Unit 1 – Chapters 1 and 4. Unit 1 Section 1.6-1.7 Section 4.1 Section 4.2 Section 4.3 Section 4.4 Section 4.5 Section 4.6-4.7 Review Ch. 4

Unit 1 – Chapters 1 and 4

Unit 1

• Section 1.6-1.7

• Section 4.1

• Section 4.2

• Section 4.3

• Section 4.4

• Section 4.5

• Section 4.6-4.7

• Review Ch. 4

Warm-Up – 1.6-1.7

3. Gabe spent $4 more than twice as much as Casey at a store. If Casey spent $6, how much did Gabe spend?

Lesson 1.6, For use with pages 35-41

ANSWER $16


1. Make a table for y = 2x + 3 with x-values of 0, 3, 6, and 9.

2. Write a rule for the function.

ANSWER

ANSWER y = 3x + 1

x 0 3 6 9

y 3 9 15 21

Input, x 0 2 4 9

Output, y 1 7 13 28

Vocabulary – 1.6-1.7• Domain

• Set of INPUTS to a function

• Sometimes these are considered the X variables

• AKA the “independent” variable

• Range

• Set of OUTPUTS of a function

• Sometimes these are considered the Y variables

• AKA the “dependent” variable

• Quadrants

• 4 regions of a coordinate plane

• Function

• A numerical relationship where ONE input has EXACTLY ONE output

• Relation

• A set of inputs and corresponding outputs

Notes – 1.6-1.7 – Functions – Intro.•To be a Function

1. Each INPUT must go to EXACTLY ONE output! 2. Graph must pass the Vertical Line Test

•Is a function a relation?• Yes

•Is a relation a function?• Sometimes!!

• Only if it passes the two tests above! •There are several ways to sketch graphs of equations, but the most common is THIS

1. GET Y BY ITSELF!!!2. Build a table with at least 3 values3. Sketch the graph

Notes – 1.6-1.7 – Functions – Intro.•4 Different Ways to view Functions

1. Verbal Rule – In English – Rarely used.2. Graphs3. Equations or Rules4. Tables

Examples 1.6-1.7

Identify the domain and range of a functionEXAMPLE 1

The input-output table shows the cost of various amounts of regular unleaded gas from the same pump. Identify the domain and range of the function.

ANSWER

19.99, 23.99, 25.99, and 33.98.

The domain is the set of inputs: 10, 12, 13, and 17. The range is the set of outputs:

10Input gallons

Output dollars

12 13 17

19.99 23.99 25.99 33.98

SOLUTION

GUIDED PRACTICE for Example 1

1. Identify the domain and range of the function.

Input 0 1 2 4Output 5 2 2 1

The domain is the set of inputs: 0, 1, 2, and 4The range is the set of outputs: 1, 2, and 5

Tell whether the pairing is a function.

Identify a functionEXAMPLE 2

a.

The pairing is not a function because the input 0 is paired with both 2 and 3.

b.

Identify a functionEXAMPLE 2

OutputInput

21

0 0

4 8

6 12

The pairing is a function because each input is pairedwith exactly one output.

SOLUTION



1221Output

12963Input2.

369

12

1221

The pairing is a function because each input is pairedwith exactly one output.

SOLUTION



3210Output

7422Input3.

The pairing is not a function because each input is notpaired with exactly one output. IT IS A RELATION, THOUGH!!

2

47

0123

The range of the function is 0, 4, 10, 14, and 16.

Make a table for a functionEXAMPLE 3

of the function. function, then identify the rangeMake a table for the

2x is 0, 2, 5, 7, and 8. The domain of the function y =

x

y

2 5 70 8

= 2x 2 2 = 4 2 5 =10 2 7 =14 2 8 =162 0 = 0

SOLUTION

Write a function ruleEXAMPLE 4

Write a rule for the function. Input

Output 2

1

6 123

4

8

60 10

SOLUTION

and let y or dependent variable. Notice that eachbe the output,

Let x be the input, or independent variable,

output is 2 more than the corresponding input. So, a rule for the function is y = x + 2.

SOLUTION

GUIDED PRACTICE for Examples 3,4 and 5

4. Make a table for the function y – x = 5 with domain 10, 12, 15, 18, and 29. Then identify the range of the function. HINT: GET Y BY ITSELF FIRST!!

x 10 12 15 18 29

y- x = 5 10 – 5 =5 12 – 5 =7 15 – 5 =10 18 – 5 =13 18 – 29 =24

The range of the function is 5,7,10,13 and 24.

GUIDED PRACTICE for Examples 3,4 and 5

5. Write a rule for the function. Identify the domain and the range.

SOLUTION

Let x be the input ,or independent variable and let y be the output, or dependent variable. Notice that each output is 8 times more than corresponding input .So as a rule of function y = 8x; domain 1, 2, 3 and 4;range 8, 16, 24 and 32.

