Exploring Quadratic Graphs

Objective:To graph quadratic

functions.

ParabolaThe graph of any quadratic

function.

It is a kind of

curve.

Where are parabolas seen in the real world?

The Golden Gate Bridge

Satellite Dishes

Headlights

Trajectory

The Arctic Poppy

Why is the parabola important?

Suspension Bridges use a parabolic design to evenly distribute the weight of the entire bridge to the supporting columns.

Why is the parabola important?

The Satellite Dish uses a parabolic shape to ensure that no matter where on the dish surface the satellite signal strikes, it is always reflected to the receiver.

Why is the parabola important?

A car’s Headlights, and common flashlights, use parabolic mirrors to project the light from the bulb into a tight beam, directing the light straight out from the car, or flashlight.

Standard Form

y = ax2 + bx + c

Examples 2 2 25 7 3y xy xx y x

http://www.mathwarehouse.com/quadratic/parabola/interactive-parabola.php or http://www-groups.dcs.st-and.ac.uk/~history/Java/Parabola.html

y = ax2 + bx + c

Positive “a” values mean the parabola will open upwards and will have a minimum. point Minimum point is also called a vertex.

y = -ax2 + bx + c

Negative “a” values mean the parabola will open downwards and will have a maximum point Maximum point is also called a vertex.

Will the graph open up or down?

2 6 4x x

Steps1. Draw a table and insert vertex of (0,0).

2. Choose two numbers greater than the x coordinate and two numbers less.

3. Solve for Y Graph

 

Ex. X Y (X,Y)

22y x

(0,0)00

22x

12

12

22(1) 2 2 (1,2)22(2) 8 8 (2,8)

22( 1) 2

22( 2) 8

28

( 1,2)( 2,8)

(X,Y)-2, 8-1,20,01,22,8

Ex. X Y (X,Y)

22 +3 y x

(0,3)30

22 +3 x

12

12

22(1) 3 5 5 (1,5)22(2) 3 11 11 (2,11)

22( 1) 3 5

22( 2) 3 11 511

( 1,5)( 2,11)

(X,Y)-2, 11-1,50,31,5

2,11

Graph the quadratic function.

Check It Out! Example 2b

y = –3x2 + 1

x

–2

–1

0

1

2

y

1

–2

–11

–2

–11

Make a table of values.Choose values of x anduse them to find valuesof y.

Graph the points. Then connect the points with a smooth curve.

Ex. X Y (X,Y)

2( ) 2 f x x

(0,0)00

22 x

12

12

22(1) 2 2 (1, 2)22(2) 8 8 (2, 8)

22( 1) 2

22( 2) 8 28

( 1, 2) ( 2, 8)

22(0) 0

(X,Y)-2,-8-1,-20,01,-22,-8

Graph each quadratic function.

Check It Out! Example 2a

y = x2 + 2

x

–2

–1

0

1

2

y

2

3

3

6

6

Make a table of values.Choose values of x anduse them to find valuesof y.

Graph the points. Then connect the points with a smooth curve.

Additional Example 3A: Identifying the Direction of a Parabola

Tell whether the graph of the quadratic function opens upward or downward. Explain.

Since a > 0, the parabola opens upward.

Identify the value of a.

Write the function in the form y = ax2 + bx + c by solving for y.

Add to both sides.

Additional Example 3B: Identifying the Direction of a Parabola

Tell whether the graph of the quadratic function opens upward or downward. Explain.

y = 5x – 3x2

y = –3x2 + 5x

a = –3 Identify the value of a.

Since a < 0, the parabola opens downward.

Write the function in the form y = ax2 + bx + c.

Check It Out! Example 3a

Tell whether the graph of each quadratic function opens upward or downward. Explain.

f(x) = –4x2 – x + 1

f(x) = –4x2 – x + 1

Identify the value of a.a = –4

Since a < 0 the parabola opens downward.

Lesson Quiz: Part I

1. Without graphing, tell whether (3, 12) is on the

graph of y = 2x2 – 5.

2. Graph y = 1.5x2.

no

Lesson Quiz: Part II

Use the graph for Problems 3-5.

3. Identify the vertex.

4. Does the function have a

minimum or maximum? What is

it?

5. Find the domain and range.

D: all real numbers;R: y ≤ –4

maximum; –4

(5, –4)