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Algebra 1 Interactive Chalkboard Copyright by The McGraw-Hill Companies, Inc.

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Lesson 5-1 Slope

Lesson 5-2 Slope and Direct Variation

Lesson 5-3 Slope-Intercept Form

Lesson 5-4 Writing Equations in Slope-Intercept Form

Lesson 5-5 Writing Equations in Point-Slope Form

Lesson 5-6Geometry: Parallel and Perpendicular Lines

Lesson 5-7 Statistics: Scatter Plots and Lines of Fit
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Example 1 Positive SlopeExample 2 Negative Slope

Example 3 Zero Slope

Example 4 Undefined SlopeExample 5 Find Coordinates Given Slope

Example 6 Find a Rate of Change

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Find the slope of the line that passes through(3 , 2) and (5 , 5) .

Let and

Substitute.

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Simplify.

Answer: The slope is

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Find the slope of the line that passes through(4, 5) and (7 , 6) .

Answer:

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Substitute.

Let and

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Answer: The slope is 4 .

Simplify.

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Answer: 2

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Let and


Substitute.

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Answer: The slope is 0.

Simplify.

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Answer: 0

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Let and


Answer: Since division by zero is undefined,the slope is undefined.

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Answer: undefined

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Find the value of r so that the line through (6, 3) and(r , 2) has a slope of

Slope formula

Substitute.

Subtract.

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Find the cross products.

Simplify.

Add 6 to each side.

Answer: Simplify.

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Find the value of p so that the line through ( p, 4) and(3, 1 ) has a slope of

Answer: 5

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Travel The graph to theright shows the number of U.S. passports issued in1991, 1995, and 1999.Find the rates of change for 1991-1995 and 1995-1999.

Use the formula for slope.

millions of passportsyears

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1991-1995:Substitute.

Simplify.

Answer: The number of passports issued increasedby 1.9 million in a 4-year period for a rate of change of 475,000 per year.

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1995-1999:Substitute.

Simplify.

Answer: Over this 4-year period, the number of U.S.passports issued increased by 1.4 million for a rate of change of 350,000 per year.

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Explain the meaning of slope in each case.

Answer: For 1991-1995, on average, 475,000 morepassports were issued each year than the last. For 1995-1999, on average, 350,000 more passports were issuedeach year than the last.

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How are the differentrates of change shownon the graph?

Answer: There is a greater

rate of change from 1991-1995 than from 1995-1999.Therefore, the section of thegraph for 1991-1995 has asteeper slope.

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Airlines The graph showsthe number of airplanedepartures in the UnitedStates in recent years. a. Find the rates of change

for 1990-1995 and1995-2000.

Answer: 240,000 per year;180,000 per year

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b. Explain the meaning of the slope in each case.

Answer: For 1990-1995, thenumber of airplane departuresincreased by about 240,000flights each year. For 1995-2000, the number of airplanedepartures increased by about180,000 flights each year.

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c. How are the differentrates of change shownon the graph?

Answer: There is a greater vertical change for 1990-1995than for 1995-2000. Therefore,the section of the graph for 1990-1995 has a steeper slope.

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Example 1 Slope and Constant of VariationExample 2 Direct Variation with k > 0

Example 3 Direct Variation with k < 0

Example 4 Write and Solve a Direct Variation Equation

Example 5 Direct Variation Equation

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Name the constant of variation for the equation.Then find the slope of theline that passes through thepair of points.

Slope formula

Simplify.

Answer: The constantof variation is 2. The slope is 2.

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Name the constant of variation for the equation.Then find the slope of theline that passes through thepair of points.

Slope formula

Simplify.

Answer: The constantof variation is 4 . The slope is 4 .

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Name the constant of variation for the equation.Then find the slope of the line that passes throughthe pair of points.

a. Answer: constant of variation: 4;slope: 4

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Answer: constant of variation: 3 ;slope: 3

Name the constant of variation for the equation.Then find the slope of the line that passes throughthe pair of points.

b.

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Step 1 Write the slope asa ratio.

Step 2 Graph (0, 0).

Step 3 From the point (0, 0), move up 1 unit and right1 unit. Draw a dot.

Step 4 Draw a line containing the points.

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Answer:

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Step 1 Write the slope asa ratio.

Step 2 Graph (0, 0).

Step 3 From the point (0, 0), move down 3 units andright 2 units. Draw a dot.

