Chapter 1: Equations and inequalities

Chapter 2:Linear equations and functionsBIG IDEAS:Representing relations and functionsGraphing linear equations and inequalities in two variablesWriting linear equations and inequalities in two variablesOn a blank notebook page, please complete the Prerequisite Skills on PG 70 #2-12 Even

Lesson 1: represent relations and functions

Essential questionHow do you graph relations and functions?Relation: A mapping, or pairing, of input values with output values

Domain: The set of input values of a relation

Range: The set of output values of a relation

Function: A relation for which each input has exactly one outputVOCABULARY SOLUTION

EXAMPLE 1Represent relationsConsider the relation given by the ordered pair (2, 3), (1, 1), (1, 3), (2, 2), and (3, 1).a. Identify the domain and range.The domain consists of all the x-coordinates: 2, 1, 1, 2, and 3. The range consists of all the y-coordinates: 3, 2, 1, and 3.6SOLUTION

EXAMPLE 1Represent relations Represent the relation using a graph and a mapping diagram.b.b. Graph

Mapping Diagram

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EXAMPLE 2Tell whether the relation is a function. Explain.Identify functionsa.

SOLUTIONThe relation is a function because each input is mapped onto exactly one output.8

EXAMPLE 2Tell whether the relation is a function. Explain.Identify functionsb.

The relation is not a function because the input 1 is mapped onto both 1 and 2.SOLUTION9SOLUTION

GUIDED PRACTICEfor Examples 1 and 21.Consider the relation given by the ordered pairs (4, 3), (2, 1), (0, 3), (1, 2), and (2, 4)a.Identify the domain and range.The domain consists of all the x-coordinates: 4, 2, 0 and 1, The range consists of all the y-coordinates: 3, 1, 2 and 4

10SOLUTION

GUIDED PRACTICEfor Examples 1 and 2b.Represent the relation using a table and a mapping diagram.

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GUIDED PRACTICEfor Examples 1 and 22.Tell whether the relation is a function. Explain.

ANSWERYes; each input has exactly one output.12SOLUTION

EXAMPLE 3

Use the vertical line test

The team graph does not represent a function because vertical lines at x = 28 and x = 29 each intersect the graph at more than one point. The graph for Kevin Garnett does represent a function because no vertical line intersects the graph at more than one point.13SOLUTION

EXAMPLE 4Graph an equation in two variablesGraph the equation y = 2x 1.STEP 1Construct a table of values.x21012y3113514

EXAMPLE 4

STEP 2Plot the points. Notice that they all lie on a line.STEP 3Connect the points with a line.Graph an equation in two variables

15SOLUTION

EXAMPLE 5

Classify and evaluate functionsTell whether the function is linear. Then evaluate the function when x = 4.a.f (x) = x2 2x + 7The function f is not linear because it has an x2-term.f (x) = x2 2x + 7Write function.f (4) = ( 4)2 2(4) + 7Substitute 4 for x.= 1Simplify.16SOLUTION

EXAMPLE 5

Classify and evaluate functionsb.g(x) = 5x + 8The function g is linear because it has the form g(x) = mx + b.g(x) = 5x + 8Write function.g(4) = 5(4) + 8Substitute 4 for x.= 12Simplify.17

GUIDED PRACTICEfor Examples 4 and 54.Graph the equation y = 3x 2.

ANSWER 18

GUIDED PRACTICEfor Examples 4 and 5Tell whether the function is linear. Then evaluate the function when x = 2.5. f (x) = x 1 x36. g (x) = 4 2x

ANSWER ANSWER Not Linear;The f(x) = 5, when x = 2Linear;The f(x) = 0, when x = 219Essential questionHow do you graph relations and functions?Make a table of domain and range values. Then plot the points from the table. If it is a function, connect the dots. An internet company had a profit of $2.6 million in retail sales over the last five years. What was its average annual profit?

