Unit 1 – Chapter 5. Unit 1 Section 5.1 – Write Linear Equations in Slope- Intercept FormSection...

Unit 1 – Chapter 5

Unit 1

• Section 5.1 – Write Linear Equations in Slope-Intercept Form

• Section 5.2

• Section 5.3

• Section 5.4

• Section 5.5

• Section 5.6

• Section 5.7

• Chapter 5 Review

Warm-Up – 5.1

Lesson 5.1, For use with pages 282-291

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

2. (–1, –3), (1, 5)

ANSWER

1. (2, –1), (4, 0)

3. A landscape architect charges $75 for a consulting fee and $30 per hour. Write an equation that shows the cost C as a function of time t (in hours).

ANSWER C = 30t + 75

Evaluate

2. f(x) = 2x-3. Find x if f(x) = 11

ANSWER

1. f(x) = - x – 2, x = 2

F(2) = -(2) -2 = -4

x = 7 because f(7) = 2(7) – 3 = 11

Vocabulary – 5.1• Y-intercept

• Where graph crosses the y-axis

• Where the story “starts”

• Slope

• How fast something changes!

• AKA

• Unit rate, steepness, rate of change, constant of variation, etc.

• Slope-Intercept Form

• Y=mx+b

• Standard Form

• Ax + By = C

Notes – 5.1 – Write LE in Slope-Int Form.•Slope – Intercept form

•Y = Mx + b •Need TWO things to write an equation in Slope-Int form. There are many ways to get them!

•Slope – •rise/run, y2-y1/x2-x1, how much over how long, draw triangles on graphs, etc.

•Y - Intercept – •Read graph, set x=0 and solve for y, find “b” in slope-intercept form

Examples 5.1

EXAMPLE 1 Use slope and y-intercept to write an equation

Write an equation of the line with a slope of – 2 and

a y -intercept of 5.y = mx + b Write slope-intercept form.

y = – 2x + 5 Substitute – 2 for m and = for b.

EXAMPLE 2 Standardized Test Practice

Which equation represents the line shown?

The line crosses the y-axis at (0, 3). So, the y-intercept is 3.

y = mx + b Write slope-intercept form.

2y = – x + 35

2Substitute – for m and 3 for b.5

= = –The slope of the line is riserun

–2 5

A y = – x + 325 B y = – x + 35

C y = – x + 125 D y = 3x +

EXAMPLE 2 Standardized Test Practice

ANSWER

The correct answer is A. B DCA

GUIDED PRACTICE for Examples 1 and 2

y = 8x –7 Substitute 8 for m and –7 for b.

1. Slope is 8; y-intercept is –7.

Write an equation of the line with the given slop and y-intercept.

SOLUTION

y = m x + b Write slope-intercept form.

2. Slope is ; y intercept is – 3. 34

y = 34

x – 3 Substitute for m and – 3 for b.34

SOLUTION

EXAMPLE 3 Write an equation of a line given two points

Write an equation of the line shown.

Write an equation of a line given two points

y = mx + b

y = x – 543

Substitute for m and 5 for b.43

STEP 1

Write an equation of the line. The line crosses the y-axis at (0, – 5). So, the y-intercept is – 5.

SOLUTION

x2 – x1 33 – 0y2 – y1=m =

– 1 – (– 5)=

Write slope-intercept form.

EXAMPLE 3

Calculate the slope.

STEP 2

SOLUTION

Write a linear function

Write an equation for the linear function f with the values f(0) = 5 and f(4) = 17.

Calculate the slope of the line that passes through (0, 5) and (4, 17).

Write f(0) = 5 as (0, 5) and f (4) = 17 as (4, 17).

EXAMPLE 4

STEP 1

x2 – x1 4 – 0y2 – y1=m =

17 – 5= 4

STEP 2

y = 3x + 5 Substitute 3 for m and 5 for b.

Write an equation of the line. The line crosses the y-axis at (0, 5). So, the y-intercept is 5.

STEP 3

EXAMPLE 4 Write a linear function

ANSWER

The function is f(x) = 3x + 5.

3. Write an equation of the line shown.

y = mx + b

Write an equation of the line. The line comes the y-axis at (0, 1). So, the y-intercept is 1.

=4 – 0

=m x2 – x1

y2 – y1 – 1 – 1=

STEP 2

STEP 1Calculate the slope.

–Substitute for m and 1 for b. 2–1

2–1 x + 1y =

SOLUTION

Calculate the slope of the line that passes through (0, –2) and (8, 4).

Write f(0) = –2 as (0, –2 ) and f (8) = 4 as (8, 4).

STEP 1

STEP 2

= ==m 86

4 3 4 – (– 2)

8 – 0

Write an equation for the linear function f with the given values.

SOLUTION

f(0) = –2, f(8) = 4

Write an equation of the line. The line comes the y-axis at (0, – 2). So, the y-intercept is – 2.

STEP 3

y = x – 234

The function is f (x) = 34

x – 2

ANSWER

Warm-Up – 5.2

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

2. (3, 2), (0, –2)

ANSWER

1. (0, –2), (1, 3)

Y = 4/3 x - 2

ANSWER Y = 5x -2

ANSWER y = –5x + 75

3. Bill wants a video game that costs $75. After borrowing $75 from his parents, he is paying back $5 per week. Write an equation that tells how much money Bill owes after x weeks.

ANSWER b = 5

Equation : y = 2x + 5

3. Using the slope-intercept form, find b if slope = 2, x = 2 and y = 9. Then find the equation

of the line.

