Precalculus Essentials Fifth Edition Chapter 1 Functions and Graphs If this PowerPoint presentation contains mathematical equations, you may need to check that your computer has the following installed: 1) MathType Plugin 2) Math Player (free versions available) 3) NVDA Reader (free versions available) Copyright © 2018, 2014, 2010 Pearson Education, Inc. All Rights Reserved

1.1 Graphs and Graphing Utilities

Objectives Plot points in the rectangular coordinate system. Graph equations in the rectangular coordinate system. Interpret information about a graphing utility’s viewing rectangle or table. Use a graph to determine intercepts. Interpret information given by graphs.

The Rectangular Coordinate System (1 of 2) We draw a horizontal line and a vertical line that intersect at right angles. The horizontal line is the x- axis. The vertical line is the y-axis. The point of intersection for these axes is their zero points, known as the origin.

The Rectangular Coordinate System (2 of 2) Positive numbers are shown to the right of the origin and above the origin. Negative numbers are shown to the left of the origin and below the origin.

Plotting Points in the Rectangular Coordinate System Each point in the rectangular coordinate system corresponds to an ordered pair of real numbers, (x, y). The first number in each pair, called the x-coordinate, denotes the distance and direction from the origin along the x-axis. The second number in each pair, called the y-coordinate, denotes the vertical distance and direction from the origin along the y-axis.

Example: Plotting Points in the Rectangular Coordinate System (1 of 2) Plot the point (−2, 4) To plot the point (−2, 4), we move 2 units to the left of the origin and 4 units up.

Example: Plotting Points in the Rectangular Coordinate System (2 of 2) Plot the point (4, −2) To plot the point (4, −2), we move 4 units to the right of the origin and 2 units down.

Graphs of Equations A relationship between two quantities can be expressed as an equation in two variables, such as A solution of an equation in two variables, x and y, is an ordered pair of real numbers with the following property: When the x-coordinate is substituted for x and the y-coordinate is substituted for y in the equation, we obtain a true statement.

Example: Graphing an Equation Using the Point-Plotting Method (1 of 3) Select integers for x, starting with −4 and ending with 2.

Example: Graphing an Equation Using the Point-Plotting Method (2 of 3) For each value of x, we find the corresponding value for y.

Example: Graphing an Equation Using the Point-Plotting Method (3 of 3) We plot the points and connect them.

Graphing Utilities Graphing calculators and graphing software packages for computers are referred to as graphing utilities or graphers. To graph an equation in x and y using a graphing utility, enter the equation and specify the size of the viewing rectangle. The size of the viewing rectangle sets minimum and maximum values for both the x-axis and the y-axis. The [−10,10,1] by [−10,10,1] viewing rectangle is called the standard viewing rectangle.

Example: Understanding the Viewing Rectangle What is the meaning of a [−100,100,50] by [−100,100,10] viewing rectangle? The minimum x-value is −100. The maximum x-value is 100. The distance between consecutive tick marks on the x-axis is 50. The minimum y-value is −100. The maximum y-value is 100. The distance between consecutive tick marks on the y-axis is 10.

Intercepts An x-intercept of a graph is the x-coordinate of a point where the graph intersects the x-axis. The y-coordinate corresponding to an x-intercept is always zero. A y-intercept of a graph is the y-coordinate of a point where the graph intersects the y-axis. The x-coordinate corresponding to a y-intercept is always zero.

Example: Identifying Intercepts Identify the x- and y-intercepts. The graph crosses the x-axis at (−3, 0). Thus, the x-intercept is −3. The graph crosses the y-axis at (0, 5). Thus, the y-intercept is 5.

Example: Interpret Information Given by Graphs (1 of 3) Divorce rates are considerably higher for couples who marry in their teens. The equation d = 4n + 5 models the percentage, d, of marriages that end in divorce after n years if the wife is under 18 at the time of marriage. Determine the percentage of marriages ending in divorce after 15 years when the wife is under 18 at the time of the marriage.

Example: Interpret Information Given by Graphs (2 of 3) To determine the percentage of marriages ending in divorce after 15 years when the wife is under 18 at the time of the marriage we will use the equation d = 4n + 5 where d is the percentage of marriages that end in divorce after n years. d = 4n + 5 d = 4(15) + 5 = 60 + 5 = 65 The percentage of marriages ending in divorce after 15 years when the wife is under 18 at the time of the marriage is 65%.

Example: Interpret Information Given by Graphs (3 of 3) The graph of d = 4n + 5 is shown to the left. We calculated that 65% of marriages would end in divorce after 15 years. How can we check our answer with the graph? Our answer is the point (15, 65) which is on the graph.