{"id":681,"date":"2025-07-14T21:01:46","date_gmt":"2025-07-14T21:01:46","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=681"},"modified":"2025-12-16T15:18:52","modified_gmt":"2025-12-16T15:18:52","slug":"module-4-polynomial-functions-cheat-sheet","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/module-4-polynomial-functions-cheat-sheet\/","title":{"raw":"Polynomial Functions: Cheat Sheet","rendered":"Polynomial Functions: Cheat Sheet"},"content":{"raw":"<h2>Essential Concepts<\/h2>\r\n<h3>Quadratic Functions<\/h3>\r\nA <strong>parabola<\/strong> is the U-shaped graph of a quadratic function with these features:\r\n<ul>\r\n \t<li>The <strong>vertex<\/strong> is the turning point of the graph; it represents the maximum (if opening down) or minimum (if opening up) value<\/li>\r\n \t<li>The <strong>axis of symmetry <\/strong>is a vertical line through the vertex where the parabola mirrors itself, given by [latex]x = -\\frac{b}{2a}[\/latex]<\/li>\r\n \t<li>The <strong>y-intercept <\/strong>is the point where parabola crosses the y-axis, all quadratics have a y-intercept<\/li>\r\n \t<li>The<strong> x-intercepts (zeros\/roots) <\/strong>are points where parabola crosses the x-axis; values where [latex]y = 0[\/latex]. Not all quadratics have x-intercepts.<\/li>\r\n<\/ul>\r\n<h4><strong>Forms of Quadratic Functions<\/strong><\/h4>\r\n<em>General Form<\/em>: [latex]f(x) = ax^2 + bx + c[\/latex]\r\n<ul>\r\n \t<li>If [latex]a &gt; 0[\/latex], parabola opens upward<\/li>\r\n \t<li>If [latex]a &lt; 0[\/latex], parabola opens downward<\/li>\r\n<\/ul>\r\n<em>Standard (Vertex) Form<\/em>: [latex]f(x) = a(x - h)^2 + k[\/latex]\r\n<ul>\r\n \t<li>Vertex is at point [latex](h, k)[\/latex]<\/li>\r\n \t<li>Makes it easy to identify transformations<\/li>\r\n<\/ul>\r\n<strong>Finding the Vertex from General Form<\/strong>\r\n<ol>\r\n \t<li>Find [latex]h = -\\frac{b}{2a}[\/latex]<\/li>\r\n \t<li>Find [latex]k = f(h)[\/latex]<\/li>\r\n \t<li>Vertex is [latex](h, k)[\/latex]<\/li>\r\n<\/ol>\r\n<h4><strong>Transformations of Quadratic Functions<\/strong><\/h4>\r\nStarting with [latex]f(x) = x^2[\/latex]:\r\n<ul>\r\n \t<li><strong>Vertical shift<\/strong>: [latex]f(x) = x^2 + k[\/latex]\r\n<ul>\r\n \t<li>[latex]k &gt; 0[\/latex]: shift up [latex]k[\/latex] units<\/li>\r\n \t<li>[latex]k &lt; 0[\/latex]: shift down [latex]|k|[\/latex] units<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li><strong>Horizontal shift<\/strong>: [latex]f(x) = (x - h)^2[\/latex]\r\n<ul>\r\n \t<li>[latex]h &gt; 0[\/latex]: shift right [latex]h[\/latex] units<\/li>\r\n \t<li>[latex]h &lt; 0[\/latex]: shift left [latex]|h|[\/latex] units<\/li>\r\n \t<li><em>Note: The sign in the formula is opposite to the direction<\/em><\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li><strong>Vertical stretch\/compression<\/strong>: [latex]f(x) = ax^2[\/latex]\r\n<ul>\r\n \t<li>[latex]|a| &gt; 1[\/latex]: narrower (vertical stretch)<\/li>\r\n \t<li>[latex]0 &lt; |a| &lt; 1[\/latex]: wider (vertical compression)<\/li>\r\n \t<li>[latex]a &lt; 0[\/latex]: reflection across x-axis<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<h4><strong>Maximum and Minimum Values<\/strong><\/h4>\r\nTo find the maximum or minimum value:\r\n<ol>\r\n \t<li>Determine if [latex]a[\/latex] is positive (minimum) or negative (maximum)<\/li>\r\n \t<li>Find the vertex [latex](h, k)[\/latex]<\/li>\r\n \t<li>The max\/min value is [latex]k[\/latex], occurring at [latex]x = h[\/latex]<\/li>\r\n<\/ol>\r\n<h3>Polynomial