NAME________________________________________ Period ________

TWO FORMS of LINEAR EQUATIONS 1) 2) EXAMPLE 1 Identify the form. a. y = 4x – 1 b. 2x – 3y = 10 c. -4.5x + 6.1y = 12 d.

€

y = −34x − 9

EXAMPLE 2 Rewrite each equation in Standard Form. a. y = 6x – 7 b.

€

y = −12x + 6

EXAMPLE 3 Write an equation in Slope-Intercept Form with the given slope and y-intercept. a. Slope = -8, y-intercept = 2 b. Slope =

€

52

, y-intercept = -3

_______________________ _______________________ c. Slope = 0, y-intercept = 12 d. Slope = 1, y-intercept = 0 _______________________ _______________________

CHAPTER 5 NOTES

LESSON 5.1 – Write Linear Equations in Slope-Intercept Form and Standard Form

2

EXAMPLE 4 Write an equation in Slope-Intercept form given a graph. a. _______________________ b. _______________________

EXAMPLE 5 Write an equation in Slope-Intercept form given 2 points. ** One of the points is the __________________________. a. (0, 1) and (4, -1)

Step1: Identify the slope. ________

Step 2: Identify the y-intercept. ________

Step 3: Write the equation. __________________________ b. (2, 5) and (0, -3) Step 1: Step 2: Step 3:

3

EXAMPLE 6 Solving a multi-step problem. A recording studio charges musicians an initial fee of $50 to record an album. Studio times costs an additional $35 per hour.

a. Write an equation for the total cost of an album (y). Step 1: Identify the rate of change and starting value. Rate of change = ______ = ___________________________ Starting value = ______ = ___________________________ Step 2: Equation: __________________________

b. Find the total cost of recording an album that takes 10 hours of studio time.

c. If the cost is $890, how many hours were spent recording?

4

LESSON 5.2 – Use Linear Equations in Slope-Intercept Form To write a linear equation in Slope-Intercept form, you need to know: 1) __________________________ 2) __________________________

EXAMPLE 1 Write an equation in Slope-Intercept form given the slope and a point on the line. a. m = –4 and passes through (–1, 3)

Step1: Identify the slope. ________

Step 2: Find the y-intercept. Use the ordered pair as (x, y).

Step 3: Write the equation. __________________________ b. . m = –2 and passes through (6, 3) Step 1: Step 2: Step 3:

5

EXAMPLE 2 Solving a multi-step problem. Your gym membership costs $33 per month after an initial membership fee. You paid a total of $228 after 6 months. Write an equation for the total cost with respect to time (in months). Find the total cost after 9 months.

d. Write an equation for the total cost (y). Step 1: Identify the rate of change and starting value. Rate of change = ______ = ___________________________

Starting value = ______ = ___________________________ Step 2: Find the starting value. Step 3: Write the equation. ___________________________ Step 4: Evaluate the equation when ______________.

6

EXAMPLE 3 Write an equation in Slope-Intercept form given 2 points.

a. (–2, 5) and (2, –1)

Step 1: Identify the slope.

Step 2: Find the y-intercept. Use the ordered pair as (x, y).

Step 3: Write the equation. __________________________ b. (1, –2) and (–5, 4)

Step 1: Step 2: Step 3: c. (1, 4) and (2, 7)

Step 1: Step 2: Step 3:

7

Point Slope Form: ________________________________________________________ m = ___________________ (x1, y1) = ___________________ EXAMPLE 1 Write an equation in point-slope form of the line that passes through the given point and has the given slope. a. (4, –3) slope = 2 b. (–8, –1) slope =

€

−56

EXAMPLE 2 Graph each equation.

1) First, identify the _____________________ and ________________________.

2) Plot the _____________________ and then use the ___________________ to find another point.

a.

€

y − 3 = 3 x − 6( ) b.

€

y + 2 =23x − 3( )

m = _________ (x, y) = _________ m = _________ (x, y) = _________

LESSON 5.3 – Write Equations in Point-Slope Form

8

c.

€

y +1= − x + 2( ) d.

€

y − 7 = −12x + 2( )

m = _________ (x, y) = _________ m = _________ (x, y) = _________

EXAMPLE 3 Write an equation in point-slope of the line shown. Step 1: Find the slope of the line.

graph Step 2: Write the equation in point-slope form.

You can use either point. CHECK: You can check that the equations are equivalent by writing them in slope-intercept form.