Pay (dollars)

1 2 43

8 16 3224

Time (hours)

Write a function rule for a real-world situationEXAMPLE 5

Concert Tickets

You are buying concert tickets that cost $15 each. You can buy up to 6 tickets. Write the amount (in dollars) you spend as a function of the number of tickets you buy. Identify the independent and dependent variables. Then identify the domain and the range of the function.


SOLUTION

So, the function rule is A = 15n. The amount spent depends on the number of tickets bought, so n is the independent variable and A is the dependent variable.

Write a verbal model. Then write a function rule. Let n represent the number of tickets purchased and A represent the amount spent (in dollars).

A 15 n=

Amount spent

(dollars)

Cost per ticket

(dollars/ticket)

Tickets purchased

(tickets)= •


Because you can buy up to 6 tickets, the domain of the function is 0, 1, 2, 3, 4, 5, and 6. Make a table to identify the range.

Amount (dollars), A

0 1 32 654

0 15 4530 7560 90

Number of tickets, n

The range of the function is 0, 15, 30, 45, 60, 75, and 90.

Graph a function EXAMPLE 1

STEP 1 Make an input-output table.

SOLUTION

Graph the function y = x with domain 0, 2, 4, 6, and 8. 12

x 0 2 4 6 8

y 0 1 2 3 4


STEP 2

Plot a point for each ordered pair (x, y).

WHY IS THE GRAPH A SCATTER PLOT AND NOT A LINE??


1. Graph the function y = 2x - 1 with domain 1, 2, 3, 4, and 5.

STEP 1 Make an input-output table.

SOLUTION

x 1 2 3 4 5

y 1 3 5 7 9

STEP 2

Plot a point for each ordered pair (x, y).


The table shows the average scores on the mathematics section of the Scholastic Aptitude Test (SAT) in the United States from 1997 to 2003 as a function of the time t in years since 1997. In the table, 0 corresponds to the year 1997, 1 corresponds to 1998, and so on. Graph the function.

519516514514511512511Average score, s

6543210Years since 1997, t

Sat Scores

STEP 2

Plot the points

EXAMPLE 2 Graph a function

Does this graph pass the straight line test? Is it a function??

Use the Vertical Line Test to determine if the graphs are Functions.


Function Function NOT a Function

Warm-Up – 4.1

ANSWER y = – x + 12

52

9. 6x + 4y = 16

Prerequisite Skills SKILLS CHECK

Write the equation so that y is a function of x.

y = – x + 432

ANSWER

10. x + 2y = 5

ANSWER y = -12x + 1211. –12x - y = –12

3. y = x + 6; when x = 0, 2, 4, 6, and 8

Prerequisite Skills SKILLS CHECK

Graph the function on a coordinate plane and give the Input/ Output table.

ANSWERInput Output

0 6

2 8

4 10

6 12

8 14

Use the Vertical Line Test to determine if the graphs are Functions.


Function Function NOT a Function

Vocabulary – 4.1• Coordinate Plane

• Two dimensional plane used to graph ordered pairs of numbers (x,y)

• X-coordinate

• The HORIZONTAL component of an ordered pair

• Sometimes called the “abscissa”

• Y-coordinate

• The VERTICAL component of an ordered pair

• Sometimes called the “ordinate”

Notes – 4.1 – Plot Pts. and Graphs• To plot points, move along the X axis first, and then the Y axis

• You have to run before you jump (or drop!).• Domain = Inputs• Range = Outputs•Usually in Tableformat•Quadrants labeledWith Roman Numerals

Examples 4.1


1. Use the coordinate plane in Example 1 to give the coordinates of points C, D, and E.

C = (0,2)

D = (3,1)

E = (-2,-3)

C.

SOLUTION


y-coordinate of any point on the x-axis is 0

2. What is the y-coordinate of any point on the x-axis?

SOLUTION

Make a table by substituting the domain values into the function.

STEP 1

SOLUTION

Graph the function y = 2x – 1 with domain – 2, – 1, 0, 1, and 2. Then identify the range of the function.


STEP 2

List the ordered pairs: (– 2, – 5),(– 1, – 3), (0, – 1), (1, 1), (2, 3).Then graph the function.


Identify the range. The range consists of the y-values from the table: – 5, – 3, – 1, 1, and 3.

GUIDED PRACTICE for Examples 2 and 3

7. Graph the function y = – x + 2 with domain – 6, –3, 0, 3, and 6. Then identify the range of the function.

13

STEP 1

Make a table by substituting the domain values into the function.


y = – (6) + 2 = 06

y = – (3) + 2 = 13

y = – (0) + 2 = 20

y = – (– 3) + 2 = 3– 3

y = – (– 6) + 2 = 4– 6

y = – x + 2x 13

13

13131313


STEP 2

List the ordered pairs: (– 6, 4),(– 3, 3), (0, 2), (3, 1), (6,0). Then graph the function.

STEP 3

Identify the range. The range consists of the y-values from the table: 0, 1, 2, 3 and 4.

Graph a function represented by a table EXAMPLE 4

VOTING

In 1920 the ratification of the 19th amendment to the United States Constitution gave women the right to vote. The table shows the number (to the nearest million) of votes cast in presidential elections both before and since women were able to vote.