Lesson 2: Find Slope and Rate of changeEssential questionHow do you determine whether two nonvertical lines are parallel or perpendicular?Slope: The ratio of vertical change (the rise) to the horizontal change (the run) for a nonvertical lineParallel: Two lines in the same plane that do not intersectPerpendicular: Two lines in the same plane that intersect to form a right angleRate of change: A comparison of how much one quantity changes, on average, relative to the change in another quantity.VOCABULARY

Find slope in real lifeEXAMPLE 1SkateboardingA skateboard ramp has a rise of 15 inches and a run of 54 inches. What is its slope?

SOLUTIONslope =riserun=ANSWERThe slope of the ramp is 518. 518=155425

Standardized Test PracticeEXAMPLE 2SOLUTION

Let (x1, y1) = (1, 3) and (x2, y2) = (2, 1).m =y2 y1x2 x1=1 32 (1)=43ANSWERThe correct answer is A.

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GUIDED PRACTICEANSWERThe correct answer is D.2. What is the slope of the line passing through the points (4, 9) and (8, 3) ?

for Examples 1 and 2

GUIDED PRACTICEFind the slope of the line passing through the given points.3. (0, 3), (4, 8)for Examples 1 and 2ANSWER544. ( 5, 1), (5, 4)ANSWER126. (7, 3), (1, 7)ANSWER125. (3, 2), (6, 1)ANSWER13

Classify parallel and perpendicular linesEXAMPLE 4Tell whether the lines are parallel, perpendicular, orneither. Line 1: through (2, 2) and (0, 1)a.Line 2: through (4, 1) and (2, 3) Line 1: through (1, 2) and (4, 3)b.Line 2: through (4, 3) and (1, 2)SOLUTION

Find the slopes of the two lines.a.m1 =1 20 (2)= 3 2=3229

GUIDED PRACTICEfor Example 4

GUIDED PRACTICETell whether the lines are parallel, perpendicular, or neither.11. Line 1: through (2, 8) and (2, 4)Line 2: through (5, 1) and (2, 2)ANSWER12. Line 1: through (4, 2) and (1, 7)Line 2: through (1, 4) and (3, 5)ANSWERperpendicularneitherEssential questionHow do you determine whether two nonvertical lines are parallel or perpendicular?Calculate the slope:Parallel lines have equal slopePerpendicular lines have slopes that are opposite reciprocalsIn 2005, Careys Pet Shop had a profit of $55,500. In 2006, profits were $38,700. In a graph of the data, is the slope of the segment between 2005 and 2006 positive or negative?

Lesson 3: Graph equations of linesEssential questionHow do you graph a linear equation using intercepts?

Y-intercept: The y-coordinate of a point where a graph intersects the y-axisX-intercept: The x-coordinate of a point where a graph intersects the x-axisSlope-intercept form: y=mx+bStandard form: Ax + By = C where a 0

VOCABULARY

Graph linear functions EXAMPLE 1Graph the equation. Compare the graph with thegraph of y = x.a.y = 2xb.y = x + 3SOLUTIONa.

The graphs of y = 2x and y = xboth have a y-intercept of 0, but the graph of y = 2x has a slope of 2 instead of 1.36

Graph linear functionsEXAMPLE 1b.

The graphs of y = x + 3 and y = x both have a slope of 1, but the graph of y = x + 3 has a y-intercept of 3 instead of 0.37

Graph an equation in slope-intercept form EXAMPLE 2Draw a line through the two points.STEP 4

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GUIDED PRACTICEfor Examples 1 and 2Graph the equation4. y = x + 2

5. y = x + 4 25

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GUIDED PRACTICEfor Examples 1 and 2Graph the equation6. y = x 3 12

7. y = 5 + x

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GUIDED PRACTICEfor Examples 1 and 2Graph the equation8. f (x) = 1 3x

9. f (x) = 10 x

41Biology

Solve a multi-step problem EXAMPLE 3 Graph the equation. Describe what the slope and y-intercept represent in this situation. Use the graph to estimate the body length of a calf that is 10 months old.The body length y (in inches) of a walrus calf can be modeled by y = 5x + 42 where x is the calfs age (in months).42SOLUTION

Solve a multi-step problemEXAMPLE 3STEP 1Graph the equation.