Vocabulary – 5.2• Slope-Intercept Form

• Y = mx + b

• Y-intercept

• Where a graph crosses the y –axis

• X = 0

• X-intercept

• Where a graph crosses the x axis

• Y=0

Notes – 5.2 – Use Linear Eqns in Slope-Int. Form.• To use slope intercept form, I need two pieces of information:

•Slope•Y-intercept

•If you have ONE point and the slope, you can find the y-intercept by “doing the dance!”•Plug in the slope and (x,y) to the slope-int form and solve for b.•To find out if 3 points are on the same line,

1.Find the lin. Eqn. for the first two points.2.Plug in the (x,y) coordinates from the third point to the eqn. and see if it works.

Examples 5.2

EXAMPLE 1 Write an equation given the slope and a point

Write an equation of the line that passes through the point (– 1, 3) and has a slope of – 4.

SOLUTION

Substitute – 4 for m, – 1 for x, and 3 for y.

3 = – 4(– 1) + b

Identify the slope. The slope is – 4.STEP 1

Find the y-intercept. Substitute the slope and the coordinates of the given point in y = m x + b. Solve for b.

STEP 2

EXAMPLE 1

– 1 = b Solve for b.

Substitute – 4 for m and – 1 for b.y = – 4x – 1

Write an equation of the line.STEP 3

Write an equation given the slope and a point

GUIDED PRACTICE for Example 1

Write an equation of the line that passes through the point (6, 3) and has a slope of 2.

SOLUTION

Substitute 3 for y, 2 for m and 6 for x.

3 = 2(6) + b

Identify the slope. The slope is 2.STEP 1

Find the y-intercept. Substitute the slope and the coordinates of the given point in y = m x + b. Solve for b.

STEP 2

Substitute 2 for m and – 9 for b.y = 2x – 9

EXAMPLE 2 Write an equation given two points

Write an equation of the line that passes through (– 2, 5) and (2, – 1).

SOLUTION

3m = y2 – y1x2 – x1

= – 1 – 52 – (– 2)

= – 64

= – 2

Find the y-intercept. Use the slope and the point (– 2, 5).

STEP 1

STEP 2

EXAMPLE 2

5 = – 32

(– 2) – b

2 = b Solve for b.

Write an equation of the line.

Substitute – 32

for m and 2 for b.y = – 32 x+ 2

STEP 3

Substitute – for m, – 2 for x, and 5 for y.

Write an equation given two points

2. Write an equation of the line that passes through (1, –2) and (–5, 4).

SOLUTION

m = y2 – y1x2 – x1

= 4 –(– 2) –5 – 1

= 6–6

= – 1

Find the y-intercept. Use the slope and the point (1, –2).

STEP 1

STEP 2

Substitute – 1 for m and – 1 for b.y = – x – 1

–2 = –1(1) + b Substitute – 2 for y and – 1 for m.

EXAMPLE 3

2. Do these three points lie on the same line? (-4,-2), (2,2.5) and (8,7).

SOLUTION

m = y2 – y1x2 – x1

= 2.5–(- 2) 2 – (-4)

= 4.56

= = 3/4

Find the y-intercept. Use the slope and the point (-4, –2).

STEP 1

STEP 2

1 = b Solve for b.

Substitute – 1 for m and – 1 for b.y = 3/4 x + 1

–2 = 3/4 (-4) + b Substitute – 2 for y and – 4 for x and ¾ for m..

Plug in (8,7) to see if it’s a solutionSTEP 4

(7) = ¾ (8) + 1 7 = 6 + 17 = 7 so all three points are on the line

Solve a multi-step problem EXAMPLE 4

Your gym membership costs $33 per month after an initial membership fee. You paid a total of $228 after 6 months Write an equation that gives the total cost as a function of the length of your gym membership (in months). Find the total cost after 9 months.

STEP 1Identify the rate of change and starting value.

GYM MEMBERSHIP

SOLUTION

Rate of change, m: monthly cost, $33 per month

STEP 2

Write a verbal model. Then write an equation.

C = t b33 +

Starting value, b: initial membership fee

Solve a multi-step problem

EXAMPLE 4

Find the starting value. Membership for 6 months costs $228, so you can substitute 6 for t and 228 for C in the equation C = 33t + b.

228 = 33(6) + b Substitute 6 for t and 228 for C.

30 = b Solve for b.

STEP 4

Write an equation. Use the function from Step 2.

STEP 3

C = 33t + 30 Substitute 30 for b.

STEP 5

Evaluate the function when t = 9.

C = 33(9) + 30 = 327 Substitute 9 for t. Simplify.

STEP 4Write an equation. Use the function from Step 2.

Your total cost after 9 months is $327.ANSWER

EXAMPLE 5 Solve a multi-step problem

BMX RACING

In Bicycle Moto Cross (BMX) racing, racers purchase a one year membership to a track. They also pay an entry fee for each race at that track. One racer paid a total of $125 after 5 races. A second racer paid a total of $170 after 8 races. How much does the track membership cost? What is the entry fee per race?

SOLUTION

STEP 1Identify the rate of change and starting value.

Rate of change, m: entry fee per race

Starting value, b: track membership costSTEP 2Write a verbal model. Then write an equation.

C = m r + b

STEP 3Calculate the rate of change. This is the entry fee per race. Use the slope formula. Racer 1 is represented by (5, 125). Racer 2 is represented by (8, 170).

