1 / 41

Vertex Form

Vertex Form. Forms of quadratics. Factored form a(x-r 1 )(x-r 2 ) Standard Form ax 2 +bx+c Vertex Form a(x-h) 2 +k. Each form gives you different information!. Factored form a(x-r 1 )(x-r 2 ) Tells you direction of opening Tells you location of x-intercepts (roots)

blythe
Télécharger la présentation

Vertex Form

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  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.

  25. The Vertex Formula • Remember the Quadratic formula

  26. What does the QF say?

  27. The Vertex Formula

  28. Example

  29. 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.

  30. 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.

  31. 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?

  32. 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)

  33. 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

  34. 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

  35. 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

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

More Related