Vertex Form

Vertex FormForms of quadraticsFactored form a(x-r1)(x-r2)

Standard Form ax2+bx+c

Vertex Form a(x-h)2+kEach form gives you different information!Factored form a(x-r1)(x-r2)Tells you direction of openingTells you location of x-intercepts (roots)Standard Form ax2+bx+cTells you direction of openingTells you location of y-interceptVertex Form a(x-h)2+kTells you direction openingTells you the location of the vertex (max or min)

Direction of openingx2 opens up

Direction of openingax2 stretches x vertically by aHere a is 1.5

Direction of openingax2 stretches x vertically by aHere a is 0.5Stretching by a fraction is a squish

Direction of openingax2 stretches x vertically by aHere a is -0.5Stretching by a negative causes a flipDirection of openinga is the number in front of the x2The value a tells you what direction the parabola is opening in.Positive a opens upNegative a opens downThe a in all three forms is the same numbera(x-r1)(x-r2)ax2+bx+ca(x-h)2+k

Factored form a(x-r1)(x-r2)a is the direction of openingr1 and r2 are the x-interceptsOr roots, or zerosExample: -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 openingr1 and r2 are the x-interceptsOr roots, or zerosExample: -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 -0.5Standard form ax2+bx+ca is the direction of openingc is the y-intercept(0)=a02+b0+c=cExample: -2x2+3x+2Opens downCrosses through the point (0,2)

Standard form ax2+bx+ca is the direction of openingc is the y-intercept(0)=a02+b0+c=cExample: -2x2+3x+2Opens downCrosses through the point (0,2)

Vertex formStart with f(x)=x2

Vertex formStretch/Flip if you wanta(x)=ax2

Vertex formShift right by ha(x-h)=a(x-h)2h

Vertex formShift up by ka(x-h)+k=a(x-h)2+khk

Vertex formDefine a new functiong(x)=a(x-h)2+k

(h,k)

Vertex form a(x-h)2+ka tells you direction of opening(h,k) is the vertex(h,k)Vertex form a(x-h)2+ka tells you direction of opening(h,k) is the vertexExample: -2(x-3/4)2+25/8Opens downHas vertex at (3/4, 25/8)

Vertex form a(x-h)2+ka tells you direction of opening(h,k) is the vertexExample: -2(x-3/4)2+25/8Opens downHas vertex at (3/4, 25/8)(3/4, 25/8)Switching between formsGives you a full pictureExample: (x)=-2(x-2)(x+0.5)(x)=-2x2+3x+2(x)=-2(x-3/4)2+25/8are all the same functionOpens downCrosses x axis at 2 and -0.5Crosses the y-axis at 2Has vertex at (3/4, 25/8)

Switching between formsGives you a full pictureExample: (x)=-2(x-2)(x+0.5)(x)=-2x2+3x+2(x)=-2(x-3/4)2+25/8are all the same functionOpens downCrosses x axis at 2 and -0.5Crosses the y-axis at 2Has vertex at (3/4, 25/8)Consider the function f(x) = -3x2+2x-9. Which of the following are true? The graph of f(x) has a negative y-interceptB) f(x) has 2 real zeros.C) The graph of f(x) attains a maximum valueD) Both (A) and (B) are trueE) Both (A) and (C) are true. Consider the function f(x) = -3x2+2x-9. Which of the following are true? Standard form: ax2+bx+c.a is negative: opens down. (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 FormulaRemember the Quadratic formula

What does the QF say?

The Vertex Formula

Example

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

How to find an equation from vertex and pointA 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 pointA 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 pointA 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,y1How to find an equation from vertex and pointA 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,y1A quadratic function has vertex at (0,2) and passes through the point (1,3). Find an equation for this parabola.y = (x+2)2y = x2+3y = x2+1y = x2None of the aboveA quadratic function has vertex at (0,2) and passes through the point (1,3). Find an equation for this parabola.

E