Unit 1Bquadratics

Day 4

Graphing a Quadratic Function

M2 Unit 1B: Day 4

Lesson 3.1B

EQ: How do we graph a quadratic function and identify all of its

characteristics?

Today, we are going to begin by reviewing what we have learned about graphing

quadratics so farLets recall how to find the following: Vertex AOS Maximum or Minimum Y-intercept

Find: a) vertex b) axis of symmetry c) state whether the vertex is a maximum or minimum. d) y - intercept

21. ( ) 2 4 2 f x x x a = -2, b = 4 c = -2

2bxa

4

2 2x

1x

2(1) 2 1 4 1 2f (1) 0f

4

a) Vertex:

b) Axis of symmetry:

c) Since a < 0 , the parabola opens down and has a: maximum

(1, 0)

x = 1

d) y-intercept: (0, -2)

a = 1, b = 0 c = 22bxa

0

2 1x 0x

2(0) 0 2f (0) 2f

22. ( ) 2f x x

5

Find: a) vertex b)axis of symmetry c) state whether the vertex is a maximum or minimum. d) y-intercept

a) Vertex:

b) Axis of symmetry:

c) Since a > 0 , the parabola opens up and has a: minimum

(0, 2)

x = 0

d) y-intercept: (0, 2)

a = -3 h = 1 k = 2

a) Vertex:

b) Axis of symmetry:

c) Since a < 0 , the parabola opens down and has a: maximum

23. ( ) 3( 1) 2f x x

6

Find: a) vertex b) axis of symmetry c) state whether the graph has a maximum or minimum.d) y - intercept

(1, 2)

x = 1

d) y-intercept: (0, -1)

Domain VS. Range Domain: (x – values) read domain from left

to right Range: (y-values) read range from bottom

to top

Last week we said that the DOMAIN of parabolas is all real numbers…unless the parabola looks like this

and has endpoint(s)

8

We say the domain is restricted, therefore it is no longer all real numbers

Find the domain of the graph below

Domain:

9

1,2

x

y

-1 < x < 2

Find the domain of the graph below

Domain:

10

2,2-2 < x < 2

x

y

Find the domain of the graph below

Domain:

11

2,

x

y

2 x

2( ) 2 4 1f x x x

Vertex:

Y-intercept:

Axis of symmetry:

One more point:

Domain:

Range:

Max or Min?

Graph the quadratic using the axis of symmetry and vertex.

1x

1,3

0,1

3, 5

maximum

All real numbers

y ≤ 3

12

Extrema: y = 3

Vertex:

Y-intercept:

Axis of symmetry:

One more point:

Domain:

Range:

Max or Min?

Graph the quadratic using the axis of symmetry and vertex.

1x

0,1

1,4

minimum

All real numbersy ≥ 0

13

2( ) ( 1)f x x (-1, 0)

Extrema: y = 0

Vertex:

Y-intercept:

Axis of symmetry:

One more point:

Domain:

Range:

Max or Min?

Graph the quadratic using the axis of symmetry and vertex.

3

2 1x

2bxa

32

x

3 5,2 4

0, 1

1,1

maximum

All real numbers

14

2( ) 3 1f x x x

54

y Extrema: y = 5/4

Stretch VS. Shrink Compare the following graphs and equations:

2y x

23y x21

3y x

What is the difference between these two

graphs when compared to the parent function?

*note: rubber bandVertical stretch Vertical shrink

Look at a couple more… 22 1 2y x 22 2 4

5y x

What we should notice and confirm at this point is that the value of “a” determines how wide or narrow the graph will be…

When a is greater than 1, we call that a vertical stretch When a is less than 1, we call that a vertical shrink

State whether the graph shows a vertical stretch or vertical shrink

Shrink…so |a| < 1

Stretch…so |a| > 1

Stretch…so |a| > 1

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Intervals of increase and decrease

To determine the intervals of increase and decrease, you must “read” the graph from left to right

What are these lines doing from left to

right?

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Let’s apply this idea to parabolas… To determine the intervals of increase and decrease,

you must “read” the graph from LEFT to RIGHT

What is this parabola doing

on the left side of the vertex?

Going downhill

What is this parabola doing

on the right side of the

vertex?Going uphill

Interval of decrease , 3 Interval of increase 3,

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One more…

What is this parabola doing on the left side of the vertex?

Going uphill

What is this parabola doing

on the right side of the

vertex?Going downhill

Interval of decrease 2, Interval of increase , 2

21

Graph each quadratic function and determine the interval of increase and decrease

3,4

Interval of decrease 3,

Interval of increase ,3

y = -2x² + 12x - 14

22

1,2

Interval of decrease , 1

Interval of increase 1,

Graph each quadratic function and determine the interval of increase and decrease

y = x² + 2x + 3

23

3 11,2 2

Interval of decrease 3,2

Interval of increase 3 ,2

23. ( ) 2 6 1f x x x 6

2 2x

2bxa

23 3 3( ) 2 6 12 2 2

f

3 11( )2 2

f

32

x

Graph each quadratic function and determine the interval of increase and decrease

24

Determine if the given interval is an interval of increase or decrease

4. 4 1x 5. 2 5x decrease increase

25

Determine if the given interval is an interval of increase or decrease

6. 5 9x 7. 5 3x - increase increase

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Assignment: Day 2 Handout