9

THREE FORMS of LINEAR EQUATIONS 1) 2) 3) EXAMPLE 4 Write an equation in Slope-Intercept form given the slope and a point on the line. (Use Point-Slope form.) a. (–5, 5) m = –2

1. Identify the slope. ________

2. Substitute the slope and the point into Point-Slope Form.

3. Solve for y. b. (6, –2) m = –1 Step 1: Step 2: Step 3:

c. (–8, 1) m =

€

34

Step 1: Step 2: Step 3:

10

EXAMPLE 5 Write an equation in Slope-Intercept form given 2 points. (Use Point-Slope form.) a. (2, 3) and (4, 4)

1. Identify the slope. ________

2. Substitute the slope and one of the points into Point-Slope Form.

3. Solve for y. b. (10, –2) and (12, –6) Step 1: Step 2: Step 3: EXAMPLE 6 In order to use an excerpt from a movie in a new television show, the television producer must pay the director of the movie $790 for the first two minutes of the excerpt and $130 per minutes after that.

a. Write an equation in slope-intercept form that gives the total cost (in dollars) of using the excerpt as a function of the length (in minutes) of the excerpt.

Step 1: Step 2: Step 3:

b. Find the total cost of using an excerpt that is 8 minutes long.

11

x 0 2 4 6 8

y 5 9 13 17 21

x 20 21 22 23 24

y -8 -5 -2 1 4

x -28 -24 -20 -16 -12

y -11 -9 -7 -5 -3

x 9 15 21 27 33

y 7 3 -1 -5 -9

In Chapter 4, we found the slope given a table. For example:

Now, we are going to write equations form looking at a table.

• First, we need to identify the _______________.

• Second, we need to identify the ______________________. Remember, the value

of x for the y-intercept is _____. EXAMPLE 1 Write a linear equation in slope-intercept form for each table. a. b. c. d.

Write Equations from a Table

x 0 1 2 3 4

y 11 9 7 5 3

x 2 4 6 8 10

y 3 6 9 12 15

Slope = Y-intercept = Equation:

Slope = Y-intercept = Equation:

12

Study the pattern below. Glencoe – Page 12 – Example 1 EXAMPLE 1 Find the next three terms in each sequence. a. 7, 13, 19, 25, … b. 243, 81, 27, 9, … c. d. Some of these patterns have a linear pattern. Therefore, we can write a rule or an ___________________ for the pattern to find any term number. EXAMPLE 2 Write a rule to find the nth term. Then use the rule to find the 99th term. a. 4, 5.5, 7, 8.5, … b.

* Use “n” to represent the number of sides used to create the triangles

Finding the Nth Term

13

___________________ lines are lines that do not intersect. They also have the ____________ slopes. ___________________ lines are lines that intersect and form _____ right angles. Their slopes are _____________________________________. EXAMPLE 1 Tell whether the graphs of the two equations are parallel, perpendicular, or neither. (WITHOUT GRAPHING) 1) Write each equation in __________________________ form. 2) Identify and compare the _________________. a. y = 5x – 7 b. y = 0.6x + 7

5x + y = 7 3y = –5x + 30 c. 3x – 7y = 1 d. y = –0.5x + 7

-7x + 3y = 4 x + 2y = 18

LESSON 5.5 – Write Equations of Parallel and Perpendicular Lines

14

EXAMPLE 2 Write an equation of a parallel line. a. Write an equation of the line that passes through (–3, –5) and is parallel to the line y = 3x – 1. Step 1: Identify the slope. Step 2: Find the y-intercept using Step 2: Substitute your slope and point SLOPE-INTERCEPT FORM using POINT-SLOPE FORM Step 3: Write an equation. Step 3: Solve for y. b. Write an equation of the line that passes through (-2, 11) and is parallel to the line y = –x + 5. Step 1: Step 2: Step 3:

OR

OR

15

EXAMPLE 3 Write an equation of a perpendicular line. a. Write an equation of the line that passes through (4, –5) and is perpendicular to the line y = –2x + 3. Step 1: Identify the slope. Step 2: Find the y-intercept using Step 2: Substitute your slope and point SLOPE-INTERCEPT FORM using POINT-SLOPE FORM Step 3: Write an equation. Step 3: Solve for y. b. Write an equation of the line that passes through (4, 3) and is perpendicular to the line y =

€

23

x – 7.

