Chapter 8 Graphs, Relations, & Functions

Chapter 8 Graphs, Relations, & Functions 8.1 Graphs of Equations 8.2 Relations 8.3 An Introduction to Functions 8.4 Functions and Their Graphs 8.5 Linear Functions & Models 8.6 Compound Inequalities 8.7 Absolute Value Equations & Inequalities

Section 8.1 Objectives 1 Graph and Equation Using the Point-Plotting Method 2 Identify the Intercepts from the Graph of an Equation 3 Interpret Graphs

y y-axis 4 3 x-axis 2 1 x 4 4 3 2 1 1 2 3 2 3 Origin 4 The Rectangular Coordinate System This coordinate system is called the rectangular or Cartesian coordinate system. Any point P can be represented by using an ordered pair (x, y) of real numbers.

y 4 3 2 1 x 4 4 3 2 1 1 2 3 2 3 4 The Rectangular Coordinate System Quadrant II x < 0, y > 0 Quadrant I x > 0, y > 0 Quadrant III x < 0, y < 0 Quadrant IV x > 0, y < 0 The coordinate system can be divided into four separate regions, or quadrants.

y (4, 3) 4 3 2 3 units up 1 x 4 4 3 2 1 1 2 3 4 units to the left on the x-axis 2 3 4 Determining Coordinates The coordinates of each point tell us how many units from the origin to travel.

Equation in Two Variables An equation in two variables, say x and y, is a statement in which the algebraic expressions involving x and y are equal. The expressions are called sides of the equation. x + 5y = 15 x2 – 2y2 = 4 y = 1 + 4x Any values of the variables that make the equation a true statement are said to satisfy the equation. The graph of an equation in two variablesx and y is the set of all ordered pairs (x, y) in the xy-plane that satisfy the equation.

Graphing Equations by Plotting Points One of the most elementary methods for graphing an equation is the point-plotting method. Values are chosen for one of the variables and the corresponding value of the remaining variable is determined by using the equation. In the equation x + 5y = 18, let x = 3. 3 + 5y = 18 Substitute. 5y = 15 Solve for y. y = 3 The point (3, 3) is on the graph of x + 5y = 18.

y (1, 4) 4 3 (1, 2) 2 (2, 1) 1 x 4 4 3 2 1 1 2 3 2 3 4 Graphing Equations by Plotting Points Example: Graph the equation y = – x + 3 by plotting points. Find three points that satisfy the equation. Plot the points and draw the line.

y (2, 7) (2, 7) 8 (1, 4) (1, 4) 6 x y = x2 + 3 y (0, 3) 4 2 y = (2)2 + 3 7 2 x 1 y = (1)2 + 3 4 8 8 6 4 2 2 4 6 0 y = (0)2 + 3 3 4 1 y = (1)2 + 3 4 6 8 2 y = (2)2 + 3 7 Graphing Equations by Plotting Points Example: Graph the equation y = x2+ 3 by plotting points. Find several points that satisfy the equation. Plot the points and draw the curve.

y 4 y-intercept 3 2 1 x 4 4 3 2 1 1 2 3 2 3 x-intercepts 4 Intercepts The intercepts are the coordinates of the points, if any, where the graph crosses or touches the coordinate axes. The x-coordinate of a point at which the graph crosses or touches the x-axis is an x-intercept,and the y-coordinate of apoint at which the graphcrosses or touches they-axis is a y-intercept.

y 4 3 2 1 x 4 4 3 2 1 1 2 3 2 3 4 Intercepts Example: Identify the x-intercepts and the y-intercept in the following graph. The x-intercepts are – 3, – 2, 1, and 3. The y-intercept is 3.

7000 6000 5000 Cost 4000 3000 2000 Minutes 1000 10000 12000 14000 16000 2000 4000 6000 8000 Interpreting Graphs Example: The following graph shows the relation between the monthly cost of a cell phone and the number of minutes used per month. a.) Find the approximate cost for using 8000 monthly minutes. b.) Approximately how many minutes can be used at a cost of $1500? Continued.

7000 6000 5000 Cost 4000 3000 2000 Minutes 1000 2000 4000 6000 8000 10000 12000 14000 16000 Interpreting Graphs Example continued: a.) It would cost approximately $3200 for 8000 minutes. b.) Approximately 3500 minutes can be used at a cost of $1500.