45m =y2 – y1x2 – x1

= 170 – 1258 – 5 = 3 = 15

STEP 4Find the track membership cost b. Use the data pair (5, 125) for racer 1 and the entry fee per race from Step 3.

Write the equation from Step 2.C = mr + b

Substitute 15 for m, 5 for r, and 125 for C.125 = 15(5) + b

Solve for b.50 = b

ANSWER

The track membership cost is $50. The entry fee per race is $15. Therefore C = 15x + 50.

Warm-Up – 5.3

Write an equation of the line.

2. passes through (–2, 2) and (1, 8)

ANSWER

1. passes through (3, 4), m = 3

y = 2x + 6

y = 3x – 5

3. Multiply both sides of the slope equation m = (y2-y1)

(x2-x1) by (x2-x1). What do you get?

ANSWER (y2-y1)= m (x2-x1)

4. Solve and graph the following inequality2|x+1| - 2 ≥ 6

ANSWER x ≤ −5 OR x ≥ 3

Vocabulary – 5.3 – Point – slope form• Point-Slope Form

• Fairly rare way of writing a linear equation that includes the slope and coordinates of one point.

Notes – 5.3 – Lin. Eqns in point-slope form• Studied TWO ways to write Lin. Eqn’s

1. Standard Form – Ax + By = C2. Slope-Intercept Form – Y = Mx + b

• The third way is the Point-Slope Form

1. Looks like (y-y1) = m(x-x1)2. Need two things:

1. Slope2. ONE point

3. Can be converted to Standard Form or Slope-Intercept form.

• CAN look different for every point (which is why it’s not used very often!)

Examples 5.3

Write an equation in point-slope formEXAMPLE 1

Write an equation in point-slope form of the line that passes through the point (4, –3) and has a slope of 2.

y – y1 = m (x – x1) Write point-slope form.

y + 3 = 2 (x – 4) Substitute 2 for m, 4 for x1, and –3 for y1.

Write an equation in point-slope formEXAMPLE 1

y – y1 = m (x – x1) Write point-slope form.

y – 4 = –2 (x +1) Substitute – 2 for m, 4 for y, and –1 for x.

Write an equation in point-slope form of the line that passes through the point (– 1, 4) and has a slope of – 2 .

Graph an equation in point-slope formEXAMPLE 2

y + 2 = (x – 3).2 3

Graph the equation.

SOLUTION

Because the equation is in point-slope form, you know that the line has a slope of and passes through the point (3, –2).

Plot the point (3, – 2). Find a second

point on the line using the slope.

Draw a line through both points.

y – 1 = (x – 2)–Graph the equation.2.

SOLUTION

Because the equation is in point-slope form, you know that the line has a slope of –1 and passes through the point (2, 1).

Plot the point (2, 1). Find a second point on the line using the slope. Draw a line through both points.

Use point-slope form to write an equationEXAMPLE 3

Write an equation in point-slope form of the line shown.

SOLUTION

STEP 1

= y1 – y2

x1 – x2

m = 3 –1– 1 –1

– 2= – 1

Find the slope of the line.

Method 1 Method 2

Use (– 1, 3). Use (1, 1).y – y1 = m(x – x1) y – y1 = m(x – x1)

y – 3 = – (x +1) y – 1 = – (x – 1)

STEP 2Write the equation in point-slope form. You can us either given point.

Check that the equations are equivalent by writing them in slope-intercept form.

y – 3 = –x – 1y = –x + 2

y – 1 = –x + 1y = –x + 2

STEP 1

= y2 – y1

x2 – x4

m = 4 –3 4 –2

Write an equation in point-slope form of the line that passes through the points (2, 3) and (4, 4).

Method 1 Method 2

Use (2, 3) Use (4, 4)y – y1 = m(x – x1) y – y1 = m(x – x1)

STEP 2Write the equation in point-slope form. You can us either given point.

y – 3 = (x – 2)1

2y – 4 = (x – 4)

Warm-Up – 5.4

ANSWER

1. (1, 4), (6, –1)

y + 2 = 3(x + 1) or y – 7 = 3(x – 2)

y – 4 = –(x – 1) or y + 1 = –(x – 6)

2. (–1, –2), (2, 7)

Write an equation in point-slope form of the line that passes through the given points.

3. Convert the equation y – 5 = 3(x – 3) to slope-intercept form AND Standard Form

ANSWERSlope Intercept y = 3x – 4

Standard Form -3x + y = -4

3. Write the following equation on your whiteboard 2x + 3y = 6 and graph it on your calculator. HINT: What form does the equation have to be in for the calculator?

4. Multiply both sides of the ORIGINAL equation by 2 and write that equation down on your whiteboard.

5. Now graph the new equation on your calculator.6. What do you notice?

ANSWER

New equation: 4x + 6y = 12

Graphs are identical!

Vocabulary – 5.4• Standard Form of a Linear Equation

• Ax + By = C

Notes – 5.4 – Write Lin. Eqns in Standard Form•Standard form is GREAT for graphing intercepts

•Y– intercept set x = 0 •X –Intercept set y = 0

•Can convert point-slope form and slope intercept for to standard for by getting all the variables on one side and the constants on the other side.•Equations that have a common multiple or factor are still equivalent.

Examples 5.4

To write another equivalent equation, multiply each side by 0.5.

4x – 12y = 8

To write one equivalent equation, multiply each side by 2.

SOLUTION

Write two equations in standard form that are equivalent to 2x – 6y = 4.