Step 1: Step 2: Step 3:

OR

OR

16

A ___________________________________ is a graph that is used to determine whether there is a relationship between paired data. CORRELATION

• If y tends to increase as x increases, the paired data are said to have a

_______________________ correlation.

• If y tends to decrease as x increases, the paired data are said to have a

_______________________ correlation.

• If x and y have no apparent relationship, the paired data are said to have

_______________________ correlation.

EXAMPLE 1 Describe the correlation of the data graphed in the scatter plot. a. b. c. EXAMPLE 2 Make a scatter plot

a. Make a scatter plot of the data in the table.

x 1.2 1.8 2.3 3.0 4.4 5.2

y 10 7 5 –1 –4 –8 b. Describe the correlation of the data.

LESSON 5.6 – Fit a Line to Data

17

A __________________________ is used to model data. This is also referred to as a _____________________________. USING A LINE OF FIT TO MODEL DATA Step 1: Make a _______________________ of the data. Step 2: Decide whether the data can be modeled by a ______________. Step 3: Draw a line that appears to ________ the data closely. There should be

approximately as many points _______________ the line as

_______________ it.

Step 4: Write an equation using ______ points on the line. The points do not have to

represent actual data pairs, but they must lie on the line of fit.

18

EXAMPLE 3 The table shows the average attendance at a school’s varsity basketball games for various years. Write an equation that models the average attendance at varsity basketball games as a function of the number of years since 2000. * *

Year 2000 2001 2002 2003 2004 2005 2006

Avg. Game Attendance 488 497 525 567 583 621 688

Step 1: Make a scatter plot of the data. Let x represent the number of years since 2000. Let y represent average game attendance. Step 2: Decide whether the data can be modeled by a line. Since the scatter plot shows a ____________________ correlation, you can fit a line to the data. Step 3: Draw a line that appears to fit the points in the scatter plot. Step 4: Write an equation using the two points bolded points in the table. Use __________ and _________. Find the slope. Find the y-intercept.

An equation of the line of fit is ___________________________.

19

EXAMPLE 4 Make a scatter plot of the data in the table. Draw a line of fit. Write an equation of the line. * *

X 10 20 30 40 50 60 Y 55 45 45 40 35 20

Step 1: Make a scatter plot of the data.

Step 2: Decide whether the data can be modeled by a line.

Since the scatter plot shows a ____________________ correlation, you can fit a line to the data.

Step 3: Draw a line that appears to fit the points in the scatter plot. Step 4: Write an equation using the two points bolded points in the table. Use __________ and _________. Find the slope. Find the y-intercept.

An equation of the line of fit is ___________________________.

20

5.7 – Predict with Linear Models Best-Fit Line: ____________________________________________________________ _________________________________________________________________________ Linear Regression: ________________________________________________________ _________________________________________________________________________ Interpolation: ____________________________________________________________ _________________________________________________________________________ Example 1 Interpolate using an equation The table shows the total number of CD singles shipped (in millions) by manufacturers for several years during the period 1993-1997.

a. Make a scatter plot of the data. Enter the data into lists on a graphing calculator. Make a scatter plot, letting the number of years since 1993 be the x-values and the number of CD singles shipped be the y-values.

b. Find an equation that models the number of CD singles shipped (in millions) as a

function of the number of years since 1993. ___________________________

21

c. Approximate the number of CD singles shipped in 1994.

Extrapolation: ___________________________________________________________ _________________________________________________________________________ Example 2 Extrapolate using an equation ** Look back at Example 1.

a. Use the equation from Example 1 to approximate the number of CD singles shipped in 1998 and in 2000.

b. In 1998 there were actually 56 million CD singles shipped. In 2000 there were actually 34 million CD singles shipped. Describe the accuracy of the extrapolations made in part (a).

About ____________________ CD singles were shipped in 1994.

22

Example 3 Predict using an equation The table shows the number of participants in U.S. youth softball during the period 1997-2001. Predict the year in which the number of youth softball participants reaches 1.2 million. Step 1: Perform linear regression. Let x represent ___________________________________ Let y represent ___________________________________ The equation for the best-fitting line is approximately: ____________________________ Step 2: Graph the equation of the best-fitting line. Trace the line until the cursor reaches y = 1.2. The corresponding x-value is shown at the bottom of the calculator screen. OR You can also predict the year by ____________________________ 1.2 for y in the equation and solving for x. Zero of a Function: _______________________________________________________ _________________________________________________________________________ Example 4 Find the zero of a function Look back at Example 3. Find the zero of the function. Explain what the zero means in the situation. Substitute _____ for _____ in the equation of the best-fitting line and solve for ______.

23

24