Vertex Form

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

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 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)

Direction of opening x2 opens up

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

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

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

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

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

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

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)

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)

Vertex form Start with f(x)=x2

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

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

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

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

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

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)

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)

Switching between forms Gives 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)

Switching between forms Gives 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)

The graph of f(x) has a negative y-intercept B) f(x) has 2 real zeros. 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.

Standard form: ax2+bx+c. 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.

The Vertex Formula Remember the Quadratic formula

What does the QF say?

The Vertex Formula

Example

x = - 9 x = 9 x = 2 x = 6 None of the above. 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.

The axis of symmetry is halfway between the roots. 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.

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?

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)

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

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

y = (x+2)2 y = x2+3 y = x2+1 y = x2 None of the above 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

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