STEP 2Interpret the slope and y-intercept. The slope, 5,represents the calfs rate of growth in inches per month. The y-intercept, 42, represents a newborn calfs body length in inches.43

Graph an equation in standard form EXAMPLE 4Identify the y-intercept.STEP 35(0) + 2y = 10y = 5Let y = 0.Solve for y.The y-intercept is 5. So, plot the point (0, 5).Draw a line through the two points.STEP 444

Graph horizontal and vertical linesEXAMPLE 5Graph (a) y = 2 and (b) x = 3.

a. The graph of y = 2 is the horizontal line that passes through the point (0, 2). Notice that every point on the line has a y-coordinate of 2.SOLUTIONb. The graph of x = 3 is the vertical line that passes through the point (3, 0). Notice that every point on the line has an x-coordinate of 3.45

GUIDED PRACTICEfor Examples 4 and 5Graph the equation.11.2x + 5y = 10

12.3x 2y = 12

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GUIDED PRACTICEfor Examples 4 and 5Graph the equation.13. x = 1

14. y = 4

47Essential questionHow do you graph a linear equation using intercepts?

To find x: Set y = 0, solve for x, (x,0)To find y: Set x = 0, solve for y,(0,y)Plot each intercept and connect the dots.48On a blank piece of paper please complete the Quiz for Lessons 2.1-2.3 on Page 96 #1-9. When finished please turn into the homework bin.

Lesson 4: Write equations of linesEssential questionHow do you write the equation of a line given the slope and a point?Point-slope form: y y1 = m(x-x1)VOCABULARY

Write an equation given the slope and y-interceptEXAMPLE 1Write an equation of the line shown.

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GUIDED PRACTICEfor Example 1Write an equation of the line that has the given slope and y-intercept.1. m = 3, b = 1y = x + 13ANSWER 2. m = 2 , b = 4 y = 2x 4ANSWER 3. m = , b =3472y = x +3472ANSWER 54

Write an equation given the slope and a pointEXAMPLE 2Write an equation of the line that passes through (5, 4) and has a slope of 3.Because you know the slope and a point on the line, use point-slope form to write an equation of the line. Let (x1, y1) = (5, 4) and m = 3.y y1 = m(x x1)Use point-slope form.y 4 = 3(x 5)Substitute for m, x1, and y1.y 4 = 3x + 15Distributive propertySOLUTIONy = 3x + 19Write in slope-intercept form.55

GUIDED PRACTICEfor Examples 2 and 3

GUIDED PRACTICE4. Write an equation of the line that passes through (1, 6) and has a slope of 4.y = 4x + 105. Write an equation of the line that passes through (4, 2) and is (a) parallel to, and (b) perpendicular to, the line y = 3x 1.y = 3x 14ANSWER ANSWER 56

Write an equation given two points EXAMPLE 4Write an equation of the line that passes through (5, 2)and (2, 10).SOLUTIONThe line passes through (x1, y1) = (5,2) and (x2, y2) = (2, 10). Find its slope.y2 y1m =x2 x110 (2) = 2 5 12 3== 457

Write an equation given two points

EXAMPLE 4You know the slope and a point on the line, so use point-slope form with either given point to write an equation of the line. Choose (x1, y1) = (4, 7).y2 y1 = m(x x1)Use point-slope form.y 10 = 4(x 2)Substitute for m, x1, and y1.y 10 = 4x + 8Distributive propertyWrite in slope-intercept form.y = 4x + 858

GUIDED PRACTICEfor Examples 4 and 5

GUIDED PRACTICEWrite an equation of the line that passes through the given points.6. (2, 5), (4, 7)y = 2x + 17. (6, 1), (3, 8)y = x 58. (1, 2), (10, 0)211x +2011y =ANSWER ANSWER ANSWER 59Essential questionHow do you write the equation of a line given the slope and a point?Plug m, x & y into the slope-intercept form: y = mx + bSolve for bRewrite the equation with known m and b. Write an equation of the line that passes through the points (0,0) and (4,8).