Step 4 Draw a line containing the points.

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Answer:

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Suppose y varies directly as x, and whenWrite a direct variation equation that relates x and y.

Find the value of k .

Direct variation formula

Replace y with 9 and x with 3 .

Divide each side by 3 .

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Simplify.

Answer: Therefore,

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Use the direct variation equation to find x whenDirect variation equation

Answer: Therefore, when

Replace y with 15 .


Simplify.

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Suppose y varies directly as x, and whena. Write a direct variation equation that relates x and y.

b. Use the direct variation equation to find x when

Answer: 15

Answer:

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Travel The Ramirez family is driving cross-countryon vacation. They drive 330 miles in 5.5 hours.

Write a direct variation equation to find the distancedriven for any number of hours.

Words The distance traveled is 330 miles, and thetime is 5.5 hours.

Variables

Distance equals rate times time.

Equation 330 mi r 5.5h

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Solve for the rate.

Answer: Therefore, the direct variation equation is

Original equation

Divide each side by 5.5 .

Simplify.

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Graph the equation.

The graph of passes through the origin with aslope of 60 .

Answer:

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Estimate how many hours it would take to drive600 miles.

Original equation

Replace d with 600 .


Simplify.

Answer: At this rate, it will take 10 hours to drive600 miles.

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Dustin ran a 26-mile marathon in 3.25 hours.

a. Write a direct variation equation to find the distanceran for any number of hours.

b. Graph theequation.

Answer:

Answer:

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c. Estimate how many hours it would take to jog 16 miles.

Answer: 2 hours

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Example 1 Write an Equation Given Slope and y -Intercept

Example 2 Write an Equation Given Two Points

Example 3Graph an Equation in Slope-Intercept Form

Example 4 Graph an Equation in Standard Form

Example 5 Write an Equation in Slope-Intercept Form

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Write an equation of the line whose slope isand whose y -intercept is 6 .

Slope-intercept form

Replace m with and b with 6 .

Answer:

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Write an equation of the line whose slope is 4and whose y -intercept is 3.

Answer:

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Write an equation of the lineshown in the graph.

Step 1 You know the coordinatesof two points on the line.Find the slope. Let

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Simplify.

The slope is 2.

Step 2 The line crosses the y -axis at (0 , 3) .So, the y -intercept is 3 .

Step 3 Finally, write the equation.


Replace m with 2 and b with 3 .

Answer: The equation of the line is

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Write an equation of the lineshown in the graph.

Answer:

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Graph

Step 1 The y -intercept is 7 .So graph (0 , 7) .

Step 2 The slope is 0.5

or

From (0 , 7) , move up 1 unit and right 2 units.Draw a dot.

Step 3 Draw a line connecting the points.

y = 0.5 x 7

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Graph

Answer:

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Graph

Step 1 Solve for y to find the slope-intercept form.

Original equation

Subtract 5 x from each side.

Simplify.

Divide each side by 4.

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Divide each term in thenumerator by 4.

Answer:

Step 2 The y -intercept of

is 2.

So graph (0, 2) .

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Step 3 The slope is

From (0, 2) , movedown 5 units andright 4 units.Draw a dot.

Step 4 Draw a lineconnecting the points.

5 x + 4y = 8

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Graph

Answer:

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Health The ideal maximum heart rate for a 25-year-old who is exercising to burn fat is 117 beats per minute. For every 5 years older than 25, that idealrate drops 3 beats per minute.Write a linear equation to find the ideal maximum

heart rate for anyone over 25 who is exercising toburn fat.

Words The rate drops 3 beats per minute every

5 years, so the rate of change is beats

per minute each year. The ideal maximum

heart rate for a 25-year-old is 117 beats

per minute.

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Variables Let R = the ideal heart rate.Let a = years older than 25.

Equationideal rate

Ideal rate of years older for 25-rate equals change times than 25 plus year-old.

R a 117

Answer:

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Graph the equation.

The graph passes through (0 , 117) with a slope of

Answer:

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Find the ideal maximum heart rate for a personexercising to burn fat who is 55 years old.

The age 55 is 30 years older than 25 . So,

Ideal heart rate equation

Replace a with 30 .

Simplify.

Answer: The ideal heart rate for a 55-year-old person is 99 beats per minute.

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The amount of money spent on Christmas gifts hasincreased by an average of $150,000 ($0.15 million) per year since 1986. Consumers spent $3 million in 1986.

a. Write a linear equation to find the average amountspent for any year since 1986.

