Vertex Form

1. Vertex Form

2. Forms of quadratics • Factored form a(x-r1)(x-r2) • Standard Form ax2+bx+c • Vertex Form a(x-h)2+k

3. Each form gives you different information! • Factored form a(x-r1)(x-r2) • Tells you direction of opening • Tells you location of x-intercepts (roots) • Standard Form ax2+bx+c • Tells you direction of opening • Tells you location of y-intercept • Vertex Form a(x-h)2+k • Tells you direction opening • Tells you the location of the vertex (max or min)

4. Direction of opening • x2 opens up

5. Direction of opening • ax2 stretches x vertically by a • Here a is 1.5

6. Direction of opening • ax2 stretches x vertically by a • Here a is 0.5 • Stretching by a fraction is a squish

7. Direction of opening • ax2 stretches x vertically by a • Here a is -0.5 • Stretching by a negative causes a flip

8. Direction of opening • a is the number in front of the x2 • The value a tells you what direction the parabola is opening in. • Positive a opens up • Negative a opens down • The a in all three forms is the same number • a(x-r1)(x-r2) • ax2+bx+c • a(x-h)2+k

9. Factored form a(x-r1)(x-r2) • a is the direction of opening • r1 and r2 are the x-intercepts • Or roots, or zeros • Example: -2(x-2)(x+0.5) • a is negative, opens down. • r1 is 2, crosses the x-axis at 2. • r2 is -0.5, crosses the x-axis at -0.5

10. Factored form a(x-r1)(x-r2) • a is the direction of opening • r1 and r2 are the x-intercepts • Or roots, or zeros • Example: -2(x-2)(x+0.5) • a is negative, opens down. • r1 is 2, crosses the x-axis at 2. • r2 is -0.5, crosses the x-axis at -0.5

11. Standard form ax2+bx+c • a is the direction of opening • c is the y-intercept • ƒ(0)=a02+b0+c=c • Example: -2x2+3x+2 • Opens down • Crosses through the point (0,2)

12. Standard form ax2+bx+c • a is the direction of opening • c is the y-intercept • ƒ(0)=a02+b0+c=c • Example: -2x2+3x+2 • Opens down • Crosses through the point (0,2)

13. Vertex form • Start with f(x)=x2

14. Vertex form • Stretch/Flip if you want • aƒ(x)=ax2

15. Vertex form • Shift right by h • aƒ(x-h)=a(x-h)2 h

16. Vertex form • Shift up by k • aƒ(x-h)+k=a(x-h)2+k k h

17. Vertex form • Define a new function • g(x)=a(x-h)2+k (h,k)

18. Vertex form a(x-h)2+k • a tells you direction of opening • (h,k) is the vertex (h,k)

19. Vertex form a(x-h)2+k • a tells you direction of opening • (h,k) is the vertex • Example: -2(x-3/4)2+25/8 • Opens down • Has vertex at (3/4, 25/8)

20. Vertex form a(x-h)2+k • a tells you direction of opening • (h,k) is the vertex • Example: -2(x-3/4)2+25/8 • Opens down • Has vertex at (3/4, 25/8) (3/4, 25/8)

21. Switching between formsGives you a full picture • Example: ƒ(x)=-2(x-2)(x+0.5) ƒ(x)=-2x2+3x+2 ƒ(x)=-2(x-3/4)2+25/8 are all the same function • Opens down • Crosses x axis at 2 and -0.5 • Crosses the y-axis at 2 • Has vertex at (3/4, 25/8)

22. Switching between formsGives you a full picture • Example: ƒ(x)=-2(x-2)(x+0.5) ƒ(x)=-2x2+3x+2 ƒ(x)=-2(x-3/4)2+25/8 are all the same function • Opens down • Crosses x axis at 2 and -0.5 • Crosses the y-axis at 2 • Has vertex at (3/4, 25/8)

23. Consider the function f(x) = -3x2+2x-9. Which of the following are true? • The graph of f(x) has a negative y-intercept B) f(x) has 2 real zeros. C) The graph of f(x) attains a maximum value D) Both (A) and (B) are true E) Both (A) and (C) are true.

24. Consider the function f(x) = -3x2+2x-9. Which of the following are true? Standard form: ax2+bx+c. a is negative: opens down. ƒ(x) attains a maximum value. (C) is true. c is my y-intercept. c is negative. My y-intercept is negative. (A) is true. E) Both (A) and (C) are true.

30. The Vertex Formula • Remember the Quadratic formula

31. What does the QF say?

32. The Vertex Formula

33. Example

34. Given the function R(x)=(2x+6)(x-12), find an equation for its axis of symmetry. • x = - 9 • x = 9 • x = 2 • x = 6 • None of the above.

35. Given the function R(x)=(2x+6)(x-12), find an equation for its axis (line) of symmetry. • The roots are x=-3 and x=12. • The axis of symmetry is halfway between the roots. • (12-3)/2=4.5, the number halfway between -3 and 12. • x=4.5 is the axis of symmetry • E) None of the above.

36. How to find an equation from vertex and point • A parabola passes has its vertex at (1,3) and passes through the point (0,1). What is the equation of this parabola?

37. How to find an equation from vertex and point • A parabola passes has its vertex at (1,3) and passes through the point (0,1). What is the equation of this parabola? • (h,k)=(1,3) • (x1,y1)=(0,1)

38. How to find an equation from vertex and point • A parabola passes has its vertex at (1,3) and passes through the point (0,1). What is the equation of this parabola? • (h,k)=(1,3) • (x1,y1)=(0,1) But to be finished, I need to know a! Use: My formula is true for every x,y including x1,y1

39. How to find an equation from vertex and point • A parabola passes has its vertex at (1,3) and passes through the point (0,1). What is the equation of this parabola? • (h,k)=(1,3) • (x1,y1)=(0,1) My formula is true for every x,y; not just x1,y1

40. A quadratic function has vertex at (0,2) and passes through the point (1,3). Find an equation for this parabola. • y = (x+2)2 • y = x2+3 • y = x2+1 • y = x2 • None of the above

41. A quadratic function has vertex at (0,2) and passes through the point (1,3). Find an equation for this parabola. E