§ 1.3 Graphing Equations The Graph of an Equation p 29-31 The graph of an equation in two variables is the set of points whose coordinates satisfy the equation. An ordered pair of real numbers (x,y) is said to satisfy the equation when substitution of the x and y coordinates into the equation makes it a true statement. For example, in the equation y = 2x + 6, the ordered pair (1,8) is a solution. When we substitute this point the sentence reads 8 = 8, which is true. The ordered pair (2,3) is not a solution. When we substitute this point, the sentence reads 3 = 10, which is not true. Blitzer, Intermediate Algebra, 5e – Slide #2 Section 1.3 Graphing an Equation p 29-31 EXAMPLE Graph y = 2|x| + 1. SOLUTION x y = 2|x| + 1 Ordered Pair (x,y) -2 y = 2|-2| + 1 = 2(2) + 1 = 4 + 1 = 5 (-2,5) -1 y = 2|-1| + 1 = 2(1) + 1 = 2 + 1 = 3 (-1,3) 0 y = 2|0| + 1 = 2(0) + 1 = 0 + 1 = 1 (0,1) 1 y = 2|1| + 1 = 2(1) + 1 = 2 + 1 = 3 (1,3) 2 y = 2|2| + 1 = 2(2) + 1 = 4 + 1 = 5 (2,5) Blitzer, Intermediate Algebra, 5e – Slide #3 Section 1.3 Graphing an Equation p 29-31 CONTINUED (-2,5) (2,5) (-1,3) (1,3) In graphing an equation, we try to get enough ordered pairs to get a good idea of what the graph looks like. Next, we plot these points. Finally, we connect these points with a smooth curve or line, always moving from left to right. This often gives us a picture of all ordered pairs that satisfy the equation. (0,1) Blitzer, Intermediate Algebra, 5e – Slide #4 Section 1.3 Graphing an Equation p 29-31 EXAMPLE Graphs in the rectangular coordinate system can also be used to tell a story. Try to select the graph that best illustrates the story of the population of the U.S.A. Years Population (c) Population (b) Population (a) Years Years SOLUTION Graph (c) Blitzer, Intermediate Algebra, 5e – Slide #5 Section 1.3 Graphing an Equation Do Check 4 on page 31 Look at Example 5 on page 32 Understanding the Viewing Rectangle Blitzer, Intermediate Algebra, 5e – Slide #6 Section 1.3 p 29-31 § 2.2 Graphs of Functions Graphs of Functions 105 The graph of a function is just the graph of its ordered pairs. For example, the graph of y = 3x is the set of points (x, y) satisfying y = 3x. Blitzer, Intermediate Algebra, 5e – Slide #8 Section 2.2 Graphs of Functions 106 EXAMPLE Graph the function f x 3x 1 . SOLUTION x y = -3x+1 Ordered Pair (x,y) -2 f(x) = -3(-2) + 1 = 6 + 1 = 7 (-2,7) -1 f(x) = -3(-1) + 1 = 3 + 1 = 4 (-1,4) 0 f(x) = -3(0) + 1 = 0 + 1 = 1 (0,1) 1 f(x) = -3(1) + 1 = -3 + 1 = -2 (1,-2) 2 f(x) = -3(2) + 1 = -6 + 1 = -5 (2,-5) Blitzer, Intermediate Algebra, 5e – Slide #9 Section 2.2 Graphs of Functions CONTINUED Blitzer, Intermediate Algebra, 5e – Slide #10 Section 2.2 The Vertical Line Test for Functions If any vertical line intersects a graph in more than one point, the graph does not define y as a function of x. (a) (b) (c) Use the vertical line test to identify graphs in which y is a function of x. Blitzer, Intermediate Algebra, 5e – Slide #11 Section 2.2 107 The Vertical Line Test 107 CONTINUED SOLUTION Here, three values of y correspond to one value of x. Whatever the exact values are, in this graph it is clear that y is not a function of x. The vertical line test is just a quick visual method for determining whether you have a function. (a) y is a function of x (b) y is not a function of x Blitzer, Intermediate Algebra, 5e – Slide #12 Section 2.2 (c) y is a function of x Graphs of Functions 107 EXAMPLE The figure shows the cost of mailing a first-class letter, f(x), as a function of its weight, x, in ounces. Use the graph to answer the following questions. Blitzer, Intermediate Algebra, 5e – Slide #13 Section 2.2 Graphs of Functions CONTINUED (a) Find f (3). What does this mean in terms of the variables in this situation? (b) Find f (4). What does this mean in terms of the variables in this situation? (c) What is the cost of mailing a letter that weighs 1.5 ounces? (d) What is the cost of mailing a letter that weighs 1.8 ounces? Blitzer, Intermediate