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) Standard Form ax 2 +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)

Direction of opening

• x 2 opens up

Direction of opening

• ax 2 stretches x vertically by a – Here a is 1.5

Direction of opening

• ax 2 stretches x vertically by a – Here a is 0.5

– Stretching by a fraction is a squish

Direction of opening

• ax 2 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 x 2 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-r 1 )(x-r 2 ) ax 2 +bx+c – a(x-h) 2 +k

Factored form a(x-r

1

)(x-r

2

)

• • • a is the direction of opening r 1 – and r 2 are the x-intercepts Or roots, or zeros Example: -2(x-2)(x+0.5) – a is negative, opens down.

– r 1 is 2, crosses the x-axis at 2.

– r 2 is -0.5, crosses the x-axis at -0.5

Factored form a(x-r

1

)(x-r

2

)

• • • a is the direction of opening r 1 – and r 2 are the x-intercepts Or roots, or zeros Example: -2(x-2)(x+0.5) – a is negative, opens down.

– r 1 is 2, crosses the x-axis at 2.

– r 2 is -0.5, crosses the x-axis at -0.5

Standard form ax

2

+bx+c

• • • a is the direction of opening c is the y-intercept – ƒ(0)=a0 2 +b0+c=c Example: -2x 2 +3x+2 – Opens down – Crosses through the point (0,2)

Standard form ax

2

+bx+c

• • • a is the direction of opening c is the y-intercept – ƒ(0)=a0 2 +b0+c=c Example: -2x 2 +3x+2 – Opens down – Crosses through the point (0,2)

Vertex form

• Start with f(x)=x 2

Vertex form

• Stretch/Flip if you want – aƒ(x)=ax 2

• Shift right by h – aƒ(x-h)=a(x-h) 2

Vertex form

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)=-2x 2 +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)=-2x 2 +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)

Consider the function f(x) = -3x 2 +2x-9. Which of the following are true? A) 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.

Consider the function f(x) = -3 x 2 + 2 x -9 . Which of the following are true?

Standard form: a x 2 + b x+ 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

when ax

2 +

bx

+

c

=

0

x

= -

b

2

a

±

b

2 -

4

ac

2

a

What does the QF say?

+

b

2 4

ac

2

a

-

b

2 4

ac

2

a

Is the distance you have to move

b

2 4

ac

2

a

from the center left and right to get to the roots

x

= 2

b a

is the line of symmetry for the curve

The Vertex Formula

when

f

(

x

) =

ax

2 +

bx

+

c

And you want to rewrite

f

(

x

) as

f

(

x

) =

a

(

x

-

h

) 2 +

k h

= -

b

2

a

and

k

=

f

(

h

)

Example

when

f

(

x

) = 2

x

2 + 3

x

+ 2 And you want to rewrite

f

(

x

) as

f

(

x

) =

a

(

x

-

h

) 2 +

k h

= 2( 3 2) = 3 4 and

k

=

f

= 2 ( ) + ( 3 4 ) 9 4 + = 2 ( ) 2 2 = 9 8 + 18 8 + 3 ( ) + + 16 8 = 25 8 2 So

f

(

x

) = 2 (

x

3 4 ) 2 + 25 8

Given the function R(x)=(2x+6)(x-12), find an equation for its axis of symmetry.

A) x = - 9 B) x = 9 C) x = 2 D) x = 6 E) 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 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) (x 1 ,y 1 )=(0,1)

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

and point

y

=

y

=

a

(

x

-

h

) 2

a

(

x

1) 2 + + 3

k

• • (h,k)=(1,3) (x 1 ,y 1 )=(0,1) But to be finished, I need to know a!

Use: My formula is true for every x,y including x 1 ,y 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) (x 1 ,y 1 )=(0,1) My formula is true for every x,y; not just x 1 ,y 1

y

=

a

(

x

-

h

) 2

y

=

a

(

x

1) 2

y

1 =

a

(

x

1 1) 2 1 = 2

a

= (0

a

1) 2 (1)

a

= 2 + 3

y

= 2(

x

1) 2 +

k

+ 3 + 3 + 3

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

A) y = (x+2) 2 B) y = x 2 +3 C) y = x 2 +1 D) y = x 2 E) 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.

Generic Formula:

y

=

a

(

x

-

h

) 2 +

k

Plug in vertex:

y

=

a

(

x

0) 2 + 2

y

=

ax

2 + 2 Find

a

: 3 =

a

1 2 1 =

a

+ 2 Plug in

a

:

y

= 1

x

2 + 2

y

=

x

2 + 2 E