Week 3.

Week 3

Due for this week… Homework 3 (on MyMathLab – via the Materials Link)  Monday night at 6pm. Read Chapter 5 (The last of the new material for MTH 208) Do the MyMathLab Self-Check for week 3. Learning team planning for week 5. Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Introduction to Graphing
3.1 Introduction to Graphing The Rectangular Coordinate System Scatterplots and Line Graphs Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

The Rectangular Coordinate System
One common way to graph data is to use the rectangular coordinate system, or xy-plane. In the xy-plane the horizontal axis is the x-axis, and the vertical axis is the y-axis. The axes intersect at the origin. The axes divide the xy-plane into four regions called quadrants, which are numbered I, II, III, and IV counterclockwise. Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Solution EXAMPLE Plotting points
Plot the following ordered pairs on the same xy-plane. State the quadrant in which each point is located, if possible. a. (4, 3) b. (3, 4) c. (1, 0) Solution a. (4, 3) Move 4 units to the right of the origin and 3 units up. Quadrant I b. (3, 4) Move 3 units to the left of the origin and 4 units down. Quadrant III c. (1, 0) Move 1 unit to the left of the origin. Not in any quadrant Try some of Q: 11-20 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Solution EXAMPLE Reading a graph
Frozen pizza makers have improved their pizzas to taste more like homemade. Use the graph to estimate frozen pizza sales in 1994 and 2000. Solution a. To estimate sales in 1994, locate 1994 on the x-axis. Then move upward to the data point and approximate its y-coordinate. b. To estimate sales in 2000, locate 2000 on the x-axis. Then move upward to the data point and approximate its y-coordinate. a. about $2.1 billion in sales b. about $3.0 billion in sales Try some of Q: 39-40 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Scatterplots and Line Graphs
If distinct points are plotted in the xy-plane, then the resulting graph is called a scatterplot. Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Making a scatterplot of gasoline prices
EXAMPLE Making a scatterplot of gasoline prices The table lists the average price of a gallon of gasoline for selected years. Make a scatterplot of the data. These price have not been adjusted for inflation. Year 1975 1980 1985 1990 1995 2000 2005 Cost (per gal in cents) 56.7 119.1 111.5 114.9 120.5 156.3 186.6 The data point (1975, 56.7) can be used to indicate the average cost of a gallon of gasoline in 1975 was 56.7 cents. Plot the data points in the xy-plane. Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Making a scatterplot of gasoline prices
EXAMPLE Making a scatterplot of gasoline prices The table lists the average price of a gallon of gasoline for selected years. Make a scatterplot of the data. These prices have not been adjusted for inflation. Year 1975 1980 1985 1990 1995 2000 2005 Cost (per gal in cents) 56.7 119.1 111.5 114.9 120.5 156.3 186.6 Try some of Q: 23-32 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Line Graphs Sometimes it is helpful to connect consecutive data points in a scatterplot with line segments. This creates a line graph. Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Use the data in the table to make a line graph.
EXAMPLE Making a line graph Use the data in the table to make a line graph. x 3 2 1 1 2 3 y 4 Plot the points and then connect consecutive points with line segments. Try some of Q: 33-38 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Linear Equations in Two Variables
3.2 Linear Equations in Two Variables Basic Concepts Tables of Solutions Graphing Linear Equations in Two Variables Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Basic Concepts Equations can have any number of variables.
A solution to an equation with one variable is one number that makes the statement true. Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Solution EXAMPLE Testing solutions to equations
Determine whether the given ordered pair is a solution to the given equation. a. y = x + 5, (2, 7) b. 2x + 3y = 18, (3, 4) Solution a. y = x + 5 b. 2x + 3y = 18 7 = 2 + 5 2(3) + 3(4) = 18 7 = 7 True 6  12 = 18 The ordered pair (2, 7) is a solution. 6  18 The ordered pair (3, 4) is NOT a solution. Try some of Q:9-18 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Tables of Solutions A table can be used to list solutions to an equation. A table that lists a few solutions is helpful when graphing an equation. Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Solution EXAMPLE Completing a table of solutions
Complete the table for the equation y = 3x – 1. x 3 1 3 y x 3 1 3 y Solution Try some of Q:19-24 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Solution EXAMPLE Graphing an equation with two variables