Answer: where D is the amount of moneyspent in millions of dollars, and n is the number of years since 1986

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b. Graph the equation.

Answer:

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c. Find the amount spent by consumers in 1999.

Answer: $4.95 million

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Example 1 Write an Equation Given Slope and One Point

Example 2 Write an Equation Given Two Points

Example 3 Write an Equation to Solve a Problem

Example 4 Linear Extrapolation

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Write an equation of a line that passes through (2 , 3)

with slope

Step 1 The line has slope To find the y -intercept,replace m with and ( x, y) with (2, 3 ) in the

slope-intercept form. Then, solve for b.

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Replace m with , y with 3 ,

and x with 2.Multiply.

Subtract 1 from each side.Simplify.

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Step 2 Write the slope-intercept form using


Replace m with and b with 4 .

Answer: The equation is

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Check You can check your result by graphing

on a graphing calculator.

Use the CALC menu to verifythat it passes through (2, 3).

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

Answer:

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Multiple-Choice Test ItemThe table of ordered pairs showsthe coordinates of two points on thegraph of a function. Which equationdescribes the function?

A BC D

x y 3 4

2 8

Read the Test ItemThe table represents the ordered pairs (3 , 4) and (2 , 8) .

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Solve the Test ItemStep 1 Find the slope of the line containing the points.

Let and .

Slope formula

Simplify.

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Step 2 You know the slope and two points. Choose onepoint and find the y -intercept. In this case, wechose (3, 4) .


Replace m with 4 , x with 3 , and y with 4 .

Multiply.

Subtract 12 fromeach side.

Simplify.

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Step 3 Write the slope-intercept form using

Answer: The equation is The answer is D.


Replace m with 4 andb with 16 .

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x y 1 3

2 6

Multiple-Choice Test ItemThe table of ordered pairs showsthe coordinates of two points on thegraph of a function. Which equationdescribes the function?

A BC D

Answer: B

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Economy In 2000, the cost of many items increasedbecause of the increase in the cost of petroleum. InChicago, a gallon of self-serve regular gasoline cost$1.76 in May and $2.13 in June. Write a linear equationto predict the cost of gasoline in any month in 2000,using 1 to represent January.

Explore You know the cost of regular gasoline in Mayand June.

Plan Let x represent the month and y represent the

cost of gasoline that month. Write an equationof the line that passes through (5, 1.76) and(6, 2.13) .

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Solve Find the slope.

Letand .

Slope formula

Simplify.

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Choose (5 , 1.76) and find the y -interceptof the line.

Slope-interceptform

Replace m with0.37 , x with 5,and y with 1.76 .

Multiply.

Subtract 1.85from each side.

Simplify.

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Write the slope-intercept form usingand

Replace m with 0.37 and b with 0.09 .


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Examine Check your result by substituting thecoordinates of the point not chosen, (6 , 2.13) ,into the equation.

Original equation

Replace y with2.13 and x with 6.

Multiply.

Simplify.

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The average cost of a college textbook in 1997 was$57.65. In 2000, the average cost was $68.15. Write alinear equation to estimate the average cost of atextbook in any given year since 1997. Let x representyears since 1997.

Answer:

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Economy The Yellow Cab Company budgeted $7000for the July gasoline supply. On average, they use3000 gallons of gasoline per month. Use the predictionequation where x represents themonth and y represents the cost of one gallon of gasoline, to determine if they will have to add to their budget. Explain.

Original equation

Replace x with 7.

Simplify.

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Answer: If gas increases at the same rate, a gallon of gasoline will cost $2.50 in July. 3000 gallons at this price is$7500, so they will have to add $500 to their budget.

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A student is starting college in 2004 and has saved$400 to use for textbooks. Use the prediction equation

where x is the years since 1997 and y is the average cost of a college textbook, todetermine whether they will have enough money for 5 textbooks.

Answer: If the cost of textbooks increases at the samerate, the average cost will be $82.15 in 2004. Fivetextbooks at this price is $410.75, so he will not haveenough money.

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Example 1Write an Equation Given Slope and a Point

Example 2 Write an Equation of a Horizontal Line

Example 3 Write an Equation in Standard Form

Example 4 Write an Equation in Slope-Intercept Form

Example 5 Write an Equation in Point-Slope Form

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Write the point-slope form of an

equation for a line that passes

through (2 , 0) with slope

Point-slopeform

Simplify.