Graph a function represented by a tableEXAMPLE 4

Years before or

since 1920

– 12 – 8 – 4 0 4 8 12

Votes (millions) 15 15 19 27 29 37 40

Explain how you know that the table represents a function.

a.

Graph the function represented by the table.b.

Describe any trend in the number of votes cast.c.

Graph a function represented by a table

EXAMPLE 4

SOLUTION

The table represents a function because each input has exactly one output.

a.

To graph the function, let x be the number of years before or since 1920. Let y be the number of votes cast (in millions).

b.

The graph of the function is shown.

Graph a function represented by a table

EXAMPLE 4

SOLUTION

In the three election years before 1920,the number of votes cast was less than 20 million. In 1920, the number of votes cast was greater than 20 million. The number of votes cast continued to increase in the three election years since 1920.

c.

Warm-Up – 4.2

You may use a calculator on every assignment from this point forward unless otherwise told not to!


1. Graph y = –x – 2 with domain –2, –1, 0, 1, and 2.

ANSWER


2. 3x + 4y = 16

Rewrite the equation so y is a function of x.

ANSWER 34

y = – x + 4

3. –6x – 2y = –12

ANSWER y = –3x + 6

Vocabulary – 4.2• Linear Equation

– The graph of the solutions to the function form a STRAIGHT LINE!

Notes – 4.2 – Graph Linear Equations• Standard Form of a Linear Equation looks like this:

•Ax + By = C, where A, B, and C are real numbers and A and B are not both = 0

• There are several ways to sketch graphs of linear equations, but the most common is THIS

1.GET Y BY ITSELF!!!2.Build a table with at least 3 values (negative #, positive #, and zero)3.Sketch the graph

•The graphs of y = constant and x = constant are special cases of linear equations.

•We’ll check those out in a minute!

Examples 4.2

Substitute 3 for x and 4 for y.

Simplify.

Write original equation.

Check whether each ordered pair is a solution of the equation.

SOLUTION

Which ordered pair is a solution of 3x – y = 7?

EXAMPLE 1 Standardized Test Practice

(3, 4)A (1, –4)B (5, –3)C (–1, –2)D

Test (3, 4):

3(3) – 4 =? 7

3x – y = 7

5 = 7

Simplify.


Standardized Test PracticeEXAMPLE 1

Test (1, – 4): 3x – y = 7

3(1) – (– 4) = ?

7 Substitute 1 for x and – 4 for y.

So, (3, 4) is not a solution, but (1, – 4) is a solution of 3x – y = 7.

ANSWER

The correct answer is B. A B DC

7 = 7

Solve the equation for y.

SOLUTION

EXAMPLE 2 Graph an equation

Graph the equation – 2x + y = – 3.

– 2x + y = – 3y = 2x – 3

STEP 1

Make a table by choosing a few values for x and finding the values of y.

x – 2 – 1 0 1 2

y – 7 – 5 – 3 – 1 1

STEP 2

Graph an equation EXAMPLE 2

Plot the points. Notice that the points appear to lie on a line.

STEP 3

Graph (a) y = 2 and (b) x = – 1.

Graph y = b and x = a

EXAMPLE 3

y = 2

x = -1


Graph the equation

2. y + 3x = – 2

Solve the equation for y.

SOLUTION

y + 3x = – 2y = – 3x – 2

STEP 1


Make a table by choosing a few values for x and finding the values of y.

x – 2 – 1 0 1 2

y 4 1 – 2 – 5 – 8

Plot the points. Notice that the points appear to lie on a line.

Connect the points by drawing a line through them. Use arrows to indicate that the graph goes on without end.

STEP 3

STEP 4

STEP 2


3. y = 2.5

SOLUTION

For every value of x, the value of y is 2.5. The graph of the equation y = 2.5 is a horizontal line 2.5 units above the x-axis.

4. x = – 4

SOLUTION

For every value of y, the value of x is – 4. The graph of the equation x = – 4 is a vertical line 4 units to the left of the y-axis.

SOLUTION

STEP 1Make a table.

x 0 2 4 6 8

y 4 3 2 1 0

EXAMPLE 4 Graph a linear function

12Graph the function y = – x + 4 with domain x > 0.

Then identify the range of the function.–

STEP 2

STEP 3

Connect the points with a ray because the domain is restricted.

STEP 4

Identify the range. From the graph, you can see that all points have a y-coordinate of 4 or less, so the range of the function is y ≤ 4.

EXAMPLE 4 Graph a linear function

Plot the points.


5. Graph the function y = – 3x + 1 with domain x < 0. Then identify the range of the function.

–

SOLUTION

STEP 1Make a table.

x 0 – 1 – 2 – 3 – 4

y 1 4 7 10 13


STEP 2

STEP 3Connect the points with a ray because the domain is restricted.

STEP 4

Identify the range. From the graph, you can see that all points have a y-coordinate of 1 or more, so the range of the function is y 1.–

Plot the points.

