Functions, Domain and Range

Date: ______

Warm Up

Use a table of values to graph y = -2x -3

Functions, Domain and Range

Function - a type of relation where for every x value there is only one y value

Domain - the set of values for which the independent variable is defined; x values

Range - the set of all values of the dependent variable; y values

(determined from the values in the domain)

Example 1: State the domain and range of each relation. Is the relation a function?

a) {(0,0), (2,2), (2,-2), (3,-2), (4,3)} b) c)

Vertical Line Test - if any vertical line drawn through the graph of a relation passes through two or more points, the relation is NOT a function

- if all vertical lines pass through at most one point, it is a function

Example 2: Is the relation a function? State the domain and range.

Practice Questions:

p 197 #1, 6, 7, 8, 10, 11, 14

Date: ______

Warm up

Consider y = 2x2 - 4x - 6

a) Complete the table of values

b) Use the table to graph the quadratic function

Features of a Parabola

Vertex:

Axis of Symmetry:

Max or Min Value:

x-intecepts:

y-intercept:

Domain:

Range:

Finite Differences

Classify each relation as linear, quadratic or neither.

x / y / x / y / X / y
-3
-2
-1
0
1
2
3 / 10
7
4
1
-2
-5
-8 / -2
-1
0
1
2
3
4 / 14
7
2
-1
-2
-1
2 / 0
1
2
3
4
5 / 0.5
1
2
4
8
16

Quadratic Regression

Use a graphing calculator to determine the equation of the curve of best fit for the data.

x / y
0
1
2
3
4
5 / 3
12
19
24
27
28

Graph the data

- STAT, 1:Edit, enter the lists

- STAT PLOT to turn it on

- ZOOM: 9 to see the graph

Make the curve of best fit

- STAT > CALC

- 5: QuadReg

- L1 , L2 , Y1 (To get Y1: VARS > Y-VARS, ENTER, ENTER)

- ENTER

Use the curve to determine the value of y if x = 7.

Find a value

- 2nd, TRACE (to calculate)

- 1:value

- type in any x-value

(You may have to adjust the window)

Date: ______

Warm up

Identify the domain and range for the following relations.

a) c)

Which of the above relations are parabolas (i.e. quadratic functions)???

Quadratic Functions

Graph y = x2

State the

vertex:

axis of symmetry:

x-intercept(s):

y-intercept:

domain:

range:

How would adding a number on the end change the graph?

What does y = x2 + k look like?

Graph y = x2 + 1 and y = x2 – 4

State the

vertex:

axis of symmetry:

x-intercept(s):

y-intercept:

domain:

range:

Practice: p. 213 #1abg, 2a, 6abe

Graphs for homework questions (if desired).

Date: ______

Warm up

Graph y = x2 and y = x2 - 9 on the same axes.

State the

vertex:

axis of symmetry:

x-intercept(s):

y-intercept:

domain:

range:

Quadratic Functions

Recall from yesterday:

What is the effect of "k" in y = x2 + k?

Today's Question:

How would putting a number in front of x2 change the graph?

What does y = ax2 look like?

Examples: Graph y = x2 y = 2x2 y = 0.5x2

State the

vertex:

axis of symmetry:

direction of opening:

max or min value:

x-intercept(s):

y-intercept:

domain:

range:

Examples: Graph y = x2 y = -2x2

State the

vertex:

axis of symmetry:

direction of opening:

max or min value:

x-intercept(s):

y-intercept:

domain:

range:

Practice: p. 213 #1efh, 2cd, 4a, 6cd

Date: ______

Warm up

Graph y = x2 and y = -3x2 on the same axes.

State the

vertex:

axis of symmetry:

direction of opening:

max or min value:

x-intercept(s):

y-intercept:

domain:

range:

Quadratic Functions

Recall from the last two lessons:

What is the effect of "k" in y = x2 + k? What is the effect of "a" in y = ax2?

Today's Question:

How would subtracting a number from x change the graph?

What does y = (x - h)2 look like?

Example 1: Graph each of the following functions by making a table of values with x values from -3 to 3.

y = x2 y = (x – 2)2 y = (x + 3)2

Example 2: Graph y = (x +5)2 and complete the information.

State the

vertex:

axis of symmetry:

direction of opening:

max or min value:

x-intercept(s):

y-intercept:

domain:

range:

Practice p. 222 #1abc, 2ab, 5a, 6a

Date: ______

Warm up

Graph on the same axes.