Lesson 5: Model direct variationEssential questionWhat is a constant of variation?Direct variation: Two variable x and y are this when y = ax where a is a nonzero constantConstant of variation: The nonzero constant z in a direct variation, inverse variation or a joint variation

VOCABULARY

Write and graph a direct variation equation EXAMPLE 1Write and graph a direct variation equation that has (4, 8) as a solution.SOLUTIONUse the given values of x and y to find the constant of variation.

Write direct variation equation.y = ax 8 = a(4) Substitute 8 for y and 4 for x.2 = aSolve for a.65

GUIDED PRACTICEfor Example 1Write and graph a direct variation equation that has the given ordered pair as a solution. (3, 9)ANSWER y = 3x.

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GUIDED PRACTICEfor Example 1Write and graph a direct variation equation that has the given ordered pair as a solution. (7, 4)ANSWERy = x7 4

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GUIDED PRACTICEfor Example 1Write and graph a direct variation equation that has the given ordered pair as a solution. (5, 3)ANSWERy = x5 3

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GUIDED PRACTICEfor Example 1Write and graph a direct variation equation that has the given ordered pair as a solution. (6, 2)ANSWERy = x3 1

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Use ratios to identify direct variation EXAMPLE 3 215 1.8

119 290 2.4

121 430 3.6

119 565 4.7

120 695 5.8

120 350 2.9121=ANSWERBecause the ratios are approximately equal, the data show direct variation. An equation relating tooth length and body length is = 120, or b = 120t.bt70Essential questionWhat is a constant of variation?

In the equation y = ax, a is the constant of variationIt represents the slope or rate of change of the direct variation equationA lines graph has slope 2/3 and contains the point (6,1). Write the equation of the line.

Lesson 6: Draw Scatter plots and best-fitting linesEssential questionHow can you tell if a set of data points can be modeled by a best fitting line?Scatter plot: A graph of a set of data pairs (x,y) used to determine whether there is a relationship between the variables x and y

Positive correlation: The paired data (x,y) are this way if y tends to increase as x increases (positive slope)

Negative correlation: The paired data (x,y) are this way if y tends to decrease as x increases (negative slope)

Best-fitting line: The line that lies as close as possible to all the data points in a scatter plotVOCABULARY

Describe correlation EXAMPLE 1TelephonesDescribe the correlation shown by each scatter plot.

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EXAMPLE 2Estimate correlation coefficientsTell whether the correlation coefficient for the data isclosest to 1, 0.5, 0, 0.5, or 1.

a.SOLUTIONa. The scatter plot shows a clear but fairly weak negative correlation. So, r is between 0 and 1, but not too close to either one. The best estimate given is r = 0.5. (The actual value is r 0.46.)

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Estimate correlation coefficients EXAMPLE 2b. The scatter plot shows approximately no correlation. So, the best estimate given is r = 0. (The actual value is r 0.02.)

SOLUTIONb.

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Estimate correlation coefficients EXAMPLE 2c.

c. The scatter plot shows a strong positive correlation. So, the best estimate given is r = 1. (The actual value is r 0.98.)

SOLUTION79

GUIDED PRACTICEfor Examples 1 and 2For each scatter plot, (a) tell whether the data have a positive correlation, a negative correlation, or approximately no correlation, and (b) tell whether the correlation coefficient is closest to 1, 0.5, 0, 0.5, or 1.

1. ANSWER (a) positive correlation(b) r = 0.580

GUIDED PRACTICEfor Examples 1 and 2For each scatter plot, (a) tell whether the data have a positive correlation, a negative correlation, or approximately no correlation, and (b) tell whether the correlation coefficient is closest to 1, 0.5, 0, 0.5, or 1.2.