Warm-Up – 4.3

Daily Homework Quiz For use after Lesson 4.2

1. Graph y + 2x = 4

ANSWER


2. The distance in miles an elephant walks in t hours is given by d = 5t. The elephant walks for 2.5 hours. Graph the function and identify its domain and range.

domain: 0 < t < 2.5 range: 0 < d < 12.5ANSWER –

––

–

Warm-up

1) Graph 5 points on the X-axis and label them.What conclusion can you make about every point on the X-axis?

2) Graph 5 points on the Y-axis and label them.What conclusion can you make about every point on the Y-axis?

Vocabulary – 4.3• X-intercept

• Where a graph crosses the X axis

• Y-intercept

• Where a graph crosses the Y axis

Notes – 4.3 – Graph using Intercepts.•The primary reason to use the Standard Form of a linear equation is b/c it does make finding the x and y intercepts VERY easy! •To find the X-intercept of a function

• Set Y=0 and solve for X• To find the Y-intercept of a function

• Set X=0 and solve for Y•Since you only need two points to make a line

•Graph the X and Y intercepts and connect them!

Examples 4.3

Substitute 0 for y.


To find the x-intercept, substitute 0 for y and solve for x.

SOLUTION

Find the x-intercept and the y-intercept of the graph of 2x + 7y = 28.

Find the intercepts of the graph of an equationEXAMPLE 1

Solve for x.

2x + 7(0) = 28

x = = 14282

2x + 7y = 28

2(0) + 7y = 28

Find the intercepts of the graph of an equation EXAMPLE 1

To find the y-intercept, substitute 0 for x and solve for y.


Substitute 0 for x.

Solve for y.

ANSWER

The x-intercept is (14,0). The y-intercept is (0,4).

2x +7y = 28

y = 287 = 4

Substitute 0 for y.


To find the x-intercept, substitute 0 for y and solve for x.

SOLUTION

Find the x-intercept and the y-intercept of the graph of the equation.

Solve for x.

3x + 2(0) = 6

x = 2

3x + 2y = 6


1. 3x + 2y = 6

3(0) + 2y = 6

Find the intercepts of the graph of an equation EXAMPLE 1

To find the y-intercept, substitute 0 for x and solve for y.


Substitute 0 for x.

Solve for y.

ANSWER

The x-intercept is (2,0). The y-intercept is (0,3).

3x +2y = 6

y =3


SOLUTION

STEP 1

Use intercepts to graph an equationEXAMPLE 2

Graph the equation x + 2y = 4.

x + 2y = 4

x = x-intercept4

Find the intercepts.

x + 2(0) = 4 0 + 2y = 4

y = y-intercept2

x + 2y = 4

X intercept = (4,0) Y intercept = (0,2)

Use intercepts to graph an equationEXAMPLE 2

STEP 2

Plot points. The x-intercept is 4, so plot the point (4, 0). The y- intercept is 2, so plot the point (0, 2). Draw a line through the points.

EVENT PLANNING

Solve a multi-step problemEXAMPLE 4

You are helping to plan an awards banquet for your school, and you need to rent tables to seat 180 people. Tables come in two sizes. Small tables seat 4 people, and large tables seat 6 people. This situation can be modeled by the equation.

4x + 6y = 180

where x is the number of small tables and y is the number of large tables.

• Find the intercepts of the graph of the equation.

SOLUTION

STEP 1


• Give four possibilities for the number of eachsize table you could rent.

• Graph the equation.

Find the intercepts.

4x + 6(0) = 180

x = x-intercept45

4x + 6y = 180

4(0) + 6y = 180

y = y-intercept30

4x + 6y = 180


Since x and y both represent numbers of tables, neither x nor y can be negative. So, instead of drawing a line, draw the part of the line that is in Quadrant I.

STEP 2

Graph the equation.

The x-intercept is 45, so plot the point (45, 0).The y-intercept is 30, so plot the point (0, 30).


STEP 3

Find the number of tables. For this problem, only whole-number values of x and y make sense. You can see that the line passes through the points (0, 30),(15,20),(30, 10), and (45, 0).


So, four possible combinations of tables that will seat 180 people are: 0 small and 30 large, 15 small and 20 large, 30 small and 10 large,and 45 small and 0 large.

Warm-Up – 4.4

1) Do pages 10-13 from the “Classified” ads packet as a group.2) You have ~15 minutes to work on this.

Vocabulary – 4.4• Rate of Change

• Ratio of How much something changed over how long did it take to change.

• Slope– The STEEPNESS of a line

– Same thing as UNIT RATE!!!!

– Same thing as Rate of Change!!!!

– HOW FAST SOMETHING IS CHANGING!!!

• Rise– Vertical or UP/DOWN change

– Change in Y’s

• Run– Horizontal or LEFT/RIGHT change

– Change in X’s

Notes – 4.4–Slope and Rate of ChangeNOTES

Slope = RISE

RUN

4 Kinds of Slope1. Positive = Slants UP

2. Negative = Slants DOWN

3. Zero = Horizontal line

4. No Slope = Vertical line

Finding slope with a Graph1. Draw a right triangle connecting the points

2. Calculate RISE and RUN

3. Use Slope = RISE/RUN

Notes – ContinuedNOTES - CONTINUED

Finding slope with a Table Slope = Change in Y = How much change = 2ND – 1ST

Change in X = How long did it take = 2ND – 1ST SAME as UNIT RATE and RATE OF CHANGE!!!

