2. Unit 10: Introduction to Quadratic Functions Foundations of Mathematics 1 Ms. C. Taylor

3. Warm-Up

4. Quadratic Function  A function of the form y=ax 2 +bx+c where a≠0 making a u-shaped graph called a parabola. Example quadratic equation:

5. Vertex- TThe lowest or highest point of a parabola. Vertex Axis of symmetry- TThe vertical line through the vertex of the parabola. Axis of Symmetry

6. Standard Form Equation y=ax 2 + bx + c  If a is positive, u opens up If a is negative, u opens down  The x-coordinate of the vertex is at  To find the y-coordinate of the vertex, plug the x- coordinate into the given eqn.  The axis of symmetry is the vertical line x=  Choose 2 x-values on either side of the vertex x- coordinate. Use the eqn to find the corresponding y- values.  Graph and label the 5 points and axis of symmetry on a coordinate plane. Connect the points with a smooth curve.

7. Example 1: Graph y=2x 2 -8x+6 aa=2 Since a is positive the parabola will open up. VVertex: use b=-8 and a=2 Vertex is: (2,-2) Axis of symmetry is the vertical line x=2 Table of values for other points: x yTable of values for other points: x y 06 06 10 10 2-2 2-2 30 30 46 46 * Graph! x=2

8. Now you try one! y=-x 2 +x+12 * Open up or down? * Vertex? * Axis of symmetry? * Table of values with 5 points?

9. (-1,10) (-2,6) (2,10) (3,6) X =.5 (.5,12)

10. Example 2: Graph y=-.5(x+3) 2 +4  a is negative (a = -.5), so parabola opens down.  Vertex is (h,k) or (-3,4)  Axis of symmetry is the vertical line x = -3  Table of values x y -1 2 -2 3.5 -3 4 -4 3.5 -5 2 Vertex (-3,4) (-4,3.5) (-5,2) (-2,3.5) (-1,2) x=-3

11. Now you try one! y=2(x-1) 2 +3  Open up or down?  Vertex?  Axis of symmetry?  Table of values with 5 points?

12. (-1, 11) (0,5) (1,3) (2,5) (3,11) X = 1

13. Example 3: Graph y=-(x+2)(x-4) SSince a is negative, parabola opens down. TThe x-intercepts are (- 2,0) and (4,0) TTo find the x-coord. of the vertex, use TTo find the y-coord., plug 1 in for x. VVertex (1,9) The axis of symmetry is the vertical line x=1 (from the x-coord. of the vertex)The axis of symmetry is the vertical line x=1 (from the x-coord. of the vertex) x=1 (-2,0)(4,0) (1,9)

14. Now you try one! y=2(x-3)(x+1)  Open up or down?  X-intercepts?  Vertex?  Axis of symmetry?

15. (-1,0)(3,0) (1,-8) x=1

16. Quadratic of the form f(x) = ax 2 Key Features Symmetry about x =0 Vertex at (0,0) The bigger the value of a the steeper the curve. -x 2 flips the curve about x - axis

17. Quadratic of the form f(x) = ax 2 + c Key Features Symmetry about x = 0 Vertex at (0,C) a > 0 the vertex (0,C) is a minimum turning point. a < 0 the vertex (0,C) is a maximum turning point.

18. Quadratic of the form f(x) = a(x - b) 2 Key Features Symmetry about x = b Vertex at (b,0) Cuts y - axis at x = 0 a > 0 the vertex (b,0) is a minimum turning point. a < 0 the vertex (b,0) is a maximum turning point.

19. Roots & Zeros of Polynomials I How the roots, solutions, zeros, x-intercepts and factors of a polynomial function are related.

20. Polynomials A Polynomial Expression can be a monomial or a sum of monomials. The Polynomial Expressions that we are discussing today are in terms of one variable. In a Polynomial Equation, two polynomials are set equal to each other.

21. Factoring Polynomials Terms are Factors of a Polynomial if, when they are multiplied, they equal that polynomial: (x - 3) and (x + 5) are Factors of the polynomial

22. Since Factors are a Product... …and the only way a product can equal zero is if one or more of the factors are zero… …then the only way the polynomial can equal zero is if one or more of the factors are zero.

23. Solving a Polynomial Equation The only way that x 2 +2x - 15 can = 0 is if x = -5 or x = 3 Rearrange the terms to have zero on one side: Factor: Set each factor equal to zero and solve:

24. Solutions/Roots a Polynomial Setting the Factors of a Polynomial Expression equal to zero gives the Solutions to the Equation when the polynomial expression equals zero. Another name for the Solutions of a Polynomial is the Roots of a Polynomial !

25. Zeros of a Polynomial Function A Polynomial Function is usually written in function notation or in terms of x and y. The Zeros of a Polynomial Function are the solutions to the equation you get when you set the polynomial equal to zero.

26. Zeros of a Polynomial Function The Zeros of a Polynomial Function ARE the Solutions to the Polynomial Equation when the polynomial equals zero.

27. Graph of a Polynomial Function Here is the graph of our polynomial function: The Zeros of the Polynomial are the values of x when the polynomial equals zero. In other words, the Zeros are the x-values where y equals zero.

28. x-Intercepts of a Polynomial The points where y = 0 are called the x-intercepts of the graph. The x-intercepts for our graph are the points... and (-5, 0) (3, 0)

29. x-Intercepts of a Polynomial When the Factors of a Polynomial Expression are set equal to zero, we get the Solutions or Roots of the Polynomial Equation. The Solutions/Roots of the Polynomial Equation are the x-coordinates for the x-Intercepts of the Polynomial Graph!

30. Factors, Roots, Zeros For our Polynomial Function: The Factors are:(x + 5) & (x - 3) The Roots/Solutions are:x = -5 and 3 The Zeros are at:(-5, 0) and (3, 0)