ANSWER (a) negative correlation(b) r = 181

GUIDED PRACTICEfor Examples 1 and 2For each scatter plot, (a) tell whether the data have a positive correlation, a negative correlation, or approximately no correlation, and (b) tell whether the correlation coefficient is closest to 1, 0.5, 0, 0.5, or 1.3.

ANSWER (a) no correlation(b) r = 082

Approximate a best-fitting line EXAMPLE 3Use point-slope form to write the equation. Choose(x1, y1) = (1,300).y y1 = m(x x1) Point-slope form y 300 = 41.3(x 1) Substitute for m, x1, and y1.Simplify.y 41.3x + 259

ANSWERAn approximation of the best-fitting line is y = 41.3x + 259.83

Use a line of fit to make a prediction EXAMPLE 4ANSWER You can predict that there will be about 796,000 alternative-fueled vehicles in use in the United States in 2010.84

GUIDED PRACTICEfor Examples 3, 4 and 54. OIL PRODUCTION: The table shows the U.S. daily oil production y (in thousands of barrels) x years after 1994.

a. Approximate the best-fitting line for the data.ANSWERy = 130x + 671085

GUIDED PRACTICEfor Examples 3, 4 and 54. OIL PRODUCTION: The table shows the U.S. daily oil production y (in thousands of barrels) x years after 1994.

b. Use your equation from part (a) to predict the daily oil production in 2009.ANSWER4760 gal86

GUIDED PRACTICEfor Examples 3, 4 and 54. OIL PRODUCTION: The table shows the U.S. daily oil production y (in thousands of barrels) x years after 1994.

c. Use a graphing calculator to find and graph an equation of the best-fitting line. Repeat the prediction from part (b) using this equation.87

GUIDED PRACTICEfor Examples 3, 4 and 5ANSWER

y = 130x + 6702

The daily oil production in 2009 will be 4752 gal.88Essential questionHow can you tell if a set of data points can be modeled by a best fitting line?

You can draw a line (either with negative or positive slope) with about as many points above it as are below it.On a blank piece of paper please complete the Quiz for Lessons 2.4-2.6 on Pg 120 #1-15 ODD. Turn into the homework bin when finished.

Lesson 7: use absolute value functions and transformationsEssential questionHow do the values of a, h and k affect the graphing of y= a*f(x-h)+k in relation to the graph of y=f(x)Absolute value function: A function that contains an absolute value expressionTransformation: changes a graphs size, shape, position or orientationTranslation: A transformation that shifts a graph horizontally and/or vertically, but does not change its size, shape or orientationReflection: A transformation that flips a graph or figure in a lineVOCABULARY

EXAMPLE 1Graph a function of the form y = | x h | + kGraph y = | x + 4 | 2. Compare the graph with the graphof y = | x |.

SOLUTIONSTEP 1 Identify and plot the vertex,(h, k) = (4, 2). STEP 2 Plot another point on the graph,such as (2, 0). Use symmetry to plot a third point, (6, 0). 94

Graph a function of the form y = | x h | + kEXAMPLE 1STEP 3 Connect the points with a V-shaped graph. STEP 4 Compare with y = | x |. The graph of y = | x + 4 | 2is the graph of y = | x | translated down 2 units and left4 units.

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Graph functions of the form y = a | x | EXAMPLE 2Graph (a) y = | x | and (b) y = 3| x |. Compareeach graph with the graph of y = | x |.12SOLUTION

The graph of y = | x | is the graph of y = | x | vertically shrunk by a factor of . The graph has vertex (0, 0) and passes through (4, 2) and (4, 2).121296

Graph functions of the form y = a | x |EXAMPLE 2The graph of y = 3| x | is the graph of y = | x | vertically stretched by a factor of 3 and then reflected in the x-axis. The graph has vertex (0, 0) and passes through (1, 3) and (1, 3).