Pay attention to positives/negatives!!

Finding slope Using Coordinates Variable for slope is usually m.

1. Slope = m = Y2 – Y1 = RISE

X2 – X1 = RUN

2. Plug in what you know and solve for what you don’t!

Examples 4.4

EXAMPLE 2 Find a negative slope

Find the slope of the line shown.

m =y2 – y1

x2 – x1

Let (x1, y1) = (3, 5) and (x2, y2) = (6, –1).

–1 – 56 – 3

=

– 63= = –2

Write formula for slope.

Substitute.

Simplify.

EXAMPLE 3 Find the slope of a horizontal line


Let (x1, y1) = (– 2, 4) and (x2, y2) = (4, 4).

m =y2 – y1

x2 – x1

4 – 44 – (– 2)

=

06= = 0


Substitute.

Simplify.

EXAMPLE 4 Find the slope of a vertical line


Let (x1, y1) = (3, 5) and (x2, y2) = (3, 1).

m =y2 – y1

x2 – x1Write formula for slope.

1 – 53 – 3

= Substitute.

Division by zero is undefined.

ANSWER

Because division by zero is undefined, the slope of a vertical line is undefined.

– 40=

EXAMPLE 2 Write an equation from a graphGUIDED PRACTICE for Examples 2, 3 and 4

Find the slope of the line that passes through the points.

4. (5, 2) and (5, – 2)

m =y2 – y1

x2 – x1

Let (x1, y1) = (5, 2) and (x2, y2) = (5, –2).

–2 – 25 – 5

=

– 40=


Substitute.

Division by zero is undefined.

ANSWER

The slope is undefined.

SOLUTION

EXAMPLE 5 Find a rate of change

INTERNET CAFEThe table shows the cost of using a computer at an Internet cafe for a given amount of time. Find the rate of change in cost with respect to time.

Time(hours) 2 4 6

Cost (dollars) 7 14 21


Rate of change =change in costchange in time

14 –74 – 2

=72= 3.5=

ANSWER

The rate of change in cost is $3.50 per hour.

SOLUTION

Time(minute) 30 60 90

Distance (miles)

1.5 3 4.5


SOLUTION

The table shows the distance a person walks for exercise. Find the rate of change in distance with respect to time.

7. EXERCISE


Rate of change =change in distance

change in time

3 – 1.560 – 30

= = 0.05

ANSWER

The rate of change in cost is $0.05 mile/minute.


EXAMPLE 6 Use a graph to find and compare rates of change

COMMUNITY THEATER

A community theater performed a play each Saturday evening for 10 consecutive weeks. The graph shows the attendance for the performances in weeks 1, 4, 6, and 10. Describe the rates of change in attendance with respect to time.

SOLUTION

EXAMPLE 6 Use a graph to find and compare rates of change

Find the rates of change using the slope formula.

Weeks 1–4: 232 – 1244 – 1 =

1083

= 36 people per week

Weeks 4–6: 204 – 2326 – 4 =

– 28 2

= – 14 people per week

Weeks 6–10: 72 – 20410 – 6 =

– 1324 = – 33 people per week

ANSWERAttendance increased during the early weeks of performing the play. Then attendance decreased, slowly at first, then more rapidly.

EXAMPLE 7 Interpret a graph

COMMUTING TO SCHOOL

A student commutes from home to school by walking and by riding a bus. Describe the student’s commute in words.

EXAMPLE 7 Interpret a graph

The first segment of the graph is not very steep, so the student is not traveling very far with respect to time. The student must be walking. The second segment has a zero slope, so the student must not be moving. He or she is waiting for the bus. The last segment is steep, so the student is traveling far with respect to time. The student must be riding the bus.

SOLUTION

Warm-Up – 4.5


1. Rewrite 6x + 2y = 8 so y is a function of x.

2. Find the slope of the line that passes through (–5, 6) and (0, 8).

ANSWER

ANSWER

y = –3x + 4

52

3. Find the intercepts of the graph of the function. (USE THE COORDINATES OF THE POINTS!)

200x + 100y = – 600.

ANSWER


y-intercept: (0,– 6), x-intercept: (- 3,0)

4a Find the slope of -2x + y = 1.

4b Find the y-intercept of this equation as well.HINT 1: GET Y BY ITSELF.HINT 2: BUILD A TABLE!

ANSWER Slope = 2 and y-intercept is 1.

Do these numbers look familiar???????

Vocabulary – 4.5

• Parallel Lines

– Lines that never intersect

– Lines that have the SAME SLOPE!