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Write the point-slope form of anequation for a line that passesthrough (4 , 3) with slope 2 .

Answer:

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Write the point-slope form of an equation for a horizontalline that passes through (0 , 5) .

Point-slopeform

Simplify.


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Write the point-slope form of an equation for a horizontal linethat passes through (3 , 4) .

Answer:

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Write in standard form.

In standard form, the variables are on the left side of theequation. A, B, and C are all integers.

Original equation

Multiply each side by 4 toeliminate the fraction.

Distributive Property

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Subtract 3 x from each side.

Simplify.

Answer: The standard form of the equationis

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Write in standard form.

Answer:

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Write in slope-intercept form.

In slope-intercept form, y is on the left side of theequation. The constant and x are on the right side.

Original equation


Add 5 to each side.

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Simplify.

Answer: The slope-intercept form of the equation

is

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Write in slope-intercept form.

Answer:

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The figure showstrapezoid ABCD withbases and

Write the point-slopeform of the linescontaining the basesof the trapezoid.

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Step 1 First find the slopes of and

Slope formula

Slope formula

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Step 2 You can use either point for ( x1, y1) in thepoint-slope form.

Method 1 Use (2 , 3). Method 2 Use (4 , 3).

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Method 1 Use (1 , 2). Method 2 Use (6 , 2).

Answer: The point-slope form of the equation containingThe point-slope form of the equation

containing

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Write each equation in standard form.

Original equation

Add 3 to each side.

Answer: Simplify.

Original equation

Subtract 2 from each side.

Answer: Simplify.

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The figure shows righttriangle ABC .

a. Write the point-slope formof the line containing the

hypotenuse

b. Write the equation instandard form.

Answer:

Answer:

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Example 1 Parallel Line Through a Given Point

Example 2Determine Whether Lines are Perpendicular

Example 3 Perpendicular Line Through a Given Point

Example 4 Perpendicular Line Through a Given Point

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Write the slope-intercept form of an equation for the

line that passes through (4 , 2) and is parallel to the

graph of

The line parallel to has the same slope,

Replace m with and ( x, y) with (4 , -2) in the

point-slope form.

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Point-slope form

Replace m with

y with 2, and x with 4.

Simplify.


Subtract 2 from each side.

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Write the equation in slope-intercept form.


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Check You can check your result by graphing both

equations. The lines appear to be parallel.

The graph of passes through (4 , 2) .

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Write the slope-intercept form of an equation for the

line that passes through (2 , 3) and is parallel to the

graph of

Answer:

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Geometry The height of atrapezoid is measured on asegment that is perpendicular to a base. In trapezoid ARTP ,

and are bases. Canbe used to measure the

height of the trapezoid? Explain.

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Find the slope of each segment.

Slope of

Slope of

Slope of

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Answer: The slope of and is 1 and the slopeof is not perpendicular to

and , so it cannot be used to measure height.

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The graph shows thediagonals of a rectangle.Determine whether is perpendicular to

Answer: The slope of is and the slope of

is Since is not perpendicular to

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Write the slope-intercept form for an equation of aline that passes through (4 , 1) and is perpendicular to the graph of

Step 1 Find the slope of the given line.

Original equation

Subtract 7 x fromeach side.

Simplify.

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Divide each side

by 2 .

Simplify.

Step 2 The slope of the given line is So, the slope

of the line perpendicular to this line is the

opposite reciprocal of or

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Step 3 Use the point-slope form to find the equation.

Point-slope form

and

Simplify.


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Subtract 1 fromeach side.

Simplify.


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Check You can check your result by graphing both

equations on a graphing calculator. Use the

CALC menu to verify that

passes through (4 , 1) .

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Write the slope-intercept form for an equation of

a line that passes through (3 , 6) and isperpendicular to the graph of

Answer:

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a line perpendicular to the graph of and passes through (0 , 6) .

Step 1 Find the slope of

Original equation

Subtract 5 x fromeach side.

Simplify.

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Divide eachside by 2.

Simplify.

Step 2 The slope of the given line is So, the slope

of the line perpendicular to this line is the

opposite reciprocal of or

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Step 3 Substitute the slope and the given point into thepoint-slope form of a linear equation. Then writethe equation in slope-intercept form.

Point-slope form

Replace x1 with 0, y1 with 6, and m with



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a line perpendicular to the graph of and passes through the x-intercept of that line.