Make a table of values for the equation y = 3x, and then use the table to graph this equation. Solution Start by selecting a few convenient values for x such as –1, 0, 1, and 2. Then complete the table. x y –1 –3 1 3 2 6 Plot the points and connect the points with a straight line. Try some of Q: 35-40 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Solution EXAMPLE Graphing linear equations Graph the linear equation.
Because this equation can be written in standard form, it is a linear equation. Choose any three values for x. x y –4 1 4 2 Plot the points and connect the points with a straight line. Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Solution EXAMPLE Graphing linear equations Graph the linear equation.
Because this equation can be written in standard form, it is a linear equation. Choose any three values for x. x y 5 2 3 Plot the points and connect the points with a straight line. Try some of Q: 41-56 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Solution EXAMPLE Solve for y and then graphing
Graph the linear equation by solving for y first. Solution Solve for y. x y –2 1 2 3 Plot the points and connect the points with a straight line. Try some of Q: 57-68 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Finding Intercepts Horizontal Lines Vertical Lines
3.3 More Graphing of Lines Finding Intercepts Horizontal Lines Vertical Lines Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Finding Intercepts The y-intercept is where the graph intersects the y-axis. The x-intercept is where the graph intersects the x-axis. Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Solution EXAMPLE Using intercepts to graph a line
Use intercepts to graph 3x – 4y = 12. Solution The x-intercept is found by letting y = 0. The y-intercept is found by letting x = 0. The graph passes through the two points (4, 0) and (0, –3). Try some of Q: 25-44 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Solution EXAMPLE Using a table to find intercepts
Complete the table. Then determine the x-intercept and y-intercept for the graph of the equation x – y = 3. Solution Find corresponding values of y for the given values of x. x 3 1 1 3 y x 3 1 1 3 y 6 4 2 The x-intercept is (3, 0). The y-intercept is (0, –3). Try some of Q: 21-24 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Solution EXAMPLE Modeling the velocity of a toy rocket
A toy rocket is shot vertically into the air. Its velocity v in feet per second after t seconds is given by v = 320 – 32t. Assume that t ≥ 0 and t ≤ 10. a. Graph the equation by finding the intercepts. b. Interpret each intercept. Try some of Q: 85-86 Solution a. Find the intercepts. b. The rocket had velocity of 0 feet per second after 10 seconds. The v-intercept indicates that the rocket’s initial velocity was 320 feet per second. Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Solution EXAMPLE Graphing a horizontal line
Graph the equation y = 2 and identify its y-intercept. Solution The graph of y = 2 is a horizontal line passing through the point (0, 2), as shown below. The y-intercept is 2. Try some of Q: 47-54a’s Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Solution EXAMPLE Graphing a vertical line
Graph the equation x = 2, and identify its x-intercept. Solution The graph of x = 2 is a vertical line passing through the point (2, 0), as shown below. The x-intercept is 2. Try some of Q: b’s Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Solution EXAMPLE Writing equations of horizontal and vertical lines
Write the equation of the line shown in each graph. a. b. Solution a. The graph is a horizontal line. The equation is y = –1. Try some of Q: 55-62 b. The graph is a vertical line. The equation is x = –1. Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Solution EXAMPLE Writing equations of horizontal and vertical lines
Find an equation for a line satisfying the given conditions. a. Vertical, passing through (3, 4). b. Horizontal, passing through (1, 2). c. Perpendicular to x = 2, passing through (1, 2). Solution a. The x-intercept is 3. The equation is x = 3. Try some of Q: 73-80 b. The y-intercept is 2. The equation is y = 2. c. A line perpendicular to x = 2 is a horizontal line with y-intercept –2. The equation is y = 2. Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Slope and Rates of Change
3.4 Slope and Rates of Change Finding Slopes of Lines Slope as a Rate of Change Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Solution EXAMPLE Calculating the slope of a line
Use the two points to find the slope of the line. Interpret the slope in terms of rise and run. Solution (–4, 1) (0, –2) The rise is 3 units and the run is –4 units. Try some of Q: 20-26 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Calculate the slope of the line passing through each pair of points. a. (3, 3), (0, 4) b. (3, 4), (3, 2) c. (2, 4), (2, 4) d. (4, 5), (4, 2) Solution Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Calculate the slope of the line passing through each pair of points. a. (3, 3), (0, 4) b. (3, 4), (3, 2) c. (2, 4), (2, 4) d. (4, 5), (4, 2) Solution Try some of Q: 35-50 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Slope Zero slope: horizontal line Undefined slope: vertical line