• slope-intercept form

– Linear equation where y = mx + b

– m = slope of the line

– b = y-intercept

Notes – 4.5 – Slope-Intercept FormSlope Intercept Form of an Equation –

y = mx + b

To use the slope intercept form

1. Solve the equation so that Y is by itself

2. The coefficient of X is the slope.

3. The constant number is the Y intercept.

To graph a function using the slope intercept form

1. Graph the Y intercept

2. Use the slope = rise/run to find the next point

3. Graph the second point and connect the two points

If two lines are parallel, their slopes are ?????

Identical!BrainPop: Slope and Intercept

http://www.glencoe.com/sec/math/brainpops/00112044/00112044.html



Examples 4.5

Examples 4.5 – Using the Graphing Calculator

1. Open up the graphing application.

2. Graph the equation f1(x) = 2x+1 and press the Graph button. What happens?

3. Graph f2(x) = 2x + 3 What happens?

4. Press the Table button (you have to find it first!). What happens?

5. On the menu, press the Window/Zoom option button and choose 5- Zoom standard. What happens?

6. Graph the function f3(x)= x^2. Look familiar?

7. Clear all the functions by pressing Actions Delete All

SOLUTION

EXAMPLE 1 Identify slope and y-intercept

Identify the slope and y-intercept of the line with the given equation.

y = 3x + 41. 3x + y = 22.

The equation is in the form y = m x + b.So, the slopeof the line is 3, and the y-intercept is 4.

a.

b. Rewrite the equation in slope-intercept form by solving for y.


3x + y = 2 Write original equation.

y= – 3x + 2 Subtract 3x from each side.

ANSWER

The line has a slope of – 3 and a y-intercept of 2.

SOLUTION



The equation is in the form y = mx + b. So, the slopeof the line is 5, and the y-intercept is –3.

y = 5x – 31.




3x – 3y = 122.



ANSWER

The line has a slope of –1 and a y-intercept of –4.


SOLUTION

Rewrite the equation in slope-intercept form by solving for y.

3x – 3y = 12 Write original equation.

Divide 3 by equation.

3x – 12 = 3y Rewrite original equation.

y 3x + 12=3

y x – 4= Simplify.



x + 4y = 63.


EXAMPLE 1 Identify slope and y-interceptGUIDED PRACTICE for Example 1

ANSWER

The line has a slope of and a y-intercept of 1 . 41–

21

SOLUTION

Rewrite the equation in slope-intercept form by solving for y.

x + 4y = 6 Write original equation.

Divide 3 by equation.

4y= – x + 6 Rewrite original equation.

Simplify.– x

4 + 6

4=

– x + 6=4

y

SOLUTION

EXAMPLE 2 Graph an equation using slope-intercept form

Graph the equation 2x + y = 3.

STEP 1

Rewrite the equation in slope-intercept form.

y – 2x + 3=


Identify the slope and the y-intercept.

STEP 2

STEP 3

Plot the point that corresponds to the y-intercept,(0, 3).

STEP 4

Use the slope to locate a second point on the line. Draw a line through the two points.

= – 2m and = 3b

ESCALATORS

EXAMPLE 3 Change slopes of lines

To get from one floor to another at a library, you can take either the stairs or the escalator. You can climb stairs at a rate of 1.75 feet per second, and the escalator rises at a rate of 2 feet per second. You have to travel a vertical distance of 28 feet. The equations model the vertical distance d (in feet) you have left to travel after t seconds.

Stairs: d = – 1.75t + 28 Escalator: d = – 2t + 28

SOLUTION


a. Graph the equations in the same coordinate plane.

b. How much time do you save by taking the escalator?

a. Draw the graph of d = – 1.75t + 28 using the fact that the d-intercept is 28 and the slope is – 1.75. Similarly, draw the graph of d = – 2t + 28. The graphs make sense only in the first quadrant.


The equation d = – 1.75t + 28 has a t-intercept of 16. The equation d = – 2t + 28 has a t-intercept of 14. So, you save 16 – 14 = 2 seconds by taking the escalator.

b.

SOLUTION


4. Graph the equation y = – 2x + 5.

STEP 1

Identify the slope and the y- intercept.


STEP 2

Plot the point that corresponds to the y-intercept, (0, 5).

= – 2m and = 5b

STEP 3

Use the slope to locate a second point on the line. Draw a line through the two points.

EXAMPLE 5 Identify parallel lines

Determine which of the lines areparallel.

Find the slope of each line.

Line a: m = – 1 – 0– 1 – 2

– 3 – (–1 )0 – 5

=– 1– 3

13=

Line b: m = – 2– 5= 2

5=

Line c: m = – 5 – (–3) – 2 – 4

– 2– 6=

1 3=

ANSWER

Line a and line c have the same slope, so they are parallel.

EXAMPLE 5 Identify parallel linesGUIDED PRACTICE for Examples 4 and 5

Determine which lines are parallel: line a through (-1, 2) and (3, 4); line b through (3, 4) and (5, 8); line c through (-9, -2) and (-1, 2).

7.

SOLUTION

Line a: m = 4 – 2 3 –(1)

8 – 4 5 – 3

= 2 3+1

12=

Line b: m = – 4– 2=

Line c: m = 2 – (–2)

– 1 – (– 9)2+2

–1+ 9= 4 8=

Find the slope of each line.