Functions<\/h3>\r\nA <strong>polynomial<\/strong> function has the form: [latex]f(x) = a_nx^n + a_{n-1}x^{n-1} + \\cdots + a_2x^2 + a_1x + a_0[\/latex]\r\n<ul>\r\n \t<li>Coefficients [latex]a_i[\/latex] are real numbers, [latex]a_n \\ne 0[\/latex]<\/li>\r\n \t<li>Powers are non-negative integers<\/li>\r\n<\/ul>\r\nThe <strong>degree <\/strong>of a polynomial is the highest power of the variable\r\n\r\nThe <strong>leading\u00a0term<\/strong> is the term with the highest degree and the\u00a0<strong>leading coefficient\u00a0<\/strong>is the coefficient of the leading term\r\n<h4><strong>End Behavior<\/strong><\/h4>\r\nEnd behavior describes what happens as [latex]x \\to \\infty[\/latex] or [latex]x \\to -\\infty[\/latex], determined by degree and leading coefficient:\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td><strong>Degree<\/strong><\/td>\r\n<td><strong>Leading Coefficient<\/strong><\/td>\r\n<td><strong>As [latex]x \\to -\\infty[\/latex]<\/strong><\/td>\r\n<td><strong>As [latex]x \\to \\infty[\/latex]<\/strong><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Even<\/td>\r\n<td>Positive<\/td>\r\n<td>[latex]f(x) \\to \\infty[\/latex]<\/td>\r\n<td>[latex]f(x) \\to \\infty[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Even<\/td>\r\n<td>Negative<\/td>\r\n<td>[latex]f(x) \\to -\\infty[\/latex]<\/td>\r\n<td>[latex]f(x) \\to -\\infty[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Odd<\/td>\r\n<td>Positive<\/td>\r\n<td>[latex]f(x) \\to -\\infty[\/latex]<\/td>\r\n<td>[latex]f(x) \\to \\infty[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Odd<\/td>\r\n<td>Negative<\/td>\r\n<td>[latex]f(x) \\to \\infty[\/latex]<\/td>\r\n<td>[latex]f(x) \\to -\\infty[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<h3>Graphs of Polynomial Functions<\/h3>\r\n<h4><strong>Intercepts and Turning Points<\/strong><\/h4>\r\n<ul>\r\n \t<li>A polynomial of degree [latex]n[\/latex] has at most [latex]n[\/latex] x-intercepts<\/li>\r\n \t<li>A polynomial of degree [latex]n[\/latex] has at most [latex]n - 1[\/latex] turning points<\/li>\r\n<\/ul>\r\n<strong>Identifying Intercepts from Factored Form<\/strong>\r\n\r\nFor [latex]f(x) = a(x - r_1)(x - r_2) \\cdots (x - r_n)[\/latex]:\r\n<ul>\r\n \t<li><strong>x-intercepts<\/strong>: Set each factor equal to zero and solve; the zeros are [latex]r_1, r_2, \\ldots, r_n[\/latex]<\/li>\r\n \t<li><strong>y-intercept<\/strong>: Substitute [latex]x = 0[\/latex] and evaluate [latex]f(0)[\/latex]<\/li>\r\n<\/ul>\r\n<strong>Multiplicity and Graph Behavior<\/strong>\r\n\r\nMultiplicity is the number of times a factor appears in factored form.\r\n\r\nAt x-intercepts:\r\n<ul>\r\n \t<li><strong>Odd multiplicity<\/strong> (1, 3, 5...): Graph crosses the x-axis\r\n<ul>\r\n \t<li>Multiplicity 1: crosses like a line<\/li>\r\n \t<li>Higher odd multiplicity: flattens as it crosses<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li><strong>Even multiplicity<\/strong> (2, 4, 6...): Graph touches and bounces off the x-axis\r\n<ul>\r\n \t<li>Multiplicity 2: bounces like a parabola<\/li>\r\n \t<li>Higher even multiplicity: flatter bounce<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\nThe sum of all multiplicities equals the degree of the polynomial.