Solution EXAMPLE Finding slope from a graph
Find the slope of each line. a. b. Solution a. The graph rises 2 units for each unit of run m = 2/1 = 2. b. The line is vertical, so the slope is undefined. Try some of Q: 15-19 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Solution EXAMPLE Sketching a line with a given slope
Sketch a line passing through the point (1, 2) and having slope ¾. Solution Start by plotting (1, 2). The slope is ¾ which means a rise (increase) of 3 and a run (horizontal) of 4. The line passes through the point (1 + 4, 2 + 3) = (5, 5). Try some of Q: 57-58 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Slope as a Rate of Change
When lines are used to model physical quantities in applications, their slopes provide important information. Slope measures the rate of change in a quantity. Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Solution EXAMPLE Interpreting slope
The distance y in miles that a boat is from the dock on a fishing expedition after x hours is shown below. a. Find the y-intercept. What does the y-intercept represent? b. The graph passes through the point (4, 15). Discuss the meaning of this point. c. Find the slope of the line. Interpret the slope as a rate of change. Solution a. The y-intercept is 35, so the boat is initially 35 miles from the dock. Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

The distance y in miles that a boat is from the dock on a fishing expedition after x hours is shown below. a. Find the y-intercept. What does the y-intercept represent? b. The graph passes through the point (4, 15). Discuss the meaning of this point. c. Find the slope of the line. Interpret the slope as a rate of change. Solution b. The point (4, 15) means that after 4 hours the boat is 15 miles from the dock. Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

The distance y in miles that a boat is from the dock on a fishing expedition after x hours is shown below. a. Find the y-intercept. What does the y-intercept represent? b. The graph passes through the point (4, 15). Discuss the meaning of this point. c. Find the slope of the line. Interpret the slope as a rate of change. Solution c. The slope is –5. The slope means that the boat is going toward the dock at 5 miles per hour. Try some of Q: 93-94 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Finding Slope-Intercept Form Parallel and Perpendicular Lines
3.5 Slope-Intercept Form Finding Slope-Intercept Form Parallel and Perpendicular Lines Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Finding Slope-Intercept Form

Solution EXAMPLE Using a graph to write the slope-intercept form
For the graph write the slope-intercept form of the line. Solution The graph intersects the y-axis at 0, so the y-intercept is 0. The graph falls 3 units for each 1 unit increase in x, the slope is –3. The slope intercept-form of the line is y = –3x . Try some of Q: 17-26 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Solution EXAMPLE Sketching a line
Sketch a line with slope ¾ and y-intercept −2. Write its slope-intercept form. Solution The y-intercept is (0, −2). Slope ¾ indicates that the graph rises 3 units for each 4 units run in x. The line passes through the point (4, 1). Try some of Q: 27-36 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Solution EXAMPLE Graphing an equation in slope-intercept form
Write the y = 4 – 3x equation in slope-intercept form and then graph it. Solution Plot the point (0, 4). The line falls 3 units for each 1 unit increase in x. Try some of Q: 51-60 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Parallel and Perpendicular Lines

Solution EXAMPLE Finding parallel lines
Find the slope-intercept form of a line parallel to y = 3x + 1 and passing through the point (2, 1). Sketch a graph of each line. Solution The line has a slope of 3 any parallel line also has slope 3. Slope-intercept form: y = 3x + b. The value of b can be found by substituting the point (2, 1) into the equation. Try some of Q: 67-74 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Solution EXAMPLE Finding perpendicular lines
Find the slope-intercept form of a line passing through the origin that is perpendicular to each line. a. y = 4x b. Solution a. The y-intercept is 0. Perpendicular line has a slope of b. The y-intercept is 0. Perpendicular line has a slope of Try some of Q: 71-74 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Derivation of Point-Slope Form Finding Point-Slope Form Applications
3.6 Point-Slope Form Derivation of Point-Slope Form Finding Point-Slope Form Applications Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

POINT-SLOPE FORM The line with slope m passing through the point (x1, y1) is given by y – y1 = m(x – x1), or equivalently, y = m(x – x1) + y1 the point-slope form of a line. Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 57

