1. 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 A function defined by f(x) = ax 2 + 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.

2. Vertex Formula to Find the Vertex of a Parabola For f(x) = ax 2 + bx + c (a ≠ 0), the vertex is given by: 3.1 – Quadratic Functions and Application Quadratic Functions Analyzing Quadratic Functions Express function in vertex formSketch the graph Open up or downAxis of symmetry Identify the vertex coordinatesMax. or min. value of the function x-intercepts: y = 0State the Domain and Range y-intercepts: x = 0

3. 3.1 – Quadratic Functions and Application Quadratic Functions Use the vertex formula to find the coordinates of the vertex for: coordinates of the vertex

4. 3.1 – Quadratic Functions and Application Quadratic Functions Analyzing Quadratic Functions Express function in vertex formSketch the graph Open up or downAxis of symmetry Identify the vertex coordinatesMax. or min. value of the function x-intercepts: y = 0State the Domain and Range y-intercepts: x = 0 coordinates of the vertex vertex form  

5. 3.1 – Quadratic Functions and Application Quadratic Functions Analyzing Quadratic Functions Express function in vertex formSketch the graph Open up or downAxis of symmetry Identify the vertex coordinatesMax./min. value of the function x-intercepts: y = 0State the Domain and Range y-intercepts: x = 0   up  

6. 3.1 – Quadratic Functions and Application Quadratic Functions

7. 8 in. 20 in. 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? 20 in. x x

8. Defn: Polynomial function 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? yesno yes no 3.2 – Introduction to Polynomial Functions

9. 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. 35 8 12 State the degree of the following polynomial functions 3.2 – Introduction to Polynomial Functions

10. End Behavior of a Function (Leading Term Test) 3.2 – Introduction to Polynomial Functions

11. End Behavior of a Function 3.2 – Introduction to Polynomial Functions

12. Defn: Real Zero of a function r is a real zero of the function. r is an x-intercept of the graph of the function. Equivalent Statements for a Real Zero x – r is a factor of the function. r is a solution to the function f(x) = 0 If f(r) = 0 and r is a real number, then r is a real zero of the function. 3.2 – Introduction to Polynomial Functions

13. Defn: The graph of the function touches the x-axis but does not cross it. Zero Multiplicity of an Even Number Multiplicity The number of times a factor (m) of a function is repeated is referred to its multiplicity (zero multiplicity of m). The graph of the function crosses the x-axis. Zero Multiplicity of an Odd Number 3.2 – Introduction to Polynomial Functions

14. 3 is a zero with a multiplicity of Identify the zeros and their multiplicity 3.-2 is a zero with a multiplicity of 1. Graph crosses the x-axis. -4 is a zero with a multiplicity of 2.7 is a zero with a multiplicity of 1. Graph crosses the x-axis. Graph touches the x-axis. -1 is a zero with a multiplicity of 1.4 is a zero with a multiplicity of 1.Graph crosses the x-axis. 2.2 is a zero with a multiplicity ofGraph touches the x-axis. 3.2 – Introduction to Polynomial Functions

15. If a function has a degree of n, then it has at most n – 1 turning points. Turning Points The point where a function changes directions from increasing to decreasing or from decreasing to increasing. If the graph of a polynomial function has t number of turning points, then the function has at least a degree of t + 1. 3-15-1 8-112-1 What is the most number of turning points the following polynomial functions could have? 24 7 11 3.2 – Introduction to Polynomial Functions

16. 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. 3.2 – Introduction to Polynomial Functions

17. Intermediate Value Theorem Do the following polynomial functions have at least one real zero in the given interval? 3.2 – Introduction to Polynomial Functions

18. Graph a possible function. Leading term: Positive coefficient w/even power End Behavior 3.2 – Introduction to Polynomial Functions Left up; Right up Zeros and Multiplicity Turning points Degree of function: 4

19. Graph a possible function 42 3.2 – Introduction to Polynomial Functions (0, -16)

21. 3.1 – Quadratic Functions and Application Quadratic Functions Analyzing Quadratic Functions Express function in vertex formSketch the graph Open up or downAxis of symmetry Identify the vertex coordinatesMax. or min. value of the function x-intercepts: y = 0State the Domain and Range y-intercepts: x = 0