= 2

=12

Warm-Up – 4.6 and 4.7


1. 4x – y = –8

2. -9x - 3y = 21

ANSWER

ANSWER

y=4x+8

Write the equation in slope intercept form and sketch the graph

y = -3x - 7


1. Slope = 1 and y-int = 2

ANSWER Y = x + 2

Write the equation and sketch the graph

3. You are traveling by bus. After 4.5 hours, the bus hastraveled 234 miles. Use the formula d = rt where d is distance, r is a rate, and t is time to find the average rate of speed of the bus.

ANSWER


52 mi/h

Find the unit rate.


1. y = 4x

ANSWER

Sketch the graph

Vocabulary – 4.6-4.7• Direct Variation

– Linear equation where y = kx

• Constant of Variation– In a Direct Variation, the

letter k is the constant of variation

– It’s the unit rate, the rate of change and …

– SAME AS SLOPE!!

• Function Notation– Different way of writing

functions

– F(x) means the “the function of x”

• Family of Functions– A group of functions

with similar characteristics (e.g. their graphs are all linear)

• Parent Linear Function– Simplest form of a

family of functions

– F(x) = x is the parent linear function

Notes – 4.6-4.7 – Direct Variations and Graphing Linear Functions

A Direct variation has the form

y = kx

k = the constant of variation and is AKA

THE SLOPE!

A direct variation graph ALWAYS goes through the origin.

A direct variation is ALWAYS PROPORTIONAL!

There are two ways to find the constant, k

1.Find the UNIT RATE

2.If they give you a coordinate (x,y),

1. Plug in the numbers to y = kx

2. Solve for k.

Notes – 4.6-4.7 – Direct Variations and Graphing Linear Functions – cont.

A “function” is usually written as

F(x) and we read it as “F of x”

OR y = F(x)

To evaluate functions

Plug in what you know and …..???

Examples 4.6-4.7

EXAMPLE 1 Identify direct variation equations

Tell whether the equation represents direct variation. If so, identify the constant of variation.

2x – 3y = 0a. – x + y = 4b.


ANSWER

Because the equation 2x – 3y = 0 can be rewritten in the form y = ax, it represents direct variation. The constant of variation is. 2

3

SOLUTION

To tell whether an equation represents direct variation, try to rewrite the equation in the form y = ax.


Subtract 2x from each side.– 3y = – 2x

y =23

x Simplify.

2x – 3y = 0


– x + y = 4 Write original equation.

Add x to each side.y = x + 4

ANSWER

Because the equation – x + y = 4 cannot be rewritten in the form y = ax, it does not represent direct variation.

b.



1. – x + y = 1


ANSWER

Because the equation – x + y = 1 cannot be rewritten in the form y = ax, it does not represent direct variation.

SOLUTION


Write original equation.– x + y = 1

y = x + 1 Add x each side.


2. 2x + y = 0

2x + y = 0 Write original equation.

y = – 2x Subtract 2x from each side.

SOLUTION

ANSWER

Because the equation 2x + y = 0 can be rewritten in the form y = ax, it represents direct variation. The constant of variation is. – 2


3. 4x – 5y = 0


4x – 5y = 0 Write original equation.

4x = 5y Subtract Add 5y each side.

y =45

x Simplify.

ANSWER

Because the equation 4x – 5y = 0 can be rewritten in the form y = ax, it represents direct variation. The constant of variation is. 4

5

SOLUTION

EXAMPLE 2 Graph direct variation equations

Graph the direct variation equation.

SOLUTION

a. y = x2

3y = – 3xb.

a. Plot a point at the origin. The slope is equal to the constant of variation, or Find and plot a second point, then draw a line through the points.

2 3


Plot a point at the origin. The slope is equal to the constant of variation, or – 3. Find and plot a second point, then draw a line through the points.

b.

EXAMPLE 3 Write and use a direct variation equation

The graph of a direct variation equation is shown.

SOLUTION

y = ax Write direct variation equation.Substitute.2 = a (– 1)

Write the direct variation equation.a.Find the value of y when x = 30.b.

Because y varies directly with x, the equation has the form y = ax. Use the fact that y = 2 when x = – 1 to find a.

a.

Solve for a.– 2 = a


ANSWER

A direct variation equation that relates x and y is y = – 2x.

b. When x = 30, y = – 2(30) = – 60.

SOLUTION


4. Graph the direct variation equation.

y = 2x

Plot a point at the origin. The slope is equal to the constant of variation, or Find and plot a second point, then draw a line through the points.

SOLUTION


Because y varies directly with x, the equation has the form y = ax. Use the fact that y = 6 when x = 4 to find a.

y = ax

6 = a (4)

64

a =32= Solve for a.

Write direct variation equation.

Substitute.

5. The graph of a direct variation on equation passes through the point (4,6). Write the direct variation equation and find the value of y when x =24.


ANSWER

A direct variation equation that relates x and y is y = x when x = 24. y = (24) = 363

232

SOLUTION

Standardized Test Practice

EXAMPLE 1

Substitute -3 for x.