EXAMPLE 1 Write equivalent equations in standard form

x – 3y = 2

SOLUTION

y – y1 = m(x – x1)

Calculate the slope.STEP 1

EXAMPLE 2 Write an equation from a graph

– 3m =1 – (–2)

1 –2=

3–1 =

Write an equation in point-slope form. Use (1, 1).

Write point-slope form.

y – 1 = – 3(x – 1) Substitute 1 for y1, 23 for m

and 1 for x1.

Write an equation in standard form of the line shown.

STEP 2

Rewrite the equation in standard form.

EXAMPLE 2 Write an equation from a graph

3x + y = 4 Simplify. Collect variable terms on one side, constants on the other.

STEP 3

2x – 2y = 6

To write one equivalent equation,multiply each side by 2.

SOLUTION

EXAMPLE 1

3x – 3y = 9

To write another equivalent equation,multiply each side by 3.

Write two equations in standard form that are equivalent to x – y = 3.

Simplify.

Find the value of A. Substitute the coordinates of the given point for x and y in the equation. Solve for A.

STEP 1

SOLUTION

EXAMPLE 4

Find the missing coefficient in the equation of the line shown. Write the completed equation.

Ax + 3y = 2A(–1) + 3(0) = 2

–A = 2A = – 2

Write equation.

Substitute – 1 for x and 0 for y.

Divide by – 1.

EXAMPLE 3EXAMPLE 4Complete an equation in standard form

EXAMPLE 4Complete an equation in standard form

Complete the equation.

– 2x + 3y = 2 Substitute – 2 for A.

STEP 2

Simplify.

Find the value of B. Substitute the coordinates of the given point for x and y in the equation. Solve for B.

STEP 1

SOLUTION

Find the missing coefficient in the equation of the line that passes through the given point. Write the completed equation.

–4x + By = 7–4(–1) + B(1) = 7

Write equation.

Substitute –1 for x and 1 for y.

EXAMPLE 3 Write an equation of a lineGUIDED PRACTICE for Examples 3 and 4

5. –4x+By = 7, (–1,1)

– 4x + 3y = 7 Substitute 3 for B.

STEP 2

Simplify.

Find the value of A. Substitute the coordinates of the given point for x and y in the equation. Solve for A.

STEP 1SOLUTION

Ax + y = –3 A(2) + 11 = –3

2A= –14

Write equation.

Substitute 2 for x and 11 for y.

EXAMPLE 3 Write an equation of a lineGUIDED PRACTICE for Examples 3 and 4

Divide each side by 2.A= –7

Find the missing coefficient in the equation of the line that passes through the given point. Write the completed equation.

6. Ax+y = –3, (2, 11)

– 7x +y = –3 Substitute –7 for A.

STEP 2

Library

EXAMPLE 5

Your class is taking a trip to the public library. You can travel in small and large vans. A small van holds 8 people and a large van holds 12 people. Your class could fill 15 small vans and 2 large vans.

b. Graph the equation from part (a).

c. List several possible combinations.

Solve a multi-step problem

Write an equation in standard form that models the possible combinations of small vans and large vans that your class could fill.

SOLUTION

a. Write a verbal model. Then write an equation.

Because your class could fill 15 small vans and 2 large vans, use (15, 2) as the s- and l-values to substitute in the equation 8s + 12l = p to find the value of p.

8(15) + 12(2) = p Substitute 15 for s and 2 for l.144 = p Simplify.

Substitute 144 for p in the equation 8s + 12l = p.

8 s l p12+ =

Substitute 0 for s.8(0) + 12l = 144

l = 12

Substitute 0 for l.

s = 188s + 12(0) = 144

ANSWER

The equation 8s + 12l = 144 models the possible combinations.

b. Find the intercepts of the graph.

Plot the points (0, 12) and (18, 0). Connect them with a line segment. For this problem only nonnegative whole-number values of s and l make sense.

The graph passes through (0, 12), (6, 8),(12, 4), and (18, 0). So, four possible combinations are 0 small and 12 large, 6 small and 8 large, 12 small and 4 large, 18 small and 0 large.

EXAMPLE 5 Solve a multi-step problemEXAMPLE 5 Solve a multi-step problemGUIDED PRACTICE for Example 5

7. WHAT IF? In Example 5, suppose that 8 students decide not to go on the class trip. Write an equation that models the possible combinations of small and large vans that your class could fill. List several possible combinations.

SOLUTION

Write a verbal model. Then write an equation.

8 students decide not to go on the class trip, so the class could fill 14 small vans and 2 large vans. Because your class could fill 14 small vans and 2 large vans, use (14, 2) as the s- and l-values to substitute in the equation 8s + 12l = p to find the value of p.

8(14) + 12(2) = p Substitute 14 for s and 2 for l.136 = p Simplify.

Substitute 136 for p in the equation 8s + 12l = p.

EXAMPLE 5 Solve a multi-step problemEXAMPLE 5 Solve a multi-step problemGUIDED PRACTICE for Example 5

8 s l p12+ =

STEP 1

Substitute 0 for s.8(0) + 12l = 136

Substitute 0 for l.

s = 178s + 12(0) = 136

ANSWERThe equation 8s + 12l = 136 models the possible combinations.

Find the intercepts of the graph.

EXAMPLE 5 Solve a multi-step problemGUIDED PRACTICE for Example 5

l = 114

STEP 2

Plot the point(0,11 )and(17, 0).connect them with a line segment. For this problem only negative whole-number values of s and l make sense.

