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3.1 – Quadratic Functions and Application Quadratic Functions A function defined by f(x) = ax2 + bx + c (a ≠ 0) is called a quadratic function. By completing the square, f(x) can be expressed in vertex form as f(x) = a(x − h)2 + k. •The graph of f is a parabola with vertex (h, k). •If a > 0, the parabola opens upward, and the vertex is the minimum point. The minimum value of f is k. •If a < 0, the parabola opens downward, and the vertex is the maximum point. The maximum value of f is k. •The axis of symmetry is x = h. This is the vertical line that passes through the vertex. 3.1 – Quadratic Functions and Application Quadratic Functions Vertex Formula to Find the Vertex of a Parabola For f(x) = ax2 + bx + c (a ≠ 0), the vertex is given by: Analyzing Quadratic Functions Express function in vertex form Sketch the graph Open up or down Axis of symmetry Identify the vertex coordinates Max. or min. value of the function x-intercepts: y = 0 State the Domain and Range y-intercepts: x = 0 3.1 – Quadratic Functions and Application Quadratic Functions Use the vertex formula to find the coordinates of the vertex for: 𝑞 𝑥 = 3𝑥 2 − 36𝑥 + 1 36 −(−36) =6 = 6 2(3) 𝑞 6 = 3(6)2 −36(6) + 1 𝑞 6 = −107 coordinates of the vertex (6, −107) 3.1 – Quadratic Functions and Application Quadratic Functions 𝑞 𝑥 = 2𝑥 2 + 8𝑥 − 1 Analyzing Quadratic Functions Express function in vertex form a Open up or down Identify the vertex coordinates Sketch the graph Axis of symmetry a x-intercepts: y = 0 Max. or min. value of the function State the Domain and Range y-intercepts: x = 0 𝑞 𝑥 = 2(𝑥 2 + 4𝑥) − 1 4 = 2 22 = 4 2 𝑞 𝑥 = 2(𝑥 + 2)2 −8 − 1 𝑞 𝑥 = 2(𝑥 2 + 4𝑥 + 4 − 4) − 1 coordinates of the vertex (−2, −9) 𝑞 𝑥 = 2((𝑥 + 2)2 −4) − 1 vertex form 𝑞 𝑥 = 2(𝑥 + 2)2 −9 3.1 – Quadratic Functions and Application Quadratic Functions 𝑞 𝑥 = 2𝑥 2 + 8𝑥 − 1 𝑞 𝑥 = 2(𝑥 + 2)2 −9 Analyzing Quadratic Functions Express function in vertex form Open up or down Identify the vertex coordinates x-intercepts: y = 0 y-intercepts: x = 0 a up a a a 𝑥 − 𝑖𝑛𝑡: 𝑦 = 0 −8 ± 82 − 4(2)(−1) 𝑥= 2(2) 𝑥 = −4.121, 0.121 Sketch the graph Axis of symmetry 𝑥 = −2 Max./min. value of the function 𝑚𝑖𝑛. : −9 State the Domain and Range 𝑦 − 𝑖𝑛𝑡: 𝑥 = 0 𝑞 𝑥 = 𝑦 = −1 3.1 – Quadratic Functions and Application Quadratic Functions 𝑞 𝑥 = 2𝑥 2 + 8𝑥 − 1 𝑥 = −2 (−4.121, 0) (0.121, 0) (0, −1) (−2, −9) 3.1 – Quadratic Functions and Application Quadratic Functions Given a sheet of aluminum that measures 20 inches by 8 inches: a) Write the equation that represents the volume of a rectangular gutter that can be formed from the sheet of aluminum. b) At what value of x does the maximum volume occur? c) What is the maximum volume? 