\r\n<h4><strong>Using Factoring to Find Zeros<\/strong><\/h4>\r\nTo find x-intercepts:\r\n<ol>\r\n \t<li>Set [latex]f(x) = 0[\/latex]<\/li>\r\n \t<li>If not factored, factor using:\r\n<ul>\r\n \t<li>Greatest common factor<\/li>\r\n \t<li>Trinomial factoring methods<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>Set each factor equal to zero and solve<\/li>\r\n \t<li>If factoring isn't possible, use technology<\/li>\r\n<\/ol>\r\n<strong>Graphing Polynomial Functions<\/strong>\r\n\r\nSteps to sketch a polynomial graph:\r\n<ol>\r\n \t<li>Determine end behavior using degree and leading coefficient<\/li>\r\n \t<li>Find intercepts:\r\n<ul>\r\n \t<li>x-intercepts: solve [latex]f(x) = 0[\/latex]<\/li>\r\n \t<li>y-intercept: evaluate [latex]f(0)[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>Identify multiplicity at each x-intercept (does it cross or bounce?)<\/li>\r\n \t<li>Plot intercepts and sketch local behavior based on multiplicity<\/li>\r\n \t<li>Connect smoothly respecting end behavior<\/li>\r\n \t<li>Add additional points if needed for accuracy<\/li>\r\n<\/ol>\r\n<h4><strong>Intermediate Value Theorem<\/strong><\/h4>\r\nIf [latex]f[\/latex] is continuous on [latex][a, b][\/latex], and [latex]f(a)[\/latex] and [latex]f(b)[\/latex] have opposite signs, then there exists at least one value [latex]c[\/latex] in [latex](a, b)[\/latex] where [latex]f(c) = 0[\/latex].\r\n\r\n<em>Use: Confirms a zero exists between two points without finding its exact location.<\/em>\r\n\r\n<strong>Writing Formulas from Graphs<\/strong>\r\n\r\nFactored form: [latex]f(x) = a(x - r_1)^{p_1}(x - r_2)^{p_2} \\cdots (x - r_n)^{p_n}[\/latex]\r\n\r\nSteps:\r\n<ol>\r\n \t<li>Identify x-intercepts from the graph<\/li>\r\n \t<li>Determine multiplicities by observing whether the graph crosses (odd) or bounces (even)<\/li>\r\n \t<li>Write the function with [latex]a[\/latex] as unknown: [latex]f(x) = a(x - r_1)^{p_1}(x - r_2)^{p_2} \\cdots[\/latex]<\/li>\r\n \t<li>Find [latex]a[\/latex] using another point on the graph (often the y-intercept)<\/li>\r\n<\/ol>\r\n<h2>Key Equations<\/h2>\r\n<table style=\"width: 81.5923%;\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 34.9398%;\"><strong>General form of a quadratic function<\/strong><\/td>\r\n<td style=\"width: 90%;\">[latex]f(x) = ax^2 + bx + c[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 34.9398%;\"><strong>Standard (vertex) form of a quadratic function<\/strong><\/td>\r\n<td style=\"width: 90%;\">[latex]f(x) = a(x - h)^2 + k[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 34.9398%;\"><strong>The quadratic formula<\/strong><\/td>\r\n<td style=\"width: 90%;\">[latex]x = \\frac{-b \\pm \\sqrt{b^2 - 4ac}}{2a}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 34.9398%;\"><strong>Axis of symmetry<\/strong><\/td>\r\n<td style=\"width: 90%;\">[latex]x = -\\frac{b}{2a}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 34.9398%;\"><strong>General form of a polynomial function<\/strong><\/td>\r\n<td style=\"width: 90%;\">[latex]f(x) = a_nx^n + a_{n-1}x^{n-1} + \\cdots + a_2x^2 + a_1x + a_0[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 34.9398%;\"><strong>Factored form of a polynomial<\/strong><\/td>\r\n<td style=\"width: 90%;\">[latex]f(x) = a(x - r_1)^{p_1}(x - r_2)^{p_2} \\cdots (x - r_n)^{p_n}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<h2>Glossary<\/h2>\r\n<strong>axis of symmetry<\/strong>\r\n<p style=\"padding-left: 40px;\">a vertical line drawn through the vertex of a parabola around which the parabola is symmetric; defined by [latex]x = -\\frac{b}{2a}[\/latex]<\/p>\r\n<strong>coefficient<\/strong>\r\n<p style=\"padding-left: 40px;\">a nonzero real number multiplied by a variable raised to an exponent<\/p>\r\n<strong>degree<\/strong>\r\n<p