Graphing QuadraticsGraphing Quadraticsy

x

1

1

2

2

3

3

4

4

5

5

6

6

– 1

– 1

– 2

– 2

– 3

– 3

– 4

– 4

– 5

– 5

– 6

– 6

– 7

– 7

– 8

– 8

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

– 1

– 1

– 2

– 2

– 3

– 3

– 4

– 4

– 5

– 5

y

x

y

x

y

x

y

x

x

y

Parabolas

We can plot this using pointsx -3 -2 -1 0 1 2 3

y=x2

y

x

1

1

2

2

3

3

4

4

5

5

6

6

– 1

– 1

– 2

– 2

– 3

– 3

– 4

– 4

– 5

– 5

– 6

– 6

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

– 1

– 1

– 2

– 2

We can see the shape looks like:Starting at the vertexOut 1 up 12

Out 2 up 22

Out 3 up 32

Out 4 up 42

y

x

Any equation, where the highest power of x, is x2

when plotted on a graph, will form a parabolaThe simplest parabola is

2y x

9 4 1 0 1 4 9

y

x

1

1

2

2

3

3

4

4

5

5

6

6

– 1

– 1

– 2

– 2

– 3

– 3

– 4

– 4

– 5

– 5

– 6

– 6

1

1

2

2

– 1

– 1

– 2

– 2

– 3

– 3

– 4

– 4

– 5

– 5

– 6

– 6

– 7

– 7

– 8

– 8

– 9

– 9

– 10

– 10

2try y x

x -3 -2 -1 0 1 2 3

y = -x2 -9 -4 -1 -0 -1 -4 -9

The graph is the same shape but upside-down

The negative sign reflects the graph in the x axis

y

x

y

x

y

x

y

x

1

1

2

2

3

3

4

4

5

5

6

6

– 1

– 1

– 2

– 2

– 3

– 3

– 4

– 4

– 5

– 5

– 6

– 6

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

– 1

– 1

– 2

– 2

2y x We can see the larger the number in front of the x2

The steeper the graphRemember the slope of linear graphs got steeper if we increased the number by the x

These graphs are best plotted using points

y

x

2We can try putting other numbers infront of the x22y x23y x

25y x

y

x

y

x

y

x

1

1

2

2

3

3

4

4

5

5

6

6

– 1

– 1

– 2

– 2

– 3

– 3

– 4

– 4

– 5

– 5

– 6

– 6

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

– 1

– 1

– 2

– 2

2y x2We can try putting other numbers infront of the x

22y x23y x

x y=x2 y =2x2 Y=3x2

0 0

1 1

2 4

3 9

0

2

8

18

0

3

12

27

y

x

1

1

2

2

3

3

4

4

5

5

6

6

– 1

– 1

– 2

– 2

– 3

– 3

– 4

– 4

– 5

– 5

– 6

– 6

1

1

2

2

– 1

– 1

– 2

– 2

– 3

– 3

– 4

– 4

– 5

– 5

– 6

– 6

– 7

– 7

– 8

– 8

– 9

– 9

– 10

– 10

– 11

– 11

– 12

– 12

– 13

– 13

– 14

– 14

– 15

– 15

y

x

2y xThe graph is still reflected in the x axisWe can see the larger the number in front of the x2

The steeper the graphRemember the gradient of linear graphs got steeper if we increased the number by the x

y

x

y

x

y

x

2We can try putting other numbers infront of the x22y x23y x25y x

y

x

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

– 1

– 1

– 2

– 2

– 3

– 3

– 4

– 4

– 5

– 5

– 6

– 6

– 7

– 7

– 8

– 8

– 9

– 9

– 10

– 10

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

– 1

– 1

– 2

– 2

– 3

– 3

– 4

– 4

– 5

– 5

– 6

– 6

– 7

– 7

– 8

– 8

– 9

– 9

– 10

– 10

y

x

y

x

y

x

y

x

y

x

y

x

y

x

y

x

The smaller the number in front of the x2 the flatter the graph

The ones with negative sign are reflected in the x axis

2y x

21

2y x

21

4y x

2We can try putting a smaller number in front of the x

21

10y x

2y x

21

2y x

21

4y x

21

10y x

y

x

1

1

2

2

3

3

4

4

5

5

6

6

– 1

– 1

– 2

– 2

– 3

– 3

– 4

– 4

– 5

– 5

– 6

– 6

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

– 1

– 1

– 2

– 2

– 3

– 3

– 4

– 4

– 5

– 5

– 6

– 6

– 7

– 7

– 8

– 8

– 9

– 9

– 10

– 10

We can see:If we add a number to the x2 the parabola shifts upIf we subtract a number from the x2 the parabola shifts down