Section 8.2 Objectives 1 Understand Relations 2 Find the Domain and Range of a Relation 3 Graph a Relation Defined by an Equation

Understanding Relations When the elements in one set are linked to elements in a second set of data, we have a relation. If x and y are two elements in these sets and if a relation exists between x and y, then we say that x corresponds to y or that ydependsonx. y “depends” on the value we put in for x. x y Inputs Outputs

Students Ages Bob 18 John 20 Bill Mark 21 Domain and Range The domain of a relation is the set of all inputs of the relation. The range is the set of all outputs of the relation. Example: Find the domain and the range of the relation. Domain: {Bob, John, Bill, Mark} Range: {18, 20, 21}

y 4 3 2 1 x 4 4 3 2 1 1 2 3 2 3 4 Domain and Range Example: Find the domain and the range of the relation. The ordered pairs in the graph are: (– 3, 0), (– 2, 2), (– 1, 4), (0, 1), (1 – 2), and (2, 1). The domain is the set of all x-coordinates: {– 3, – 2, – 1, 0, 1, 2}. The range is the set of all y-coordinates: {0, 2, 4, 1, – 2, 1}.

y 4 3 2 1 x 4 4 3 2 1 1 2 3 2 3 4 Domain and Range Example: Find the domain and the range of the relation. The domain is {x| x is a real number}. The range is {y| y  – 1}.

x y y 3 4 4 3 0 2 2 1 3 0 x 4 4 3 2 1 1 2 3 2 3 4 Relations Defined by Equations Example: Graph the relation Determine the domain and range. The domain is {x| x is a real number}. The range is {y| y is a real number}.

Section 8.3 Objectives 1 Determine Whether a Relation Expressed as a Map or Ordered Pairs Represents a Function 2 Determine Whether a Relation Expressed as an Equation Represents a Function 3 Determine Whether a Relation Expressed as a Graph Represents a Function 4 Find the Value of a Function 5 Work with Applications of Functions

Students Ages Bob 18 John 20 Bill Mark 21 Relations as Functions A function is a relation in which each element in the domain (the inputs) of the relation corresponds to exactly one element in the range (the outputs) of the relation. This is a function because each element in the domain (students) corresponds to exactly one element in the range (ages).

Students Ages Bob 18 John 20 Bill Mark 21 Relations as Functions Example: Determine if the following relation represents a function. This is a NOT function because there is an element in the domain (Bob) that does not correspond to exactly one element in the range. (There is no way that Bob can be 18 and 20 at the same time.)

Relations as Functions Example: Determine if each relation represents a function. If it is a function, state the domain and range. a.) {(1, 3), (0,5), (– 5, 2), (1, 8), (10, – 5)} b.) {(1, 2), (0,5), (– 5, 2), (11, 6), (9, – 4), (– 2, 0)} a.) This is NOT a function because there are two ordered pairs, (1, 3), (1, 8) with the same first coordinate, but different second coordinates. b.) This is a function because there are no ordered pairs, with the same first coordinate, but different second coordinates. The domain is {– 5, – 2, 0, 1, 9, 11}. The range is {– 4, 0, 2, 5, 6}.

a.) The rule for getting from x to y is to multiply x by and add 2. Since there is only one output y that can result by performing these operations on any given input x, the equation is a function. Equations as Functions Example: Determine if each equation represents a function. b.) Notice that for any single value of x, two values of y will result. If x = 2, then y = (2)2 – 6, (– 10 or – 2). Since a single x corresponds to more than one y, the equation is not a function.

y y y x x x Vertical Line Test Vertical Line Test A set of points in the xy-plane is the graph of a function if and only if every vertical line intersects the graph in at most one point. Not a Function Function Not a Function

x is the independent variable. (It can be assigned any number in the domain.) f(x) = y is the dependentvariable. (It depends on the value of x.) f(x) Functions are often denoted by letters such as f, g, F, G, and so on. If f is a function, then for each number x in its domain, the corresponding value in the range is denoted f(x), read “f of x” or “f at x.” f(x) is the value of f at the number x.

Finding the Value of a Function Example: For the function defined by f(x) = x2 – 3x – 1, evaluate: a.) f(–2) b.) f(x + 4). a.) b.) f(x) = x2 – 3x – 1 f(x) = x2 – 3x – 1 f(–2) = (–2)2 – 3(–2) – 1 f(x + 4) = (x + 4)2 – 3(x + 4) – 1 = 4 + 6 – 1 = x2 + 8x + 16 – 3x – 12 – 1 = 9 = x2 + 5x + 3

Applications of Functions Example: The function C(x) = 1500 + 35x represents the daily cost for a local film company to produce x DVDs. a.) Identify the dependent and independent variables. b.) Evaluate C(200). Provide a verbal explanation of the meaning of C(200). a.) Because cost depends on the number of DVDs produced, the dependent variable is the cost, C, and the independent variable is the number of DVDs, x. b.) C(200) = 1500 + 35(200) = 1500 + 7000 = 8500 This is the cost (in dollars) for producing 200 DVDs.