Solution y – y1 = m(x – x1) y − 1 = 2(x – 3) 3 – 1 ? 2(4 – 3) 2 = 2
EXAMPLE Finding a point-slope form Find the point-slope form of a line passing through the point (3, 1) with slope 2. Does the point (4, 3) lie on this line? Solution Let m = 2 and (x1, y1) = (3,1) in the point-slope form. y – y1 = m(x – x1) y − 1 = 2(x – 3) To determine whether the point (4, 3) lies on the line, substitute 4 for x and 3 for y. The point (4, 3) lies on the line because it satisfies the point-slope form. 3 – 1 ? 2(4 – 3) 2 = 2 Try some of Q: 9-24 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 58

Solution EXAMPLE Finding an equation of a line
Use the point-slope form to find an equation of the line passing through the points (−2, 3) and (2, 5). Solution Before we can apply the point-slope form, we must find the slope. Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 59

y – y1= m(x – x1) EXAMPLE continued
We can use either (−2, 3) or (2, 5) for (x1, y1) in the point-slope form. If we choose (−2, 3), the point-slope form becomes the following. y – y1= m(x – x1) If we choose (2, 5), the point-slope form with x1 = 2 and y1 = 5 becomes Try some of Q: 25-30 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 60

Solution EXAMPLE Finding equations of lines
Find the slope-intercept form of the line perpendicular to passing through the point (4, 6). Solution The line has slope m1 = 1. The slope of the perpendicular line is m2 = −1. The slope-intercept form of a line having slope −1 and passing through (4, 6) can be found as follows. Try some of Q: 45-54 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 61

Modeling water in a pool
EXAMPLE Modeling water in a pool A swimming pool is being emptied by a pump that removes water at a constant rate. After 1 hour the pool contains 8000 gallons and after 4 hours it contains 2000 gallons. How fast is the pump removing water? Find the slope-intercept form of a line that models the amount of water in the pool. Interpret the slope. Find the y-intercept and the x-intercept. Interpret each. Sketch the graph of the amount of water in the pool during the first 5 hours. The point (2, 6000) lies on the graph. Explain its meaning. Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 62

Solution y – y1= m(x – x1) y – 8000 = −2000(x – 1)
EXAMPLE continued The pump removes 8000 − 2000 gallons of water in 3 hours, or 2000 gallons per hour. The line passes through the points (1,8000) and (4, 2000), so the slope is Solution Use the point-slope form to find the slope-intercept form. y – y1= m(x – x1) A slope of −2000, means that the pump is removing 2000 gallons per hour. y – 8000 = −2000(x – 1) y – 8000 = −2000x y = −2000x + 10,000 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 63

The x-intercept of 5 indicates that the pool is emptied after 5 hours.
EXAMPLE continued The y-intercept is 10,000 and indicates that the pool initially contained 10,000 gallons. To find the x-intercept let y = 0 in the slope-intercept form. The x-intercept of 5 indicates that the pool is emptied after 5 hours. Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 64

EXAMPLE continued d. The x-intercept is 5 and the y-intercept is 10,000. Sketch a line passing through (5, 0) and (0, 10,000). e. The point (2, 6000) indicates that after 2 hours the pool contains 6000 gallons of water. Try some of Q: 63-64 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 65

Introduction to Modeling
3.7 Introduction to Modeling Basic Concepts Modeling Linear Data Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Generally, mathematical models are not exact representations of data.

Determining whether a model is exact
EXAMPLE Determining whether a model is exact A person can vote in the United States at age 18 or over. The table shows the voting-age population P in millions for selected years x. Does the equation P = 3.25x – 6297 model the data exactly? Explain. x 2000 2002 2004 P 203 211 216 Solution Source: U.S. Census Bureau To determine whether the equation models the data exactly, let x = 2000, 2002, and 2004 in the given equation. The model is not exact because it does not predict the voting-age population of 211 million in 2002. x = 2000: P = 3.25(2000) – 6297 = 203 x = 2002: P = 3.25(2002) – 6297 = 209.5 x = 2004: P = 3.25(2004) – 6297 = 216 Try some of Q: 15-20 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 68