style=\"padding-left: 40px;\">the highest power of the variable that occurs in a polynomial<\/p>\r\n<strong>end behavior<\/strong>\r\n<p style=\"padding-left: 40px;\">the behavior of the graph of a function as the input decreases without bound and increases without bound<\/p>\r\n<strong>general form of a quadratic function<\/strong>\r\n<p style=\"padding-left: 40px;\">the function that describes a parabola, written as [latex]f(x) = ax^2 + bx + c[\/latex], where [latex]a[\/latex], [latex]b[\/latex], and [latex]c[\/latex] are real numbers and [latex]a \\ne 0[\/latex]<\/p>\r\n<strong>global maximum<\/strong>\r\n<p style=\"padding-left: 40px;\">highest turning point on a graph; [latex]f(a)[\/latex] where [latex]f(a) \\ge f(x)[\/latex] for all [latex]x[\/latex]<\/p>\r\n<strong>global minimum<\/strong>\r\n<p style=\"padding-left: 40px;\">lowest turning point on a graph; [latex]f(a)[\/latex] where [latex]f(a) \\le f(x)[\/latex] for all [latex]x[\/latex]<\/p>\r\n<strong>Intermediate Value Theorem<\/strong>\r\n<p style=\"padding-left: 40px;\">for two numbers [latex]a[\/latex] and [latex]b[\/latex] in the domain of [latex]f[\/latex], if [latex]a &lt; b[\/latex] and [latex]f(a) \\ne f(b)[\/latex], then [latex]f[\/latex] takes on every value between [latex]f(a)[\/latex] and [latex]f(b)[\/latex]; specifically, when a polynomial function changes from a negative value to a positive value, the function must cross the x-axis<\/p>\r\n<strong>leading coefficient<\/strong>\r\n<p style=\"padding-left: 40px;\">the coefficient of the leading term<\/p>\r\n<strong>leading term<\/strong>\r\n<p style=\"padding-left: 40px;\">the term containing the highest power of the variable<\/p>\r\n<strong>local maximum\/minimum<\/strong>\r\n<p style=\"padding-left: 40px;\">highest\/lowest point on a graph in an open interval around [latex]x = a[\/latex]<\/p>\r\n<strong>multiplicity<\/strong>\r\n<p style=\"padding-left: 40px;\">the number of times a given factor appears in the factored form of a polynomial; if a polynomial contains a factor [latex](x - h)^p[\/latex], then [latex]x = h[\/latex] is a zero of multiplicity [latex]p[\/latex]<\/p>\r\n<strong>polynomial function<\/strong>\r\n<p style=\"padding-left: 40px;\">a function that consists of either zero or the sum of a finite number of non-zero terms, each of which is a product of a number (coefficient) and a variable raised to a non-negative integer power<\/p>\r\n<strong>standard form of a quadratic function<\/strong>\r\n<p style=\"padding-left: 40px;\">the function that describes a parabola, written as [latex]f(x) = a(x - h)^2 + k[\/latex], where [latex](h, k)[\/latex] is the vertex<\/p>\r\n<strong>term of a polynomial function<\/strong>\r\n<p style=\"padding-left: 40px;\">any [latex]a_ix^i[\/latex] of a polynomial function in the form [latex]f(x) = a_nx^n + \\cdots + a_2x^2 + a_1x + a_0[\/latex]<\/p>\r\n<strong>turning point<\/strong>\r\n<p style=\"padding-left: 40px;\">the location at which the graph of a function changes direction; a point where the function changes from increasing to decreasing or decreasing to increasing<\/p>\r\n<strong>vertex<\/strong>\r\n<p style=\"padding-left: 40px;\">the point at which a parabola changes direction, corresponding to the minimum or maximum value of the quadratic function<\/p>\r\n<strong>zeros\/roots<\/strong>\r\n<p style=\"padding-left: 40px;\">in a given function, the values of [latex]x[\/latex] at which [latex]y = 0[\/latex], also called roots<\/p>","rendered":"<h2>Essential