The graph passes through (17, 0), (14, 2), (11, 4), (8, 6), (5, 8) and (2, 10). So, several combinations are 17 small, 0 large; 14 small 2 large; 11 small, 4 large; 18 small, 6 large; 5 small, 8 large; 2 small, 10 large.

STEP 3

Warm-Up – 5.5

Are the lines parallel? Explain.

2. –x = y + 4, 3x + 3y = 5

ANSWER

1. y – 2 = 2x, 2x + y = 7

Yes; both slopes are –1.

No; one slope is 2 and the other is –2.

2. Graph x + y > 1

ANSWER

ANSWER $6

3. You play tennis at two clubs. The total cost C (in dollars) to play for time t (in hours) and rent equipment is given by C = 15t + 23 at one club andC = 15t + 17 at the other. What is the difference in total cost after 4 hours of play?

3. You play tennis at a club. The first two hours are free for members if you pay the $50 yearly fee. After that you pay $10 per hour. Your friend isn’t a member, so she can play for $15 an hour. Write two equations for the cost as a function of time played.

ANSWER

Members: C(t)= 10(t-2) + 50

Non-Members: C(t) = 15t

2. Write equation in POINT-SLOPE FORM of the line that goes through (2,-6) and (-3, 4)

ANSWER

6. Find Equation in SLOPE INTERCEPT FORM of the line that goes through (2,8) and (5,17)

Y+6 = -2(x-2) OR y-4 = -2(x+3)

Y = 3x + 2

2. Write an equation in STANDARD FORM of the line that passes through (1,3) and (3,13)

ANSWER -5X + Y = -2

5.5 - Warmup1. Take out your graph paper, patty paper, ruler, protractor,

and pencil.2. Graph two points that are on gridlines and connect them.

MAKE SURE YOUR SLOPE DOES NOT EQUAL 1.3. Find the slope of your line.4. Take one of the two points, and using the protractor make

a point that is 90 degrees from your point. Connect these points.

5. Find the slope of your new line.6. What do we call these types of lines?7. What do you notice about the slopes of perpendicular

lines?8. But I bet it only works once!!!! *grin*

Vocabulary – 5.5• Conditional Statement

• A statement with a hypothesis and a conclusion

• Frequently posed as an “if-then” statement

• Example: If Haley is at a volleyball game, then she’s not here.

• Is this statement always true?

• Converse

• A statement that swaps the hypothesis and conclusion of a conditional statement.

• Example: If Haley’s not here, then she’s at a volleyball game.

• Is this statement always true?

• Perpendicular lines

• Lines that form a right angle at their intersection.

Notes – 5.5 – Parallel and Perpendicular Lines•If two lines are Parallel, then their slopes are???

• Equal•What’s the converse?? And is it always true??•If two lines have equal slopes, then they are ???

•Parallel OR•IDENTICAL!!

•If two lines are Perpendicular, then their slopes are ?• Negative Reciprocals

•What’s the converse?? Is it always true??•If two non-vertical lines have slopes that are negative reciprocals, they are ???

•Perpendicular•Yes, it’s always true!

Examples 5.5

SOLUTION

EXAMPLE 1 Write an equation of a parallel line

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. The graph of the given equation has a slope of 3. So, the parallel line through (– 3, – 5) has a slope of 3.

STEP 2Find the y-intercept. Use the slope and the given point.

EXAMPLE 1 Write an equation of a parallel line

y = mx + b

– 5 = 3(– 3) + b

Substitute 3 for m, 23 for x, and 25 for y.

Solve for b.

STEP 3

Write an equation. Use y = mx + b.

y = 3x + 4 Substitute 3 for m and 4 for b.

SOLUTION

STEP 1

Identify the slope. The graph of the given equation has a slope of – 1.So, the parallel line through (– 2, 11) has a slope of – 1.

STEP 2

Find the y-intercept. Use the slope and the given point.

1. Write an equation of the line that passes through

(–2, 11) and is parallel to the line y = – x + 5.

y = mx + b

11 = (–1 )(– 2) + b

Substitute 11 for y, – 1 for m, and – 2 for x.

Solve for b.

STEP 3

Write an equation. Use y = m x + b.

y = – x + 9 Substitute – 1 for m and 9 for b.

SOLUTION

EXAMPLE 2 Determine whether lines are parallel or perpendicular

Determine which lines, if any, are parallel or perpendicular.Line a: y = 5x – 3

Line b: x +5y = 2

Line c: –10y – 2x = 0

Find the slopes of the lines.

Line a: The equation is in slope-intercept form. The slope is 5. Write the equations for lines b and c in slope-intercept form.

EXAMPLE 2

Line b: x + 5y = 2

5y = – x + 2

Line c: – 10y – 2x = 0

– 10y = 2x

y = – x15

Determine whether lines are parallel or perpendicular

xy =25

EXAMPLE 2

ANSWER

Lines b and c have slopes of – , so they are

parallel. Line a has a slope of 5, the negative reciprocal

of – , so it is perpendicular to lines b and c.

Determine whether lines are parallel or perpendicular

Determine which lines, if any, are parallel or perpendicular.Line a: 2x + 6y = – 3

Line b: 3x – 8 = y

Line c: –1.5y + 4.5x = 6

6y = –2x – 3

Line a: 2x + 6y = – 3

Find the slopes of the lines.

xy =12

––

– 1.5y = 4.5x – 6

Line b: 3x – 8 = y

Line c: –1.5y + 4.5x = 6

y = 3x – 4

Lines b and c have slopes of 3, so they are parallel. Line a has a slope of , the negative reciprocal

of 3, so it is perpendicular to lines b and c.