Determining gas mileage
EXAMPLE Determining gas mileage The table shows the number of miles y traveled by a motorhome on x gallons of gasoline. Plot the data in the xy-plane. Be sure to label each axis. Sketch a line that models the data. Find the equation of the line and interpret the slope of the line. How far could this motorhome travel on 15 gallons of gasoline? x 3 6 9 P 24 48 72 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 69

Solution EXAMPLE Determining gas mileage
3 6 9 P 24 48 72 Solution a. Plot the points (3, 24), (6, 48), and (9, 72). b. Sketch a line through the points. Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 70

Solution EXAMPLE continued c. Find the slope of the line.
Now find the equation of the line passing through (3, 24) with the slope of 8. y – y1 = m(x – x1) The data are modeled by the equation y = 8x. Slope 8 indicates that the mileage of the motorhome is 8 miles per gallon. y – 24 = 8(x – 3) y – 24 = 8x – 24 y = 8x d. On 15 gallons of gasoline, the motorhome could go y = 8(15) = 120 miles. Try Q: 53 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 71

Solution EXAMPLE Modeling linear data
The table contains ordered pairs that can be modeled approximately by a line. Plot the data. Could a line pass through all five points? Sketch a line that models the data and determine its equation. x 1 2 3 4 5 y 7 9 10 11 Try some of Q: 33-40 Solution b. One possibility of the line is shown. a. y – y1 = m(x – x1) y – 5 = 3/2(x – 1) Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 72

Solution EXAMPLE Modeling with linear equations
Find a linear equation in the form y = mx + b that models the quantity y after x days. A quantity y is initially 750 and increases at a rate of 4 per day. A quantity y is initially 2300 and decreases at a rate of 50 per day. A quantity y is initially 17,875 and remains constant. Try some of Q: 27-32 Solution In the equation y = mx + b, the y-intercept b represents the initial amount and the slope m represents the rate of change. a. y = 4x + 750 b. y = −50x c. y = 17,875 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 73

Functions and Their Representations
8.1 Basic Concepts Representations of a Function Definition of a Function Identifying a Function Graphing Calculators (Optional) Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

FUNCTION NOTATION The notation y = f(x) is called function notation. The input is x, the output is y, and the name of the function is f. Name y = f(x) Output Input Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

The variable y is called the dependent variable and the variable x is called the independent variable. The expression f(4) = 28 is read “f of 4 equals 28” and indicates that f outputs 28 when the input is 4. A function computes exactly one output for each valid input. The letters f, g, and h, are often used to denote names of functions. Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Representations of a Function Verbal Representation (Words) Numerical Representation (Table of Values) Symbolic Representation (Formula) Graphical Representation (Graph) Diagrammatic Representation (Diagram) Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Solution EXAMPLE Calculating sales tax
Let a function f compute a sales tax of 6% on a purchase of x dollars. Use the given representation to evaluate f(3). Solution a. Verbal Representation Multiply a purchase of x dollars by 0.06 to obtain a sales tax of y dollars. b. Numerical Representation x f(x) $1.00 $0.06 $2.00 $0.12 $3.00 $0.18 $4.00 $0.24 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

c. Symbolic Representation f(x) = 0.06x
EXAMPLE continued c. Symbolic Representation f(x) = 0.06x d. Graphical Representation e. Diagrammatic Representation 1 ● 2 ● 3 ● 4 ● ● 0.06 ● 0.12 ● 0.18 ● 0.24 f(3) = 0.18 Try some of Q: 51-60 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 79

Solution EXAMPLE Evaluating symbolic representation (formulas)
Evaluate the function f at the given value of x. f(x) = 5x – x = −4 Solution f(−4) = 5(−4) – 3 = −20 – 3 = −23 Try some of Q: 19-30 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

A function receives an input x and produces exactly one output y, which can be expressed as an ordered pair: (x, y). Input Output A relation is a set of ordered pairs, and a function is a special type of relation. Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Function A function f is a set of ordered pairs (x, y), where each x-value corresponds to exactly one y-value. The domain of f is the set of all x-values, and the range of f is the set of all y-values. Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 82

Solution EXAMPLE Finding the domain and range graphically
Use the graph of f to find the function’s domain and range. Solution The arrows at the ends of the graph indicate that the graph extends indefinitely. Thus the domain includes all real numbers. The smallest y-value on the graph is y = −4. Thus the range is y ≥ −4. Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Try some of Q: 79-86 Slide 84

