5.1 Quadratic Function

11/30/12

Graph is a parabola

VocabularyQuadratic Function :

a function that is written in the standard form: y = ax2 + bx + c where a ≠ 0

Vertex: The highest or lowest point of the parabola.

Vertex

Vertex

the line that divides a parabola into mirror images and passes through the vertex.

Axis of symmetry:

Axis of symmetry

STEPS FOR GRAPHING y = ax2 + bx + c

Step 1: Find and plot the vertex. The x –coordinate of the vertex is

Substitute this value for x in the equation and evaluate to find the y -coordinate of the vertex.

Step 2: Draw the axis of symmetry. It is a vertical line through the vertex. Equation is x = # (x-coordinate of the vertex).

Step 3: Make an x-y chart. Choose 2 (or more) values for x to the right or left of the line of symmetry. Plug them in the equation and solve for y.

Step 4: Graph the points. Mirror the points on the other side of the line of symmetry. Draw a parabola through the points.

a

bx

2

Graph y = x2 ax2 + bx + c where a = 1, b= 0 and c = 0

Simplest quadratic equation

1. Find the vertex:

0)1(2

02

x

a

bx To find y, plug x

in the equation and solve for y.

0

02

y

yVertex: (0, 0)

2. Draw the line of symmetry: x=03. Make an x-y chart. Choose 2 (or more) values for x to the right or left of the line of symmetry. Plug them in the equation and solve for y.4. Graph the points. Mirror the points on the other side of the line of symmetry.Draw a U-shaped curve through the points.

x y

1 1

2 4

x= 1, y = 12

y= 1x = 2, y = 22

y = 4Plot (1,1) & (2, 4)

Graph y = - x2Think y=-1x2 where a = -1, b= 0 and c = 0

1. Find the vertex:

0)1(2

02

x

a

bx To find y, plug x

in the equation and solve for y.

0

02

y

yVertex: (0, 0)

2. Draw the line of symmetry:x=03. Make an x-y chart. Choose 2 (or more) values for x to the right or left of the line of symmetry. Plug them in the equation and solve for y.4. Graph the points. Mirror the points on the other side of the line of symmetry. Draw a U-shaped curve through the points.

x y

1 -1

2 -4

x= 1, y = -1•12

y= -1x = 2, y = -1•22

y = - 4Plot (1,-1) & (2, -4)

Graph of y = 1x2

Graph of y = -1x2

When a is positive, the parabola opens up.When a is negative, the parabola opens down.

Graph a Quadratic Function in Standard FormExample 2

Graph = x 2 6xy 5+– .

Graph a Quadratic Function in Standard FormExample 2

Graph = x 2 6xy 5+–

SOLUTION

The function is in standard form y ax 2 bx c where a 1, b 6, and c 5. Because a > 0, the parabola opens up.

= + += = – =

STEP 1 Find and plot the vertex.

.

= 3x2a

b– = –

2( )1=

6–

= x 2 6xy 5+– = 6 5+–( )23 ( )3 = 4–

The vertex is .( )3, 4–

Graph a Quadratic Function in Standard FormExample 2

= x 2 6xy 5+– = x 2 6xy 5+–

= 5+–( )20 6( )0 = 5 = 5+–( )21 6( )1 = 0

STEP 3 Plot two points to the left of the axis of symmetry. Evaluate the function for two x-values that are less than 3, such as 0 and 1.

STEP 2 Draw the line of symmetry. x=3

Graph a Quadratic Function in Standard FormExample 2

Plot the points and . Plot their mirror images by counting the distance to the axis of symmetry and then counting the same distancebeyond the axis of symmetry.

( )0, 5 ( )1, 0

STEP 4 Draw a parabola through the points.

Checkpoint Graph a Quadratic Function in Standard Form

Graph the function. Label the vertex and the axis of symmetry.

4. = x 2 6xy 2– –

Checkpoint Graph a Quadratic Function in Standard Form

Graph the function. Label the vertex and the axis of symmetry.

ANSWER

5. = x 2 2xy 1–– +

Checkpoint Graph a Quadratic Function Using a Table

Graph the function using a table of values.

ANSWER1. y = – 3x 2

Checkpoint Graph a Quadratic Function Using a Table

Graph the function using a table of values.

ANSWER2. y = – x 2 – 2

Homework

5.1 p.225 #20-25, 30-32, 33-37 (5 graphs)

mathematical expressions with 2 terms.

Ex. x + 2, 2x2 -5, x3 - 1

VocabularyBinomials:

Multiplying Binomials:

FOILFirst, Outside, Inside, Last

Ex. (x + 3)(x + 5) ( x + 3)( x + 5)

(x + 3)(x + 5) = x2 + 5x + 3x +15 = x2 + 8x + 15

I

O

L

First: x •x = x2

Outside: x • 5 = 5xInside: 3•x = 3xLast: 3•5 = 15

F

Example 3 Multiply Binomials

Find the product .( )3+2x ( )7x –

Write products of terms.

SOLUTION

( )7–( )3+2x ( )7x – = 2x( )x + 2x + 3x + 3( )7–

= 2x 2 14x + 3x– 21– Multiply.

= 2x 2 11x– 21– Combine like terms.

F LIO

Checkpoint Multiply Binomials

Find the product.

7. ( )4x – ( )6x + ANSWER x 2 + 2x 24–

8. ( )1x –( )13x + ANSWER 3x 2 2x 1––

9. –( )52x ( )2x – ANSWER 2x 2 9x 10– +

Example 4 Write a Quadratic Function in Standard Form

Write the function in standard form.

Write original function.

SOLUTION

( )22x –y = 2 5+

( )22x –y = 2 5+

( ) 2x –= 2 5+( ) 2x – Rewrite as .

( )22x –( )2x – ( )2x –

( ) 2xx 2 –= 2 4+2x– 5+ Multiply using FOIL.

( ) 4xx 2 –= 2 4+ 5+ Combine like terms.

8x2x 2 –= 8+ 5+ Use the distributive property.

8x2x 2 –= 13+ Combine like terms.

Checkpoint Write a Quadratic Function in Standard Form

Write the function in standard form.

10. ( )3x –( )1x +y = 2

11. ( )6x –y = 3( )4x –

12. ( )21x –y = 3– –

ANSWER 2x 2 4x 6––y =

ANSWER y = 3x 2 30x 72– +

ANSWER y = x 2 2x 4–+–