SOLUTION

EXAMPLE 3 Determine whether lines are perpendicular

Line a: 12y = – 7x + 42

Line b: 11y = 16x – 52

Find the slopes of the lines. Write the equations in slope-intercept form.

The Arizona state flag is shown in a coordinate plane. Lines a and b appear to be perpendicular. Are they?

STATE FLAG

EXAMPLE 3 Determine whether lines are perpendicular

Line a: 12y = – 7x + 42

Line b: 11y = 16x – 52

y = –

x + 1242 7

y = x –1611

ANSWER

The slope of line a is – . The slope of line b is .

The two slopes are not negative reciprocals, so lines a and b are not perpendicular.

SOLUTION

EXAMPLE 4 Write an equation of a perpendicular line

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. The graph of the given equation has a slope of 2. Because the slopes of perpendicular lines are negative reciprocals, the slope of the perpendicular line through (4, –5) is .1

EXAMPLE 4

STEP 2 Find the y-intercept. Use the slope and thegiven point.

– 5 = – (4) + b12

Substitute – for m, 4 for x, and

– 5 for y.

y = mx + b

STEP 3 Write an equation.

y = – x – 312 Substitute – for m and – 3 for b.1

Write an equation of a perpendicular line

3. Is line a perpendicular to line b? Justify your answer using slopes

Line a: 2y + x = – 12

Line b: 2y = 3x – 8

SOLUTION

Find the slopes of the lines. Write the equations in slope-intercept form.

Line a: 2y + x = 12

y = –

GUIDED PRACTICE

Line b: 2y = 3x – 8

ANSWER

The slope of line a is – . The slope of line b is .

The two slopes are not negative reciprocals, so lines a and b are not perpendicular.

for Examples 3 and 4

GUIDED PRACTICE

STEP 1

Identify the slope. The graph of the given equation has a slope of 4.Because the slopes of perpendicular lines are negative reciprocals, the slope of the perpendicular line through (4, 3) is .1

4. Write an equation of the line that passes through (4, 3) and is perpendicular to the line y = 4x – 7.

SOLUTION

GUIDED PRACTICE

STEP 3 Write an equation.

y = – x + 414 Substitute – for m and 4 for b.

STEP 2 Find the y-intercept. Use the slope and the given point.

3 = – (4) + b14

y = mx + b

4 = b Solve for b.

Substitute 3 for y, 3 for x, and for y.1

Warm-Up – 5.6

1. (–4, 1) and (6, –4)

Find the slopes of the line that passes through the point.

2. (2, –3) and (–1, 6)

ANSWER –3

ANSWER – 12

3. Your commission c varies with the number s of pair of shoes you sell. You made $180 when you sold 15 pairs of shoes. Write a direct variation equation that relates c to s.

ANSWER c = 12s

Find the slopes of the line that passes through the point.

3. Graph 3x + 4y <= 12

ANSWER

Vocabulary – 5.6• Scatter Plot

• Points that show relationships or trends in data

• Correlation between data

• Shows a relationship between data (if it exists)

• Line of fit

• AKA Linear Regression

• Line that “approximates” the data by modeling the trend.

Notes – 5.6 – Fit Data to a Line• Three types of correlation

1. Positive – Trends Up – As x gets larger, y gets larger

2. Negative – Trends Down – As x gets larger, y gets smaller

3. No Trend – Little to no “pattern” or correlation

Positive Negative No Trend

Notes – 5.6 – Fit Data to a Line• To construct a regression line, about half the points should be above the line and about half below.•The “accuracy” of the line or how close is models the data is given by the r2 value (r is short for residuals.)

•If the r2 value is VERY close to 1 the regression line accurately models the data.•If the r2 value is not close, the data isn’t showing a very strong correlation and the regression line is not very accurate.

•Once you have the line, you can use points or the graph to determine the equation of the line.

Examples 5.6

EXAMPLE 1 Describe the correlation of data

Describe the correlation of the data graphed in the scatter plot.

a. The scatter plot shows a positive correlation between hours of studying and test scores. This means that as the hours of studying increased, the test scores tended to increase.

EXAMPLE 1 Describe the correlation of data

b. The scatter plot shows a negative correlation between hours of television watched and test scores. that as the hours of television This means that as the hours of television watched

Increased, the test scores tended to decrease.

Using the scatter plots in Example 1, predict a reasonable test score for 4.5 hours of studying and 4.5 hours of television watched.

Sample answer: 72, 77

ANSWER

Swimming Speeds

EXAMPLE 2 Make a scatter plot

The table shows the lengths (in centimeters) and swimming speeds (in centimeters per second) of six fish.

a. Make a scatter plot of the data.

b. Describe the correlation of the data.

b. The scatter plot shows a positive correlation, which means that longer fish tend to swim faster.

SOLUTION

a. Treat the data as ordered pairs. Let x represent the fish length (in centimeters),and let y represent the speed

(in centimeters per second). Plot the ordered pairs as points in a coordinate plane.

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

ANSWER

The scatter plot shows a positive correlation.

BIRD POPULATIONS

EXAMPLE 3 Write an equation to model data

The table shows the number of active red-cockaded woodpecker clusters in a part of the De Soto National Forest in Mississippi. Write an equation that models the number of active clusters as a function of the number of years since 1990.