Solution EXAMPLE Finding the domain of a function
Use f(x) to find the domain of f. a. f(x) = 3x b. Solution a. Because we can multiply a real number x by 3, f(x) = 3x is defined for all real numbers. Thus the domain of f includes all real numbers. b. Because we cannot divide by 0, the input x = 4 is not valid. The domain of f includes all real numbers except 4, or x ≠ 4. Try some of Q: Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Solution EXAMPLE Determining whether a table represents a function
Determine whether the table of values represents a function. x f(x) 2 −6 3 4 −1 1 Solution The table does not represent a function because the input x = 3 produces two outputs; 4 and −1. Try some of Q: Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Vertical Line Test If every vertical line intersects a graph at no more than one point, then the graph represents a function. Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 87

Solution EXAMPLE Determining whether a graph represents a function
Determine whether the graph represents a function. Solution Any vertical line will cross the graph at most once. Therefore the graph does represent a function. Try some of Q: Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 88

8.2 Linear Functions Basic Concepts
Representations of Linear Functions Modeling Data with Linear Functions The Midpoint Formula (Optional) Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Solution EXAMPLE Identifying linear functions
Determine whether f is a linear function. If f is a linear function, find values for a and b so that f(x) = ax + b. a. f(x) = 6 – 2x b. f(x) = 3x2 – 5 Solution a. Let a = –2 and b = 6. Then f(x) = −2x + 6, and f is a linear function. b. Function f is not linear because its formula contains x2. The formula for a linear function cannot contain an x with an exponent other than 1. Try some of Q: 9-16 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Solution EXAMPLE Determining linear functions
Use the table of values to determine whether f(x) could represent a linear function. If f could be linear, write the formula for f in the form f(x) = ax + b. x 1 2 3 f(x) 4 11 18 25 Solution For each unit increase in x, f(x) increases by 7 units so f(x) could be linear with a = 7. Because f(0) = 4, b = 4. thus f(x) = 7x + 4. Try some of Q: 21-28 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 92

Solution EXAMPLE Graphing a linear function by hand
Sketch the graph of f(x) = x – 3 . Use the graph to evaluate f(4). Solution Begin by creating a table. Plot the points and sketch a line through the points. x y −1 −4 −3 1 −2 2 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 93

Solution EXAMPLE continued
Sketch the graph of f(x) = x – 3 . Use the graph to evaluate f(4). Solution To evaluate f(4), first find x = 4 on the x-axis. Then find the corresponding y-value. Thus f(4) = 1. Try some of Q: 49-58 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 94

MODELING DATA WITH A LINEAR FUNCTION
The formula f(x) = ax + b may be interpreted as follows. f(x) = ax b (New amount) = (Change) + (Fixed amount) When x represents time, change equals (rate of change) × (time). f(x) = a × x b (Future amount) = (Rate of change) × (Time) + (Initial amount) Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 95

Solution EXAMPLE Modeling the cost of a truck rental
Suppose that a moving truck costs $0.25 per mile and a fixed rental fee of $20. Find a formula for a linear function that models the rental fees. Solution Total cost is found by multiplying $0.25 (rate per mile) by the number of miles driven x and then adding the fixed rental fee (fixed amount) of $ Thus f(x) = 0.25x + 20. Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 96

Elapsed Time (hours) Temperature
EXAMPLE Modeling with a constant function The temperature of a hot tub is recorded at regular intervals. Elapsed Time (hours) 1 2 3 Temperature 102°F a. Discuss the temperature of the water during this time interval. b. Find a formula for a function f that models these data. c. Sketch a graph of f together with the data. Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 97

Solution EXAMPLE continued Elapsed Time (hours) 1 2 3 Temperature
1 2 3 Temperature 102°F a. The temperature appears to be a constant 102°F. b. Because the temperature is constant, the rate of change is 0. Thus f(x) = 0x or f(x) = 102. c. Graphing the data points, gives the following constant function. Try some of Q: Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 98

End of week 3 You again have the answers to those problems not assigned Practice is SOOO important in this course. Work as much as you can with MyMathLab, the materials in the text, and on my Webpage. Do everything you can scrape time up for, first the hardest topics then the easiest. You are building a skill like typing, skiing, playing a game, solving puzzles. NEXT TIME: Exponents and Polynomials

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Week 3.

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