Answer:

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Example 1 Analyze Scatter Plots

Example 2 Find a Line of Fit

Example 3 Linear Interpolation

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Determine whether the

graph shows a positivecorrelation , a negativecorrelation , no correlation .If there is a positive or negative correlation,describe it.The graph shows averagepersonal income for U.S. citizens.

Answer: The graph shows a positive correlation. Witheach year, the average personal income rose.

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Determine whether the

graph shows a positivecorrelation , a negativecorrelation , no correlation .If there is a positive or negative correlation,describe it.The graph shows the averagestudents per computer in U.S.public schools.

Answer: The graph shows a negative correlation. Witheach year, more computers are in the schools, making thestudents per computer rate smaller.

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Determine whether each

graph shows a positivecorrelation , a negativecorrelation , no correlation .If there is a positive or negative correlation,describe it.a. The graph shows the

number of mail-order prescriptions.

Answer: Positive correlation; with each year, the number of mail-order prescriptions has increased.

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Determine whether each

graph shows a positivecorrelation , a negativecorrelation , no correlation .If there is a positive or negative correlation,describe it.

Answer: no correlation

b. The graph shows thepercentage of voter participation in

Presidential Elections.

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The table shows the world population growing at a

rapid rate.

Year Population (millions)

1650 500

1850 10001930 20001975 4000

1998 5900

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Draw a scatter plot

and determine whatrelationship exists,if any, in the data.

Let the independent

variable x be the year and let the dependentvariable y be thepopulation (in millions).

The scatter plot seems to indicate that as the year increases, the population increases. There is a positivecorrelation between the two variables.

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Draw a line of fit for

the scatter plot.No one line will passthrough all of the datapoints. Draw a line that

passes close to thepoints. A line is shownin the scatter plot.

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Write the slope-intercept form of an equation for

equation for the line of fit.The line of fit shown passes through the data points(1850 , 1000 ) and ( 1998 , 5900 ).

Step 1 Find the slope.

Slope formula

Letand

Simplify.

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Step 2 Use m = 33.1 and either the point-slope form

or the slope-intercept form to write the equation.You can use either data point. We chose(1850 , 1000) .

Point-slope form Slope-intercept form

Answer: The equation of the line is.

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Check Check your result by substituting (1998 , 5900) into

Line of fitequation

Subtract.

The solution checks.

Replace x with1998 and y with5900 .

Multiply.

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The table shows the number of bachelors degrees

received since 1988. Years Since 1988 2 4 6 8 10Bachelors DegreesReceived (thousands)

1051 1136 1169 1165 1184

Source: National Center for Education Statistics

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a. Draw a scatter plot and

determine what relationshipexists, if any, in the data.

Answer: The scatter plotseems to indicate that as the

number of years increase, thenumber of bachelors degreesreceived increases. There is apositive correlation betweenthe two variables.

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b. Draw a line of best fit for the

scatter plot.c. Write the slope-intercept

form of an equation for theline of fit.

Answer: Using (4 , 1137) and(10 , 1184) ,

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Use the prediction equation

where x is the year and y is the population (inmillions), to predict the world population in 2010.

Original equation

Replace x with 2010 .

Simplify.

Answer: 6,296,000,000

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Use the equation where x is the years

since 1988 and y is the number of bachelors degrees(in thousands), to predict the number of bachelorsdegrees that will be received in 2005.

Answer: 1,204,000

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Explore online information about theinformation introduced in this chapter.

Click on the Connect button to launch your browser and go to the Algebra 1 Web site. At this site, youwill find extra examples for each lesson in theStudent Edition of your textbook. When you finishexploring, exit the browser program to return to thispresentation. If you experience difficulty connectingto the Web site, manually launch your Web browser and go to www.algebra1.com/extra_examples .
http://opt/scribd/conversion/tmp/scratch2332/fscstart%20/al1%20/3%20101http://opt/scribd/conversion/tmp/scratch2332/fscstart%20/al1%20/5%20/104

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Click the mouse button or press theClick the mouse button or press theSpace Bar to display the answers.Space Bar to display the answers.

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Click the mouse button or press theClick the mouse button or press the

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End of Custom ShowsWARNING! Do Not Remove

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WARNING! Do Not RemoveThis slide is intentionally blank and is set to auto-advance to end

custom shows and return to the main presentation.

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