, and rational exponents.<\/li>\n<li class=\"whitespace-normal break-words\">Remember that any finite combination of algebraic functions is still algebraic.<\/li>\n<li class=\"whitespace-normal break-words\">Transcendental functions often involve trigonometric ratios, e, pi, or logarithms.<\/li>\n<li class=\"whitespace-normal break-words\">Practice identifying functions by their form:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Fraction of polynomials? Rational function.<\/li>\n<li class=\"whitespace-normal break-words\">Variable under a root? Root function.<\/li>\n<li class=\"whitespace-normal break-words\">Variable in the exponent? Likely exponential (transcendental).<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Be cautious with composite functions &#8211; they may combine algebraic and transcendental elements.<\/li>\n<li class=\"whitespace-normal break-words\">When in doubt, ask yourself: &#8220;Can this be broken down into basic algebraic operations?&#8221;<\/li>\n<\/ul>\n<p><strong>Transformations of Functions<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Practice identifying transformations from equations.<\/li>\n<li class=\"whitespace-normal break-words\">Sketch basic function graphs and apply transformations step-by-step.<\/li>\n<li class=\"whitespace-normal break-words\">Remember that horizontal transformations work in the &#8220;opposite&#8221; direction of what you might expect.<\/li>\n<li class=\"whitespace-normal break-words\">Use the order of operations for multiple transformations: parentheses, exponents, multiplication\/division, addition\/subtraction.<\/li>\n<li class=\"whitespace-normal break-words\">Familiarize yourself with basic toolkit functions as starting points for transformations.<\/li>\n<li class=\"whitespace-normal break-words\">When dealing with complex transformations, break them down into individual steps.<\/li>\n<li class=\"whitespace-normal break-words\">Use technology to verify your hand-drawn transformed graphs.<\/li>\n<\/ul>\n","protected":false},"author":15,"menu_order":1,"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":"cheat_sheet","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\/3309"}],"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":12,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/3309\/revisions"}],"predecessor-version":[{"id":4067,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/3309\/revisions\/4067"}],"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\/3309\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/media?parent=3309"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapter-type?post=3309"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/contributor?post=3309"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/license?post=3309"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}