𝑎) 𝑐) 𝑉=𝑙𝑤ℎ 𝑉 = 20 8 − 2𝑥 𝑥 𝑉(𝑥) = 𝑏) x 8 − 2𝑥 x −40𝑥 2 + 160𝑥 max 𝑣𝑜𝑙𝑢𝑚𝑒 𝑉(𝑥) = −40𝑥 2 + 160𝑥 𝑉(2) = −40 2 2 + 160(2) 𝑝𝑎𝑟𝑎𝑏𝑜𝑙𝑎, 𝑜𝑝𝑒𝑛𝑠 𝑑𝑜𝑤𝑛 𝑉(2) = 160 ∴ 𝑡ℎ𝑒𝑟𝑒 𝑖𝑠 𝑎 𝑚𝑎𝑥𝑖𝑚𝑢𝑚 max 𝑣𝑜𝑙𝑢𝑚𝑒 = 160 𝑖𝑛3 max 𝑜𝑐𝑐𝑢𝑟𝑠 𝑎𝑡 𝑡ℎ𝑒 𝑣𝑒𝑟𝑡𝑒𝑥 −𝑏 −160 𝑥= = =2 2𝑎 2(−40) 3.2 – Introduction to Polynomial Functions Defn: Polynomial function In the form of: 𝑓 𝑥 = 𝑎𝑛 𝑥 𝑛 + 𝑎𝑛−1 𝑥 𝑛−1 + ⋯ 𝑎1 𝑥 + 𝑎0 . The coefficients are real numbers. The exponents are non-negative integers. The domain of the function is the set of all real numbers. Are the following functions polynomials? 𝑓 𝑥 = 5𝑥 + 2𝑥 2 − 6𝑥 3 + 3 yes ℎ 𝑥 = 2𝑥 3 (4𝑥 5 + 3𝑥) yes 𝑔 𝑥 = 2𝑥 2 − 4𝑥 + 𝑥 − 2 no 2𝑥 3 + 3 𝑘 𝑥 = 5 4𝑥 + 3𝑥 no 3.2 – Introduction to Polynomial Functions Defn: Degree of a Function The largest degree of the function represents the degree of the function. The zero function (all coefficients and the constant are zero) does not have a degree. State the degree of the following polynomial functions 𝑓 𝑥 = 5𝑥 + 2𝑥 2 − 6𝑥 3 + 3 𝑔 𝑥 = 2𝑥 5 − 4𝑥 3 + 𝑥 − 2 3 ℎ 𝑥 = 2𝑥 3 (4𝑥 5 + 3𝑥) 8 5 𝑘 𝑥 = 4𝑥 3 + 6𝑥 11 − 𝑥 10 + 𝑥 12 12 3.2 – Introduction to Polynomial Functions End Behavior of a Function (Leading Term Test) If 𝑓 𝑥 = 𝑎𝑛 𝑥 𝑛 + 𝑎𝑛−1 𝑥 𝑛−1 + ⋯ 𝑎1 𝑥 + 𝑎0 , then the end behaviors of the graph will depend on the first term of the function, 𝑎𝑥 𝑛 . If 𝑓 𝑥 = 𝑎𝑥 𝑛 and n is even, then both ends will approach +. If 𝑓 𝑥 = −𝑎𝑥 𝑛 and n is even, then both ends will approach –. If 𝑓 𝑥 = 𝑎𝑥 𝑛 and n is odd, then as x – , 𝑓 𝑥 – and as x , 𝑓 𝑥 . If 𝑓 𝑥 = −𝑎𝑥 𝑛 and n is odd, then as x – , 𝑓 𝑥 and as x , 𝑓 𝑥 –. 3.2 – Introduction to Polynomial Functions End Behavior of a Function 𝑓 𝑥 = 𝑎𝑥 𝑛 and n is even 𝑓 𝑥 = −𝑎𝑥 𝑛 and n is even 𝑓 𝑥 = 𝑎𝑥 𝑛 and n is odd 𝑓 𝑥 = −𝑎𝑥 𝑛 and n is odd 3.2 – Introduction to Polynomial Functions Defn: Real Zero of a function If f(r) = 0 and r is a real number, then r is a real zero of the function. Equivalent Statements for a Real Zero r is a real zero of the function. r is an x-intercept of the graph of the function. x – r is a factor of the function. r is a solution to the function f(x) = 0 3.2 – Introduction to Polynomial Functions Defn: Multiplicity The number of times a factor (m) of a function is repeated is referred to its multiplicity (zero multiplicity of m). Zero Multiplicity of an Even Number The graph of the function touches the x-axis but does not cross it. Zero Multiplicity of an Odd Number The graph of the function crosses the x-axis. 3.2 – Introduction to Polynomial Functions Identify the zeros and their multiplicity 𝑓 𝑥 = 𝑥−3 𝑥+2 3 3 is a zero with a multiplicity of 1. Graph crosses the x-axis. -2 is a zero with a multiplicity of 3. Graph crosses the x-axis. 