Note the shape is the basic 2y x

Remember the correct word for shift is translate

2y x2 3y x 2 1y x 2 5y x 2 7y x

y

x

y

x

y

x

y

x

y

x

We can also add numbers to the x before it is squared

2( 4)y x

We can see the shape is the basic x2 shapeIf we add a number in the brackets the graph is translated to the left by the magnitude of the numberIf we subtract a number in the brackets the graph is translated to the right by the magnitude of the number

intercept, 0x y 2( 3) 0x

( 3) 0x

3x

?Why

2y x

2( 3)y x

2( 5)y x

2( 1)y x

y

x

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

– 1

– 1

– 2

– 2

– 3

– 3

– 4

– 4

– 5

– 5

– 6

– 6

– 7

– 7

– 8

– 8

– 9

– 9

– 10

– 10

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

11

12

12

13

13

14

14

15

15

– 1

– 1

– 2

– 2

y

x

y

x

y

x

y

x

y

x

If we add a number in the brackets and at the end the graph is translated vertically and horizontally

y

x

1

1

2

2

3

3

4

4

5

5

6

6

– 1

– 1

– 2

– 2

– 3

– 3

– 4

– 4

– 5

– 5

– 6

– 6

– 7

– 7

– 8

– 8

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

– 1

– 1

– 2

– 2

– 3

– 3

– 4

– 4

– 5

– 5

2( 4) 1y x

2( 3) 4y x

2( 2) 5y x

2( 1) 2y x

y

x

y

x

y

x

y

x

y

x

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

– 1

– 1

– 2

– 2

– 3

– 3

– 4

– 4

– 5

– 5

– 6

– 6

1

1

2

2

3

3

4

4

5

5

– 1

– 1

– 2

– 2

– 3

– 3

– 4

– 4

– 5

– 5

– 6

– 6

– 7

– 7

– 8

– 8

– 9

– 9

– 10

– 10

We can also translate the reflected graphs

2 2In general ( ) translates the basic graph

and vertically to

k

horizontally to

Verte ( )

x h,

y

k

x

h

k y xh

2 4y x

2( 5)y x

2( 2) 3y x

y

x

y

x

y

x

To draw Parabolas:

• Decide how it has been translated (find vertex)

• Decide if reflected in x axis (a<0)

• Stretched or compressed?

• Draw appropriate y = ax2 type from new vertex.

This gives the x coordinate, sub it back into

the original equation to find the y coordinate

( 3 1)( 3 5)

( 2)(2)

4 Vertex ( 3, 4)

y

y

x

1

1

– 1

– 1

– 2

– 2

– 3

– 3

– 4

– 4

– 5

– 5

– 6

– 6

– 7

– 7

– 8

– 8

1

1

2

2

3

3

4

4

5

5

– 1

– 1

– 2

– 2

– 3

– 3

– 4

– 4

– 5

– 5

2) Cuts axis 0

(0 1)(0 5)

1 5

5

y x

y

y

y

Parabolas in factorised form

1) Cuts axis 0

( 1)( 5) 0

1 0, 5 0

1, 5

x y

x x

x x

x x

y

x

1

1

– 1

– 1

– 2

– 2

– 3

– 3

– 4

– 4

– 5

– 5

– 6

– 6

– 7

– 7

– 8

– 8

1

1

2

2

3

3

4

4

5

5

– 1

– 1

– 2

– 2

– 3

– 3

– 4

– 4

– 5

– 5

3) To find the vertex.

This must be half way between the x intercepts

-5 -1 63

2 2x x x

Features

1) intercept 1, 5

2) int ercept 5

3) ( 3, 4)

4) Line of symmetry 3

x x x

y y

Min

x

Remember calculator will find max or min

( 1)( 5)y x x

y

x

1

1

– 1

– 1

– 2

– 2

– 3

– 3

– 4

– 4

– 5

– 5

– 6

– 6

– 7

– 7

– 8

– 8

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

– 1

– 1

Features:

1) -intercept, 3

2) intercept, 9

3) ( 3,0)

4) Line of symmetry, 3

x x

y y

Min

x

2Eg

2

1) Cuts axis 0

( 3) 0

3 0

3

x y

x

x

x

2( 3)y x

2

2) Cuts axis 0

(0 3)

3

9

y x

y

y

y

3) Vertex is at( 3,0)

as there is only one intercept

y

x

1

1

– 1

– 1

– 2

– 2

– 3

– 3

– 4

– 4

– 5

– 5

– 6

– 6

– 7

– 7

– 8

– 8

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

– 1

– 1

E.g. 3

Features:

1)x-intercept, 0, 3

2) intercept, 0

3) ( 1.5,2.25)

4)Line of symmetry 1.5

x x

y y

Max

x

y

x

1

1

2

2

3

3

– 1

– 1

– 2

– 2

– 3

– 3

– 4

– 4

– 5

– 5

– 6

– 6

1

1

2

2

3

3

– 1

– 1

– 2

– 2

– 3

– 3

– 4

– 4

– 5

– 5

( 3)y x x

1) Cuts axis 0

( 3) 0

0, 3 0,

0, 3

x y

x x

x x

x x

2) Cuts y axis 0

(0)((0) 3)

0

x

y

y

3) To find the vertex

0 -3

23

2

x

x

sub it back into the original

equation to find the y coordinate

( 1.5)( 1.5 3)

(1.5)(1.5)

2.25

Vertex ( 1.5,2.25)

y

y

x

1

1

2

2

3

3

– 1

– 1

– 2

– 2

– 3

– 3

– 4

– 4

– 5

– 5

– 6

– 6

1

1

2

2

3

3

– 1

– 1

– 2

– 2

– 3

– 3

– 4

– 4

– 5

– 5

End Lesson Graphing End Lesson Graphing QuadraticsQuadratics

Homework: Page 47 #1-4, 9, 10.Homework: Page 47 #1-4, 9, 10.