Write original function.f (x) 3x – 15=

= (– 3) 3(– 3) – 15f

ANSWER

The correct answer is A. A B C D

Simplify.= 24


1. Evaluate the function h(x) = – 7x when x = 7.

Substitute 7 for x.

Write original function.h(x) = – 7x

(7) = – 7(7)h

= – 49 Simplify.

SOLUTION

Find an x-valueEXAMPLE 2

Write original function.

Substitute 6 for f(x).

8 x= Solve for x.

ANSWER

When x = 8, f(x) = 6.

6 2x – 10=

=f(x) 6.For the function f(x) 2x – 10, find the value of x so that=

= 2x – 10f(x)

Graph a function

The gray wolf population in central Idaho was monitored over several years for a project aimed at boosting the number of wolves. The number of wolves can be modeled by the function f(x) = 37x + 7 where x is the number of years since 1995. Graph the function and identify its domain and range.

GRAY WOLF

EXAMPLE 3

Review – Chap. 4


1. Write the equation 15 = 5y – 4x so that y is a function of x

y = x + 345

ANSWER

2. Solve C = 2r for r

ANSWER r = C2

3. Solve V = Bh for B.13

ANSWER B = 3Vh


4. On a round-trip bicycle trip from Santa Barbara to Canada, Phil rode 2850 miles in 63 days. Find his average miles per day for the trip. Use the formula d = rt where d is distance, r is rate, and t is time.Solve for r to find the rate in miles per day to the nearest mile.

about 45 miles per dayANSWER


Give the coordinates of the points. Plot the points in a coordinate plane

1. and 2.

ANSWER A(-3,2) B(0,-1)ANSWER


3. A(-2,-4)

ANSWER


4. B(3,0)

ANSWER

5. Graph y = x – 1 with domain – 4, – 2, 0, 2, 4. Then identify the range

12

– 3, – 2, – 1, 0, 1ANSWER


1. Graph y + 2x = 4

ANSWER


2. The distance in miles an elephant walks in t hours is given by d = 5t. The elephant walks for 2.5 hours. Graph the function and identify its domain and range.

domain: 0 < t < 2.5 range: 0 < d < 12.5ANSWER –

––

–


1. Find the x-intercept and the y-intercept of the graph of 3x – y = 3

x-int: 1, y-int: – 3 ANSWER


2. A recycling company pays $1 per used ink jet cartridge and $2 per used cartridge. The company paid a customer $14.This situation is given by x + 2y = 14 where x is the number of inkjet cartridges and y the number of laser cartridges. Use intercepts to graph the equation. Give four possibilities for the number of each type of cartridge that could have been recycled.

ANSWER (0,7),(6,4),(10,2),(14,0)


Find the slope of the line that passes through the points

1. (12, – 1) and (– 3, – 1)

ANSWER 0

2. (–2, 6) and (4, –3)

ANSWER 32–


3. The graph shows the ticket sales for a school dance on day 1, day 3, day 6, and day 9,of ticket sales. Describe the rates of change in ticket sales with respect to time.

ANSWER

Ticket sales grew moderately, declined slightly, and then had another moderate rate of increase.


1.

Identify the slope and y-intercept of the line 2x + 4y = –16.

2. y = + 123

x

ANSWER

12

–Slope: ,y-intercept: –4ANSWER


3. Determine which of the lines are parallel.

ANSWER Lines a and c


1.

Identify the slope and y-intercept of the line 2x + 4y = –16.

2. y = + 123

x

ANSWER

12

–Slope: ,y-intercept: –4ANSWER


3. Determine which of the lines are parallel.

ANSWER Lines a and c



1. 5x – 6y = 2

ANSWER no

2. x + y = 0

ANSWER yes, – 1


3. The number p of parts a machine produces varies directly with the time t (in minutes) the machine is in operation. The machine produces 84 parts in 14 minutes. Write a direct variation equation that relates t and p. how many parts does the machine produce in 25 minutes?

ANSWER p = 6t; 150 parts


Evaluate f (x) = 8x – 4 when x = –3, 0, and 2. 1.

ANSWER –28, –4, 12

Find the value of x so g (x) = –2x+ 1 has the value –3.2.

ANSWER 2


A stable charges $25 for feed and $50 per day to stable horses. The cost is given by f(x) = 50x + 25. Recently, the stable raised its fee for food to $50. The new fee is given by g(x) = 50x + 50.Graph the functions and then compare the two graphs.

3.

ANSWER

The graphs have the same slope.The y-intercept of g is 25 units greater than that of f, so g is a vertical translation.

Warm-Up – X.X

Vocabulary – X.X• Holder

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Notes – X.X – LESSON TITLE.• Holder•Holder•Holder•Holder•Holder

Examples X.X

Documents

Unit 1 – Chapters 1 and 4. Unit 1 Section 1.6-1.7 Section 4.1 Section 4.2 Section 4.3 Section 4.4 Section 4.5 Section 4.6-4.7 Review Ch. 4