𝑔 𝑥 =5 𝑥+4 𝑥−7 2 -4 is a zero with a multiplicity of 1. Graph crosses the x-axis. 7 is a zero with a multiplicity of 2. Graph touches the x-axis. 𝑔 𝑥 = 𝑥 + 1 (𝑥 − 4) 𝑥 − 2 2 -1 is a zero with a multiplicity of 1. Graph crosses the x-axis. 4 is a zero with a multiplicity of 1. Graph crosses the x-axis. 2 is a zero with a multiplicity of 2. Graph touches the x-axis. 3.2 – Introduction to Polynomial Functions Turning Points The point where a function changes directions from increasing to decreasing or from decreasing to increasing. If a function has a degree of n, then it has at most n – 1 turning points. If the graph of a polynomial function has t number of turning points, then the function has at least a degree of t + 1 . What is the most number of turning points the following polynomial functions could have? 𝑓 𝑥 = 5𝑥 + 2𝑥 2 − 6𝑥 3 + 3 3-1 2 ℎ 𝑥 = 2𝑥 3 (4𝑥 5 + 3𝑥) 8-1 7 𝑔 𝑥 = 2𝑥 5 − 4𝑥 3 + 𝑥 − 2 5-1 4 𝑘 𝑥 = 4𝑥 3 + 6𝑥 11 − 𝑥 10 + 𝑥 12 12-1 11 3.2 – Introduction to Polynomial Functions Intermediate Value Theorem In a polynomial function, if a < b and f(a) and f(b) are of opposite signs, then there is at least one real zero between a and b. (𝑏, 𝑓 𝑏 ) (𝑎, 𝑓 𝑎 ) 𝑟𝑒𝑎𝑙 𝑧𝑒𝑟𝑜 𝑟𝑒𝑎𝑙 𝑧𝑒𝑟𝑜 (𝑎, 𝑓 𝑎 ) (𝑏, 𝑓 𝑏 ) 3.2 – Introduction to Polynomial Functions Intermediate Value Theorem Do the following polynomial functions have at least one real zero in the given interval? 𝑓 𝑥 = 2𝑥 3 − 3𝑥 2 − 2 𝑓 𝑥 = 2𝑥 3 − 3𝑥 2 − 2 [0, 2] [3, 6] 𝑓 0 = −2 𝑓 2 = 2 𝑦𝑒𝑠 𝑛𝑜𝑡 𝑒𝑛𝑜𝑢𝑔ℎ 𝑖𝑛𝑓𝑜𝑟𝑚𝑎𝑡𝑖𝑜𝑛 𝑓 𝑥 = 𝑥 4 − 2𝑥 2 − 3𝑥 − 3 [−5, −2] 𝑓 −5 = 587 𝑓 6 = 322 𝑓 3 = 25 𝑓 −2 = 11 𝑛𝑜𝑡 𝑒𝑛𝑜𝑢𝑔ℎ 𝑖𝑛𝑓𝑜𝑟𝑚𝑎𝑡𝑖𝑜𝑛 𝑓 𝑥 = 𝑥 4 − 2𝑥 2 − 3𝑥 − 3 [−1, 3] 𝑓 −1 = −1 𝑓 3 = 51 𝑦𝑒𝑠 3.2 – Introduction to Polynomial Functions Graph a possible function. 𝑔 𝑥 = 𝑥 + 1 (𝑥 − 4) 𝑥 − 2 2 End Behavior Leading term: 𝑥 4 Positive coefficient w/even power Left up; Right up Zeros and Multiplicity 𝑥 = −1, zero mult. of 1 𝑐𝑟𝑜𝑠𝑠𝑒𝑠 𝑎𝑡 𝑥 = −1 𝑥 = 4, zero mult. of 1 𝑐𝑟𝑜𝑠𝑠𝑒𝑠 𝑎𝑡 𝑥 = 4 𝑥 = 2, zero mult. of 2 𝑡𝑜𝑢𝑐ℎ𝑒𝑠 𝑎𝑡 𝑥 = 2 y-int. → 𝑥 = 0 𝑔 𝑥 = 𝑦 = 0 + 1 (0 − 4) 0 − 2 𝑦 = −16 Turning points Degree of function: 4 4−1=3 𝐴𝑡 𝑚𝑜𝑠𝑡, 3 𝑡𝑢𝑟𝑛𝑖𝑛𝑔 𝑝𝑜𝑖𝑛𝑡𝑠 2 3.2 – Introduction to Polynomial Functions Graph a possible function 𝑔 𝑥 = 𝑥 + 1 (𝑥 − 4) 𝑥 − 2 -1 (0, -16) 2 4 2 3.1 – Quadratic Functions and Application Quadratic Functions 𝑞 𝑥 = −2 𝑥 + 3 2 +8 Analyzing Quadratic Functions Express function in vertex form Sketch the graph Open up or down Axis of symmetry Identify the vertex coordinates Max. or min. value of the function x-intercepts: y = 0 State the Domain and Range y-intercepts: x = 0