Year 1992 1993 1994 1995 1996 1997 1998 1999 2000

Active clusters

22 24 27 27 34 40 42 45 51

STEP 1

SOLUTION

Make a scatter plot of the data. Let x represent the number of years since 1990. Let y represent the number of active clusters.

STEP 3

STEP 4

STEP 2Decide whether the data can be modeled by a line. Because the scatter plot shows a positive correlation, you can fit a line to the data.

Draw a line that appears to fit the points in the scatter plot closely.

Write an equation using two points on the line. Use(2, 20) and (8, 42).

m = 113

42 – 208 – 2 = 22

6 =y2 – y1

x2 – x1=

Find the y-intercept of the line. Use the point (2, 20).

y = mx + b

20 = (2) + b113

Substitute for m, 2 for

x, and 20 for y.

An equation of the line of fit is y =

= b Solve for b.

x + 383

The number y of active woodpecker clusters can be modeled by the y = where x is the number of years since 1990.

ANSWER

x + 383

3. Use the data in the table to write an equation that models y as a function of x.

ANSWER

y = 1.6x + 2.3

a. Describe the domain and range of the function.

EXAMPLE 4 Interpret a model

Refer to the model for the number of woodpecker clusters in Example 3.

b. At about what rate did the number of active woodpecker clusters change during the period 1992–2000?

EXAMPLE 4 Interpret a model

SOLUTION

b. The number of active woodpecker clusters increased at a rate of or about 3.7 woodpecker clusters per year.

a. The domain of the function is the the period from 1992 to 2000,or 2 x 10. The range is the the

number of active clusters given by the function for 2 x 10, or 20 y 49.3.<– <–<– <–

<– <–

EXAMPLE 4GUIDED PRACTICE for Example 4

In Guided Practice Exercise 2, at about what rate does y change with respect to x

ANSWER

y changes with respect to x at the rate of about 1.6

Warm-Up – 5.7

2. The table shows the profit of a company. Write an modeling the profit y as the function of the number

of years x since 1998.

ANSWER

15.5; 20.5

y = 2.8x + 16.4

1. Evaluate f(x) = 2.5x + 8 when x is 3 or 5.

Vocabulary – 5.7• Linear Regression

• Line of “best fit” – Approximates the data in plot

• Interpolation

• Using a linear regression line to APPROXIMATE a point that is BETWEEN TWO KNOWN VALUES!

• Extrapolation

• Using a linear regression line to APPROXIMATE a point that is OUTSIDE of the data range (OR THE KNOWN VALUES).

• Tells the future!!

• Zero of a function

• Where a FUNCTION equals zero.

Notes –5.7 – Predict with Lin. Models• You can use your knowledge of Linear functions and the calculator to predict the future or the past.•IMPORTANT CALCULATOR FUNCTIONS:

•Enter Lists – STATEDIT•Plot Lists –

1. 2nd Y= (Stat Plot) 2. Turn Plot on and configure it (SET WINDOW!!)3. Remember ZOOMSTATPLOT as well.

•Determine points on a line (e.g. to find y1(x) using function notation or predict the future!)

• VARSY-VarsFunctionY1 Enter• Press ( put value here ) and press Enter• Line should look like Y1(5)

Notes –5.7 – Continued•IMPORTANT CALCULATOR FUNCTIONS:

•To find a Regression Line1. StatCalcLinreg (ax+b)2. Type in the lists where you put the data3. Tell “Calli” where you want to store the eqn.

4. EX: LinReg (ax+b) L1,L2,Y1

5. Turn Plot on and configure it (SET WINDOW!!)6. Remember ZOOMSTATPLOT as well.

Examples 5.7 – TURN TO PAGE 335

IN BOOK!

Interpolate using an equation EXAMPLE 1

CD SINGLES

The table shows the total number of CD single shipped (in millions) by manufacturers for several years during the period 1993–1997.

Make a scatter plot of the data.

b. Find an equation that models the number of CD singles shipped (in millions) as a function of the number of years since 1993.

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

SOLUTION

a. Enter the data into lists on a graphing calculator. Make a scatter plot, letting the number of years since 1993 be the x-values (0, 2, 3, 4) and the number of CD singles shipped be the y-values.

b. Perform linear regression using the paired data. The equation of the best-fitting line is approximately y = 14x + 2.4.

c. Graph the best-fitting line.Use the trace feature and the arrow keys to find the value of the equation when x = 1.

ANSWER

About 16 million CD singles were shipped in 1994.

Extrapolate using an equation EXAMPLE 2

CD SINGLES

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).

SOLUTION

a. Evaluate the equation of the best-fitting line from Example 1 for x = 5 and x = 7. The model predicts about 72 million CD singles shipped in 1998 and about 100 million CD singles shipped in 2000.

b. The differences between the predicted number of CD singles shipped and the actual number of CD singles shipped in 1998 and 2000 are 16 million CDs and 66 million CDs, respectively. The difference in the actual and predicted numbers increased from 1998 to 2000. So, the equation of the best-fitting line gives a less accurate prediction for the year that is farther from the given years.

1. HOUSE SIZE The table shows the median floor area of new single-family houses in the United States during the period 1995–1999.

a. Find an equation that models the floor area (in square feet) of a new single-family house as a function of the number of years since 1995.

ANSWER y = 26.6x + 1921.4

c. Which of the predictions from part (b) would you expect to be more accurate? Explain your reasoning.

b. Predict the median floor area of a new single-family house in 2000 and in 2001.