Concepts<\/h2>\n<h3>Quadratic Functions<\/h3>\n<p>A <strong>parabola<\/strong> is the U-shaped graph of a quadratic function with these features:<\/p>\n<ul>\n<li>The <strong>vertex<\/strong> is the turning point of the graph; it represents the maximum (if opening down) or minimum (if opening up) value<\/li>\n<li>The <strong>axis of symmetry <\/strong>is a vertical line through the vertex where the parabola mirrors itself, given by [latex]x = -\\frac{b}{2a}[\/latex]<\/li>\n<li>The <strong>y-intercept <\/strong>is the point where parabola crosses the y-axis, all quadratics have a y-intercept<\/li>\n<li>The<strong> x-intercepts (zeros\/roots) <\/strong>are points where parabola crosses the x-axis; values where [latex]y = 0[\/latex]. Not all quadratics have x-intercepts.<\/li>\n<\/ul>\n<h4><strong>Forms of Quadratic Functions<\/strong><\/h4>\n<p><em>General Form<\/em>: [latex]f(x) = ax^2 + bx + c[\/latex]<\/p>\n<ul>\n<li>If [latex]a > 0[\/latex], parabola opens upward<\/li>\n<li>If [latex]a < 0[\/latex], parabola opens downward<\/li>\n<\/ul>\n<p><em>Standard (Vertex) Form<\/em>: [latex]f(x) = a(x - h)^2 + k[\/latex]<\/p>\n<ul>\n<li>Vertex is at point [latex](h, k)[\/latex]<\/li>\n<li>Makes it easy to identify transformations<\/li>\n<\/ul>\n<p><strong>Finding the Vertex from General Form<\/strong><\/p>\n<ol>\n<li>Find [latex]h = -\\frac{b}{2a}[\/latex]<\/li>\n<li>Find [latex]k = f(h)[\/latex]<\/li>\n<li>Vertex is [latex](h, k)[\/latex]<\/li>\n<\/ol>\n<h4><strong>Transformations of Quadratic Functions<\/strong><\/h4>\n<p>Starting with [latex]f(x) = x^2[\/latex]:<\/p>\n<ul>\n<li><strong>Vertical shift<\/strong>: [latex]f(x) = x^2 + k[\/latex]\n<ul>\n<li>[latex]k > 0[\/latex]: shift up [latex]k[\/latex] units<\/li>\n<li>[latex]k < 0[\/latex]: shift down [latex]|k|[\/latex] units<\/li>\n<\/ul>\n<\/li>\n<li><strong>Horizontal shift<\/strong>: [latex]f(x) = (x - h)^2[\/latex]\n<ul>\n<li>[latex]h > 0[\/latex]: shift right [latex]h[\/latex] units<\/li>\n<li>[latex]h < 0[\/latex]: shift left [latex]|h|[\/latex] units<\/li>\n<li><em>Note: The sign in the formula is opposite to the direction<\/em><\/li>\n<\/ul>\n<\/li>\n<li><strong>Vertical stretch\/compression<\/strong>: [latex]f(x) = ax^2[\/latex]\n<ul>\n<li>[latex]|a| > 1[\/latex]: narrower (vertical stretch)<\/li>\n<li>[latex]0 < |a| < 1[\/latex]: wider (vertical compression)<\/li>\n<li>[latex]a < 0[\/latex]: reflection across x-axis<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<h4><strong>Maximum and Minimum Values<\/strong><\/h4>\n<p>To find the maximum or minimum value:<\/p>\n<ol>\n<li>Determine if [latex]a[\/latex] is positive (minimum) or negative (maximum)<\/li>\n<li>Find the vertex [latex](h, k)[\/latex]<\/li>\n<li>The max\/min value is [latex]k[\/latex], occurring at [latex]x = h[\/latex]<\/li>\n<\/ol>\n<h3>Polynomial Functions<\/h3>\n<p>A <strong>polynomial<\/strong> function has the form: [latex]f(x) = a_nx^n + a_{n-1}x^{n-1} + \\cdots + a_2x^2 + a_1x + a_0[\/latex]<\/p>\n<ul>\n<li>Coefficients [latex]a_i[\/latex] are real numbers, [latex]a_n \\ne 0[\/latex]<\/li>\n<li>Powers are non-negative integers<\/li>\n<\/ul>\n<p>The <strong>degree <\/strong>of a polynomial is the highest power of the variable<\/p>\n<p>The <strong>leading\u00a0term<\/strong> is the term with the highest degree and the\u00a0<strong>leading coefficient\u00a0<\/strong>is the coefficient of the leading term<\/p>\n<h4><strong>End Behavior<\/strong><\/h4>\n<p>End behavior describes what happens as [latex]x \\to \\infty[\/latex] or [latex]x \\to -\\infty[\/latex], determined by degree and leading coefficient:<\/p>\n<table>\n<tbody>\n<tr>\n<td><strong>Degree<\/strong><\/td>\n<td><strong>Leading Coefficient<\/strong><\/td>\n<td><strong>As [latex]x \\to -\\infty[\/latex]<\/strong><\/td>\n<td><strong>As [latex]x \\to \\infty[\/latex]<\/strong><\/td>\n<\/tr>\n<tr>\n<td>Even<\/td>\n<td>Positive<\/td>\n<td>[latex]f(x) \\to \\infty[\/latex]<\/td>\n<td>[latex]f(x) \\to \\infty[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Even<\/td>\n<td>Negative<\/td>\n<td>[latex]f(x) \\to -\\infty[\/latex]<\/td>\n<td>[latex]f(x) \\to -\\infty[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Odd<\/td>\n<td>Positive<\/td>\n<td>[latex]f(x) \\to -\\infty[\/latex]<\/td>\n<td>[latex]f(x) \\to \\infty[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Odd<\/td>\n<td>Negative<\/td>\n<td>[latex]f(x) \\to \\infty[\/latex]<\/td>\n<td>[latex]f(x) \\to -\\infty[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h3>Graphs of Polynomial Functions<\/h3>\n<h4><strong>Intercepts and Turning Points<\/strong><\/h4>\n<ul>\n<li>A polynomial of degree [latex]n[\/latex] has at most [latex]n[\/latex] x-intercepts<\/li>\n<li>A polynomial of degree [latex]n[\/latex] has at most [latex]n - 1[\/latex] turning points<\/li>\n<\/ul>\n<p><strong>Identifying Intercepts from Factored Form<\/strong><\/p>\n<p>For [latex]f(x) = a(x - r_1)(x - r_2) \\cdots (x - r_n)[\/latex]:<\/p>\n<ul>\n<li><strong>x-intercepts<\/strong>: Set each factor equal to zero and solve; the zeros are [latex]r_1, r_2, \\ldots, r_n[\/latex]<\/li>\n<li><strong>y-intercept<\/strong>: Substitute [latex]x = 0[\/latex] and evaluate [latex]f(0)[\/latex]<\/li>\n<\/ul>\n<p><strong>Multiplicity and Graph Behavior<\/strong><\/p>\n<p>Multiplicity is the number of times a factor appears in factored form.<\/p>\n<p>At x-intercepts:<\/p>\n<ul>\n<li><strong>Odd multiplicity<\/strong> (1, 3, 5&#8230;): Graph crosses the x-axis\n<ul>\n<li>Multiplicity 1: crosses like a line<\/li>\n<li>Higher odd multiplicity: flattens as it crosses<\/li>\n<\/ul>\n<\/li>\n<li><strong>Even multiplicity<\/strong> (2, 4, 6&#8230;): Graph touches and bounces off the x-axis\n<ul>\n<li>Multiplicity 2: bounces like a parabola<\/li>\n<li>Higher even multiplicity: flatter bounce<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p>The sum of all multiplicities equals the degree of the polynomial.<\/p>\n<h4><strong>Using Factoring to Find Zeros<\/strong><\/h4>\n<p>To find x-intercepts:<\/p>\n<ol>\n<li>Set [latex]f(x) = 0[\/latex]<\/li>\n<li>If not factored, factor using:\n<ul>\n<li>Greatest common factor<\/li>\n<li>Trinomial factoring methods<\/li>\n<\/ul>\n<\/li>\n<li>Set each factor equal to zero and solve<\/li>\n<li>If factoring isn&#8217;t possible, use technology<\/li>\n<\/ol>\n<p><strong>Graphing Polynomial Functions<\/strong><\/p>\n<p>Steps to sketch a polynomial graph:<\/p>\n<ol>\n<li>Determine end behavior using degree and leading coefficient<\/li>\n<li>Find intercepts:\n<ul>\n<li>x-intercepts: solve [latex]f(x) = 0[\/latex]<\/li>\n<li>y-intercept: evaluate [latex]f(0)[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li>Identify multiplicity at each x-intercept (does it cross or bounce?)<\/li>\n<li>Plot intercepts and sketch local behavior based on multiplicity<\/li>\n<li>Connect smoothly respecting end behavior<\/li>\n<li>Add additional points if needed for accuracy<\/li>\n<\/ol>\n<h4><strong>Intermediate Value Theorem<\/strong><\/h4>\n<p>If [latex]f[\/latex] is continuous on [latex][a, b][\/latex], and [latex]f(a)[\/latex] and [latex]f(b)[\/latex] have opposite signs, then there exists at least one value [latex]c[\/latex] in [latex](a, b)[\/latex] where [latex]f(c) = 0[\/latex].