Section 8.4 Objectives 1 Find the Domain of a Function 2 Graph a Function 3 Obtain Information from the Graph of a Function 4 Graph Functions in the Library of Functions 5 Interpret Graphs of Functions

b.) Domain of a Function When only the equation of a function is given, we agree that the domain of f is the largest set of real numbers for which f(x) is a real number. Example: Find the domain of the functions: a.) h(x) = x2 – 4 a.) The function h tells us to square a number x and then subtract 4. These operations can be performed on any real number, so the domain of the function is {x| x is a real number}. b.) The function f tells us to divide 12 by x – 5. Since division by 0 is not defined, the denominator x – 5 cannot be 0. Therefore, x can never equal 5. The domain is {x| x  5}.

y 8 6 4 (2, 0) (2, 0) 2 x 8 8 6 4 2 2 4 6 (0, – 4) 4 6 8 Domain & Range from a Graph Example: Determine the domain and range of the function. Identify the intercepts. Because the graph exists for all real numbers x, the domain of the function is {x| x is a real number}. Because the graph exists for all real numbers y greater than or equal to – 4, the range of the function is {y| y  – 4}. The x-intercepts are – 2 and 2; the y-intercept is – 4.

y 8 6 4 2 x 8 8 6 4 2 2 4 6 4 6 8 Graphing a Function When a function is defined by an equation in x and y, the graph of the function is the set of all ordered pairs (x, y) such that y = f(x). Example: Graph the function f(x) = x3.

h 8 6 4 2 t 0 15 20 5 10 Obtaining Information from a Graph Example: A young boy is swinging on a swing. In the graph below, let h be the distance he is above ground as a function of time t (in seconds). a.) What are h(2.5) and h(5)? Interpret these values. (2.5, 8) (12.5, 8) b.) What is the domain of h? What is the range of h? c.) For what values of t does h(t) = 8? Continued.

h 8 6 4 2 t 0 15 20 5 10 Obtaining Information from a Graph Example continued: a.) h(2.5) = 8 and h(5) = 0 After 2.5 seconds, the boy is 8 feet in the air. After 5 seconds, he is 0 feet high. 5 is a zero of the function h. (2.5, 8) (12.5, 8) b.) The domain of h is {t| 0  t  15}. The range of h is {h| 0  h  8. c.) The points on the graph for which h(t) = 8 are (2.5, 8), (7.5, 8) and (12.5,8).

Graphs in the Library of Functions

Interpreting Graphs of Functions Example: A college student is heating a jar of spaghetti sauce to put on his pasta. Because the jar has already been opened, he takes it out of the refrigerator, dumps the contents into a bowl and microwaves it for 60 seconds. After taking it out of the microwave, he lets it cool for 30 seconds before putting it on his pasta and eating it. Draw a graph that represent the temperature of the sauce as a function of time. Continued.

Microwave Temperature T Eating Temperature Temperature (F°) Refrigerator Temperature t 90 120 30 60 Time (seconds) Interpreting Graphs of Functions Example continued: The sauce is first taken out of the refrigerator. It is then heated for 60 seconds in the microwave. It is then cooled for 30 seconds before eating.

Section 8.5 Objectives 1 Graph Linear Functions 2 Find the Zero of a Linear Function 3 Build Linear Models from Verbal Descriptions 4 Build Linear Models from Data

Graphs of Linear Functions Continued.

Graphs of Linear Functions

Linear Functions A linear function is a function of the form f(x) = mx + b where m and b are real numbers. The graph of a linear function is called a line. Example: Graph the linear function f(x) = 3x – 5. Continued.

Find the Zero: Linear Function Example continued: Comparing f(x) = 3x – 5 with f(x) = 3x – 5, the slope m is 3 and the y-intercept is –5. Plot (0, –5). Because from (0, –5), go right 1 and up 3 and end at (1, –2) Draw a line through these two points.

Find the Zero: Linear Function Example Find the zero of f(x) = –4x + 12. We find the zero by solving f(x) = 0. f(x) = 0 –4x + 12 = 0 –4x = –12 x = 3 Since f(3) = –4(3)+ 12 = 0, the zero of f is 3.

Chapter 8 Graphs, Relations, & Functions