ANSWER about 2054.4 ft2, about 2081 ft2

ANSWER

The prediction for 2000 because the farther removed an x-value is from the known x-values, the less confidence you can have in the accuracy of the predicted y-value.

SOFTBALL

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.

Year 1997 1998 1999 2000 2001

Participants (millions) 1.44 1.4 1.411 1.37 1.355

SOLUTION

Perform linear regression. Let x represent the number of years since 1997, and let y represent the number of youth softball participants (in millions). The equation for the best-fitting line is approximately y = – 0.02x + 1.435.

STEP 1

STEP 2Graph 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.

ANSWERThere will be 1.2 million participants about 12 years after 1997, or in 2009.

2. SOFTBALL In Example 3, in what year will there be 1.25 million youth softball participants in the U.S?

ANSWER

There will be 1.25 million youth softball participants in 2006.

SOLUTION

SOFTBALL

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 this situation.

Substitute 0 for y in the equation of the best-fitting lineand solve for x.

y = – 0.02x + 1.435 Write the equation.

0 = – 0.02x + 1.435 Substitute 0 for y.

x 72 Solve for x.

EXAMPLE 4 Find the zero of a function

ANSWERThe zero of the function is about 72. The function has a negative slope, which means that the number of youth softball participants is decreasing. According to the model, there will be no youth softball participants 72 years after 1997, or in 2069.

JET BOATS The number y (in thousands) of jet boats purchased in the U.S. can be modeled by the function y = – 1.23x + 14 where x is the number of years since 1995. Find the zero of the function. Explain what the zero means in this situation.

SOLUTION

Substitute 0 for y in the equation of the best-fitting lineand solve for x.

y = – 1.23x + 14 Write the equation.

0 = – 1.23x + 14 Substitute 0 for y.

x 11.4 Solve for x.

EXAMPLE 4

ANSWERThe function has a negative slope, which means that the number of jet boats purchased in the U.S. is decreasing. According to the model, there will be no jet boats purchased 11.4 years after 1995, or in 2006.

Review – Ch. 5

Daily Homework Quiz For use after Lesson 5.1

ANSWER y = – 4x + 1

Write an equation of the line with a slope of – 4 and a y-intercept of 1.

Write an equation of the line that passes through the given points.

(–9,1), (0, –8)2.

ANSWER y = – x –8

ANSWER y = 3x + 6

(–4, –6), (0, 6)3.

An electronics game store sells used games for $12.99 with a $20 membership fee. Write an equation that gives the total cost to become a member and buy games as a function of the number of games that are purchased. Then find the cost or 6 games.

Write an equation for the linear function f with f (0) = – 4 and f(–1) = –9.

ANSWER y = 5x – 4

ANSWERC =12.99g + 20 where C is total cost and g is the number of games; $97.94

ANSWER y = – x + 3

(4, –1), m = –11.

Write an equation of the line that passes through the given point with given slope.

ANSWER y = 4x – 8

(2, 0), m = 42.

ANSWER $770

ANSWER y = –2x – 3

ANSWER y = 2x – 1

(2, 3), (4,7)3.

(–5, 7), (2, –7)4.

A camp charges a registration fee and a daily amount. If the total bill for one camper was $338 for 12 days and the total bill for another camper was $506 for 19 days and the total bill be for a camper who enrolls for 30 days?

Write an equation of the line that passes through the given point.

ANSWER y + 4 = –2(x –6)

Write an equation in point-slope form of the line that passes through (6, – 4) and has slope 2.

Write an equation in point-slope form of the line that passes through (–1, –6) and (3,10).

ANSWER y + 6 = 4(x + 1) or y –10 = 4(x–3)

A travel company offers guided rafting trips for $875 for the first three days and $235 for each additional day. Write an equation that gives the total cost (in dollars) of a rafting trip as a function of the length of the trip. Find the cost for a 7-day trip.

ANSWER

C = 235t + 170, where C is total cost and t is time (in days); $1815

Write an equation in standard form of the line that passes through the given point and has the given slope m or that passes through the two given points.

ANSWER 2x + y = – 4

(1, – 6), m = – 2

2. (– 4, – 3), (2, 9)

ANSWER – 2x + y = 5

ANSWER 8p + 6c = 96; 16 h kayak and 0 h paddleboat; 12 h paddleboat and 0 h kayak;6 h paddleboat and 8 h kayak.

You have $96 to spend on campground activities. You can rent a paddleboat for $8 per hour and a kayak for $6 per hour. Write an equation in standard form that models the possible hourly combinations of activities you can afford. List three possible combinations.

1. Write an equation of the line that passes through the point (–1,4) and is parallel to the line y = 5x –2.

y = 5x + 9

ANSWER

Write an equation of the line that passes through the point (–1, –1) and is perpendicular to the line y = x +2.1

y = 4x + 3

ANSWER

3. Path a, b and c are shown in the co ordinate grid. Determine which paths,if any, are parallel or perpendicular. Justify your answer using slopes.

ANSWER

Paths a and b are perpendicular because their slopes,and 2 are negative reciprocals. No paths are parallel.

1. Tell whether x and y show a positive correlation, a negative correlation, or relatively no correlation.

ANSWER negative correlation.

y = 3.1x – 10.3, where x is body length and y is wingspan.

ANSWER

The table shows the body length and wingspan (both in inches) of seven birds. Write an equation that models the wingspan as a function of body length.

Warm-Up – X.X

Vocabulary – X.X• Holder

• Holder 2

• Holder 3

• Holder 4

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

Examples X.X

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