Clearly label each function.

y = x2

y = x2 – 3

y = -3x2

y = (x – 3)2

Transformations of Quadratic Functions

How does the graph of y=a(x-h)2+k compare with the graph of y=x2 ?

Example 1:

Graph the relation by applying transformations to y=x2

y = (x – 4)2 - 7

vertex:

axis of symmetry:

direction of opening:

max/min value:

domain:

range:

Example 2:

Graph the relation by applying transformations to y=x2

y = 2(x + 3)2

vertex:

axis of symmetry:

direction of opening:

max/min value:

domain:

range:

Example 3: Graph the relation using the vertex and graphing pattern.

vertex:

axis of symmetry:

direction of opening:

max/min value:

domain:

range:

Practice

p. 222: #1de, 2cd, 3, 5, 6

Date: ______

Warm up: Review Graphing

1) Graph y = 7(x – 5)2 + 6

2) Another quadratic is far to the left of, a lot wider than, and somewhat lower than the one in part 1. What could its equation be?

3) Graph the quadratic you came up with in part 2.

Writing the Equation of a Quadratic

Example 1:

A parabola has vertex (2,5), it is stretched by a factor of 2.5 and opens down.

1) What is its equation?

2) Graph it.

3) State the

Axis of symmetry:

Max or min:

Direction of opening:

Domain:

Range:

Example 2:

A parabola has vertex (-4, -6) and is congruent to y=x2. What is its equation?

Example 3:

Write an equation of a parabola that opens downward and is compressed by a factor of 3. It is also shifted right 2 units and up 6 units compared to y=x2.

Note: Congruent means the same shape (no stretch or compression)

p. 224: #9, 10 p. 255: #13 Quiz Review: p. 252: #1, 4, 5, 6

Date: ______

Warm up

Graph and state the features.

vertex:

A of S:

max or min value:

direction of opening:

domain:

range:

Writing Equations of Parabolas

Determine the equation of the parabola

p 226 #21, 22, 23

Date: ______

Warm up

Graph y = 2(x – 3)2 - 4

vertex:

axis of symmetry:

max or min:

domain:

range:

Standard and Vertex Form

vertex form standard form

Example 1: Covert to vertex form by "completing the square."

y = 2x2 + 12x + 11

Step 1: Factor "a" out of the first two terms

Step 2: Add and subtract the special number

Step 3: Pull the special number out of the brackets

Step 4: Factor the perfect square trinomial

State the vertex, axis of symmetry,

max or min value, domain and range.

Graph it.

Example 2: Covert to vertex form by "completing the square."

y = x2 + 8x + 7

Example 3: Covert to vertex form by "completing the square."

y = 5x - 0.5x2

p. 234: #2ade, 5ad, 8ade, 9adeg

Date: ______

Warm up

Convert the equation to vertex form and then graph it.

y = -2x2 - 16x + 1

Finding the Vertex of a Parabola

Given y = ax2 + bx + c

Example: Find the vertex of y = -2x2 - 16x + 1

Example: Find the maximum or minimum value and the x-value when it occurs.

y = -3x2 + 12x - 11

Why is the x-value of the vertex?

Practice: p234 #7, 9bcfh

Date: ______

Warm up

1) Convert to vertex form 2) Determine the equation of the parabola

y = - 4x2 - 28x + 1

Word Problems

There are 3 types of word problems

1) Find the maximum or minimum value

2) Given x (or t or d), find y (or height or profit)

3) Given y, find x

Practice p. 236: #14, 15, 17

Date: ______

Warm up

1) Write the equation in standard form 2) Write the equation in vertex form by

by expanding completing the square

y = -3(x + 1)2 – 7 y = -5x2 - 30x + 11

Optimization Problems

Example 1: Maximize Revenue

Alex runs a snowboard rental business that charges $12 per snowboard and averages 36 rentals per day. She discovers that for each $0.50 decrease in price, her business rents out two additional snowboards per day. At what price can Alex maximize her revenue?

Example 2: Maximize Area

The perimeter of a rectangle is 100m. What is the maximum possible area? What dimensions produce the maximum area?

Example 3: Maximize Area

You have 4 m of fencing to make a rectangular dog run on the side of your house. Three sides will be enclosed by the fence and the fourth side will be the wall of your house. What dimensions will produce the maximum possible area?

Practice Questions p. 18, 19, 20