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EXAMPLE 3Graph a function of the form y = a x h + kGraph y = 2 x 1 + 3. Compare the graph with the graph of y = x .SOLUTIONIdentify and plot the vertex, (h, k) = (1, 3).

Plot another point on the graph, such as (0, 1). Use symmetry to plot a third point, (2, 1).STEP 1STEP 298

EXAMPLE 3Graph a function of the form y = a x h + kSTEP 3Connect the points with a V-shaped graph.Compare with y = x . The graph of y = 2 x 1 + 3 is the graph of y = x stretched vertically by a factor of 2, then reflected in the x-axis, and finally translated right 1unit and up 3 units.STEP 499

GUIDED PRACTICEfor Examples 1, 2 and 3Graph the function. Compare the graph with the graph of y = |x|.1. y = |x 2| + 5The graph is translated right 2 units and up 5 units

ANSWER 100

GUIDED PRACTICEfor Examples 1, 2 and 3Graph the function. Compare the graph with the graph of y = |x|.2. y = |x|14The graph is shrunk vertically by a factor of14

ANSWER 101

GUIDED PRACTICEfor Examples 1, 2 and 3Graph the function. Compare the graph with the graph of y = |x|.3. f (x) = 3| x + 1| 2The graph is reflected over the x-axis,stretched by a factor of 3, translated left 1 unit and down 2 units

ANSWER 102Essential questionHow do the values of a, h and k affect the graphing of y= a*f(x-h)+k in relation to the graph of y=f(x)

- When the absolute value of a is not equal to 1, the value of a causes a vertical shrink or stretchIf a = -1, the graph is reflected in the x-axisH translates the graph horizontallyK translates the graph vertically

Tell whether each statement is true or false when x = -2 and y =1.

2x-y 0

Lesson 8: Graph linear inequalities in two variablesEssential questionWhat does a dashed boundary line on the graph of an inequality represent?Absolute value: the distance an umber is from 0 on a number line; always positive

Extraneous solution: an apparent solution that must be rejected because it does not satisfy the original equationVOCABULARY

Standardized Test Practice EXAMPLE 1SOLUTIONOrdered pairSubstituteConclusion(6, 3)(6, 3) is not a solution(0, 2) is not a solution(0, 2)(2 , 1) is not a solution(2, 1)(3, 5) is a solution3(3) + 4(5) = 11 > 8(3, 5)

3(2) + 4(1) = 10 > 83(6) + 4( 3) = 6 > 83(0) + 4(2) = 8 > 8108

GUIDED PRACTICEfor Example 1Tell whether the given ordered pair is a solution of 5x 2y 6.Ordered pairConclusion (0, 4)(0, 4 ) is not a solution (2, 2)(2, 2 ) is a solution (3, 8)(3, 8 ) is a solution (1, 7)( 1, 7 ) is not a solutionANSWER

GUIDED PRACTICEfor Examples 2 and 3Graph the inequality in a coordinate plane. y > 1

GUIDED PRACTICEfor Examples 2 and 3Graph the inequality in a coordinate plane. x > 4

GUIDED PRACTICEfor Examples 2 and 3Graph the inequality in a coordinate plane. y > 3x

GUIDED PRACTICEfor Examples 2 and 3Graph the inequality in a coordinate plane. y < 2x +3

GUIDED PRACTICEfor Examples 2 and 3Graph the inequality in a coordinate plane. x + 3y < 9

GUIDED PRACTICEfor Examples 2 and 3Graph the inequality in a coordinate plane. 2x 6y > 9

GUIDED PRACTICEfor Examples 4 and 5Graph the inequality in a coordinate plane. y < x 2 + 1

ANSWER 116

GUIDED PRACTICEfor Examples 4 and 5Graph the inequality in a coordinate plane. y > x + 3 2

ANSWER 117

GUIDED PRACTICEfor Examples 4 and 5Graph the inequality in a coordinate plane. y < 3 x 1 3

ANSWER 118Essential questionWhat does a dashed boundary line on the graph of an inequality represent?

The points on that line are not solutions of the inequality.