<\/p>\n<p><em>Use: Confirms a zero exists between two points without finding its exact location.<\/em><\/p>\n<p><strong>Writing Formulas from Graphs<\/strong><\/p>\n<p>Factored form: [latex]f(x) = a(x - r_1)^{p_1}(x - r_2)^{p_2} \\cdots (x - r_n)^{p_n}[\/latex]<\/p>\n<p>Steps:<\/p>\n<ol>\n<li>Identify x-intercepts from the graph<\/li>\n<li>Determine multiplicities by observing whether the graph crosses (odd) or bounces (even)<\/li>\n<li>Write the function with [latex]a[\/latex] as unknown: [latex]f(x) = a(x - r_1)^{p_1}(x - r_2)^{p_2} \\cdots[\/latex]<\/li>\n<li>Find [latex]a[\/latex] using another point on the graph (often the y-intercept)<\/li>\n<\/ol>\n<h2>Key Equations<\/h2>\n<table style=\"width: 81.5923%;\">\n<tbody>\n<tr>\n<td style=\"width: 34.9398%;\"><strong>General form of a quadratic function<\/strong><\/td>\n<td style=\"width: 90%;\">[latex]f(x) = ax^2 + bx + c[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 34.9398%;\"><strong>Standard (vertex) form of a quadratic function<\/strong><\/td>\n<td style=\"width: 90%;\">[latex]f(x) = a(x - h)^2 + k[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 34.9398%;\"><strong>The quadratic formula<\/strong><\/td>\n<td style=\"width: 90%;\">[latex]x = \\frac{-b \\pm \\sqrt{b^2 - 4ac}}{2a}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 34.9398%;\"><strong>Axis of symmetry<\/strong><\/td>\n<td style=\"width: 90%;\">[latex]x = -\\frac{b}{2a}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 34.9398%;\"><strong>General form of a polynomial function<\/strong><\/td>\n<td style=\"width: 90%;\">[latex]f(x) = a_nx^n + a_{n-1}x^{n-1} + \\cdots + a_2x^2 + a_1x + a_0[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 34.9398%;\"><strong>Factored form of a polynomial<\/strong><\/td>\n<td style=\"width: 90%;\">[latex]f(x) = a(x - r_1)^{p_1}(x - r_2)^{p_2} \\cdots (x - r_n)^{p_n}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h2>Glossary<\/h2>\n<p><strong>axis of symmetry<\/strong><\/p>\n<p style=\"padding-left: 40px;\">a vertical line drawn through the vertex of a parabola around which the parabola is symmetric; defined by [latex]x = -\\frac{b}{2a}[\/latex]<\/p>\n<p><strong>coefficient<\/strong><\/p>\n<p style=\"padding-left: 40px;\">a nonzero real number multiplied by a variable raised to an exponent<\/p>\n<p><strong>degree<\/strong><\/p>\n<p style=\"padding-left: 40px;\">the highest power of the variable that occurs in a polynomial<\/p>\n<p><strong>end behavior<\/strong><\/p>\n<p style=\"padding-left: 40px;\">the behavior of the graph of a function as the input decreases without bound and increases without bound<\/p>\n<p><strong>general form of a quadratic function<\/strong><\/p>\n<p style=\"padding-left: 40px;\">the function that describes a parabola, written as [latex]f(x) = ax^2 + bx + c[\/latex], where [latex]a[\/latex], [latex]b[\/latex], and [latex]c[\/latex] are real numbers and [latex]a \\ne 0[\/latex]<\/p>\n<p><strong>global maximum<\/strong><\/p>\n<p style=\"padding-left: 40px;\">highest turning point on a graph; [latex]f(a)[\/latex] where [latex]f(a) \\ge f(x)[\/latex] for all [latex]x[\/latex]<\/p>\n<p><strong>global minimum<\/strong><\/p>\n<p style=\"padding-left: 40px;\">lowest turning point on a graph; [latex]f(a)[\/latex] where [latex]f(a) \\le f(x)[\/latex] for all [latex]x[\/latex]<\/p>\n<p><strong>Intermediate Value Theorem<\/strong><\/p>\n<p style=\"padding-left: 40px;\">for two numbers [latex]a[\/latex] and [latex]b[\/latex] in the domain of [latex]f[\/latex], if [latex]a < b[\/latex] and [latex]f(a) \\ne f(b)[\/latex], then [latex]f[\/latex] takes on every value between [latex]f(a)[\/latex] and [latex]f(b)[\/latex]; specifically, when a polynomial function changes from a negative value to a positive value, the function must cross the x-axis<\/p>\n<p><strong>leading coefficient<\/strong><\/p>\n<p style=\"padding-left: 40px;\">the coefficient of the leading term<\/p>\n<p><strong>leading term<\/strong><\/p>\n<p style=\"padding-left: 40px;\">the term containing the highest power of the variable<\/p>\n<p><strong>local maximum\/minimum<\/strong><\/p>\n<p style=\"padding-left: 40px;\">highest\/lowest point on a graph in an open interval around [latex]x = a[\/latex]<\/p>\n<p><strong>multiplicity<\/strong><\/p>\n<p style=\"padding-left: 40px;\">the number of times a given factor appears in the factored form of a polynomial; if a polynomial contains a factor [latex](x - h)^p[\/latex], then [latex]x = h[\/latex] is a zero of multiplicity [latex]p[\/latex]<\/p>\n<p><strong>polynomial function<\/strong><\/p>\n<p style=\"padding-left: 40px;\">a function that consists of either zero or the sum of a finite number of non-zero terms, each of which is a product of a number (coefficient) and a variable raised to a non-negative integer power<\/p>\n<p><strong>standard form of a quadratic function<\/strong><\/p>\n<p style=\"padding-left: 40px;\">the function that describes a parabola, written as [latex]f(x) = a(x - h)^2 + k[\/latex], where [latex](h, k)[\/latex] is the vertex<\/p>\n<p><strong>term of a polynomial function<\/strong><\/p>\n<p style=\"padding-left: 40px;\">any [latex]a_ix^i[\/latex] of a polynomial function in the form [latex]f(x) = a_nx^n + \\cdots + a_2x^2 + a_1x + a_0[\/latex]<\/p>\n<p><strong>turning point<\/strong><\/p>\n<p style=\"padding-left: 40px;\">the location at which the graph of a function changes direction; a point where the function changes from increasing to decreasing or decreasing to increasing<\/p>\n<p><strong>vertex<\/strong><\/p>\n<p style=\"padding-left: 40px;\">the point at which a parabola changes direction, corresponding to the minimum or maximum value of the quadratic function<\/p>\n<p><strong>zeros\/roots<\/strong><\/p>\n<p style=\"padding-left: 40px;\">in a given function, the values of [latex]x[\/latex] at which [latex]y = 0[\/latex], also called roots<\/p>\n","protected":false},"author":67,"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":74,"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\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/681"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/users\/67"}],"version-history":[{"count":9,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/681\/revisions"}],"predecessor-version":[{"id":5101,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/681\/revisions\/5101"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/74"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/681\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=681"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=681"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=681"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=681"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}