Algebra, 5e – Slide #14 Section 2.2 Graphs of Functions CONTINUED SOLUTION (a) Find f (3). What does this mean in terms of the variables in this situation? f (3) = 0.83. This means that when a first-class letter weighs 3 ounces, postage costs 83 cents. Blitzer, Intermediate Algebra, 5e – Slide #15 Section 2.2 Graphs of Functions CONTINUED (b) Find f (4). What does this mean in terms of the variables in this situation? f (4) = 1.06. This means that when a first-class letter weighs 4 ounces, postage costs $1.06. Blitzer, Intermediate Algebra, 5e – Slide #16 Section 2.2 Graphs of Functions CONTINUED (c) What is the cost of mailing a letter that weighs 1.5 ounces? f (1.5) = 0.60. This means that when a first-class letter weighs 1.5 ounces, postage costs $0.60. Blitzer, Intermediate Algebra, 5e – Slide #17 Section 2.2 Graphs of Functions CONTINUED (d) What is the cost of mailing a letter that weighs 1.8 ounces? f (1.8) = 0.60. This means that when a first-class letter weighs 1.8 ounces, postage costs $0.60. Blitzer, Intermediate Algebra, 5e – Slide #18 Section 2.2 Graphs of Functions bottom of 107 Obtaining Information from Graphs A closed dot indicates that the graph does not extend beyond this point and the point belongs to the graph. An open dot indicates that the graph does not extend beyond this point and the point does not belong to the graph. An arrow indicates that the graph extends indefinitely in the direction in which the arrow points. Blitzer, Intermediate Algebra, 5e – Slide #19 Section 2.2 Graphs of Functions Problem 2 on page 111 (similar to 1 and 3 in homework) graph the given functions, f and g, in the same rectangular coordinate system. Select integers for x, starting with -2 and ending with 2. Describe how the graph of g is related to the graph of f. Blitzer, Intermediate Algebra, 5e – Slide #20 Section 2.2 106, 111 f ( x) x and g ( x) x 4 Graphs of Functions Problem 4 on page 112 (similar to 1 and 3 in homework) 106, 112 f ( x) 2 x and g ( x) 2 x 3 graph the given functions, f and g, in the same rectangular coordinate system. Select integers for x, starting with -2 and ending with 2. Describe how the graph of g is related to the graph of f. Blitzer, Intermediate Algebra, 5e – Slide #21 Section 2.2 Graphs of Functions Problem 6 on page 112 (similar to 5 in homework) graph the given functions, f and g, in the same rectangular coordinate system. Select integers for x, starting with -2 and ending with 2. Describe how the graph of g is related to the graph of f. Blitzer, Intermediate Algebra, 5e – Slide #22 Section 2.2 106, 112 f ( x) x 2 and g ( x) x 2 2 Graphs of Functions Problem 8 on page 112 (similar to 7 in homework) graph the given functions, f and g, in the same rectangular coordinate system. Select integers for x, starting with -2 and ending with 2. Describe how the graph of g is related to the graph of f. Blitzer, Intermediate Algebra, 5e – Slide #23 Section 2.2 106, 112 f ( x) x and g ( x) x 1 Graphs of Functions Problem 10 on page 112 (similar to 9 in homework) Blitzer, Intermediate Algebra, 5e – Slide #24 Section 2.2 106, 112 f ( x) x3 and g ( x) x3 1 Domain and Range 110-111 The graph of a function can be used to determine the function’s domain and range. Domain: set of inputs (found on the x axis – the collection of all x values in the graph) Range: set of outputs (found on the y axis – the collection of all y values in the graph) Blitzer, Intermediate Algebra, 5e – Slide #25 Section 2.2 Domain and Range EXAMPLE (similar to 31, 35, and 37 in homework) Do Check Point 4 on page 111 (picture in the textbook) a. domain: x | 2 x 1 b. domain: x | 2 x 1 c. domain: x | 3 x 0 range: y | 0 y 3 range: y | 1 y 2 range: y | y 3,2,1 Blitzer, Intermediate Algebra, 5e – Slide #26 Section 2.2 Domain and Range Problem 42 on page 113 Use the graph of f to determine each of the following a. dom ainof f x | x 6 b. range of f y | y 1 c. f (4) 1 Blitzer, Intermediate Algebra, 5e – Slide #27 Section 2.2 DONE