Precalculus by Richard Wright

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Do not let your heart envy sinners, but always be zealous for the fear of the Lord. There is surely a future hope for you, and your hope will not be cut off. ‭‭Proverbs‬ ‭23‬:‭17‬-‭18‬ ‭NIV

1-02 Graphs

A picture is worth a thousand words. Equations are useful and descriptive, but a picture of the equation is far more descriptive. For example, the stock market is described by numbers of how stocks or indices are changing. However, a graph of the stocks or indices such as in in figure 1 more easily shows trends which are important for stock traders.

Graph Equations by Plotting Points

An equation in two variables can be represented by a graph. The most basic way to create a graph is to make a table. To create a table, start by solving the equation for y. Then choose values for the x-variable, and calculate the corresponding value for y. Finally, plot all the points and connect them with a line.

The x-values chosen for the table are arbitrary, but usually include both positive and negative numbers. However, if the the equation models a real-world situation, then consider what values for x are reasonable such as if x represents time, then it should probably only be positive numbers.

Example 1: Graph an Equation by Plotting Points

Graph y = 2x − 4.

Solution

Create a table with x and y rows. Pick both positive and negative values for x.

x	−6	−5	−4	−3	−2	−1	0	1	2	3	4	5	6
y

Substitute each x-value into the equation and calculate the corresponding y-value. For example, substitute x = −6.

y = 2x − 4

y = 2(−6) − 4 = −16

Fill that in the table and calculate the next number.

x	−6	−5	−4	−3	−2	−1	0	1	2	3	4	5	6
y	−16	−14	−12	−10	−8	−6	−4	−2	0	2	4	6	8

Some of the points are much farther away from the origin than others. Make the axes scale something reasonable. For this and many graphs in this book, making the scale go from −6 to 6 will produce a nice graph. Some problems will need a bigger scale, however. For this graph, plot the points that fit between −6 and 6 and draw a line through the points.

Try It 1

Graph y = −x + 2.

Answer

Graph with a Graphing Utility

Graphing calculators will make a graph of an equation. For most graphing calculators, the equation will need to be solved for y. The most commonly used graphing calculator in schools is the TI-84 Plus, but other good graphing calculators are made by other companies such as Casio and NumWorks.

TI-84

To graph on a TI-84, press the Y= button on the top left. Then enter your equation (figure 3a). To see the graph, next press the GRAPH button on the top right (figure 3b).

The viewing window may need adjusted to see the graph well. This can done by pressing the WINDOW button (figure 4a) or by choosing a zoom option in the ZOOM button menu (figure 4b). The standard window goes from −10 to 10 in both the x and the y. This can easily be set by choosing ZStandard in the Zoom menu.

To use the graphing calculator to draw the graph on your paper, use the TABLE button. Press 2ND then TABLE which is above the GRAPH button. Use that table to plot points on your paper and draw the line connecting them. Then check to make sure your graph matches the one on the calculator.

NumWorks

To graph on a NumWorks calculator, press the HOME button and select Functions. In the Functions tab, add a new function or edit the existing function. Then move to the Graph tab.

The viewing window may need adjusted to see the graph well. By default, the graph zooms to an auto window that is often huge. Use the arrow pad to select Axes. To get a window with the same scale in the x and y, set the Values of X to −10; 10 and the Values of Y to −6; 6, then select Confirm. The Zoom level of the screen can be changed by pressing the + and − buttons.

To use the graphing calculator to draw the graph on your paper, use the Table tab. By default the table starts at 0, but can be changed by selecting Set the interval or the desired x-values can simply be entered into the table.

The other brands of graphing calculators work in a similar method, but have different buttons. You may need to look at your calculator's manual to learn how to make a graph and table.

Example 2: Use a Graphing Calculator

Use a graphing calculator to graph \(y = -\frac{1}{2}x + 2\).

Solution (TI-84)

On a TI-84, start by pressing the Y= button and enter the equation.

Press GRAPH to see the graph.

To draw your graph, press 2ND TABLE to get the table of points.

Plot several of the points on your paper and draw a line through them.

Solution (NumWorks)

On a NumWorks, start by selecting Functions from the home screen. In the Functions tab, enter the equation.

Select the Graph tab to see the graph.

To draw your graph, select the Table tab. Either Set the interval or enter the desired x-values to get the table of points.

Plot several of the points on the paper and draw a line through them.

Try It 2

Use a graphing calculator to graph y = 3x − 1.

Answer

x- and y-intercepts

Two important points on a graph are the x-intercept and the y-intercept. The x-intercept is where the graph intersects the x-axis. Points on the x-axis are not up or down, so the y-coordinate is 0. To find the x-intercept, let the y be 0 and solve for x.

The y-intercept is where the graph intersects the y-axis. Points on the y-axis are not left or right, so the x-coordinate is 0. To find the y-intercept, let the x be 0 and solve for y.

Example 3: Find x- and y-intercepts

Find the (a) x-intercept and (b) y-intercept of y = 4x − 2.

Solution

1. To find the x-intercept, substitute y = 0 and solve for x.

   y = 4x − 2

   0 = 4x − 2

   2 = 4x

   $$ x = \frac{1}{2} $$

   \(\left(\frac{1}{2}, 0\right)\)

2. To find the y-intercept, substitute x = 0 and solve for y.

   y = 4x − 2

   y = 4(0) − 2

   y = −2

   (0, −2)

Try It 3

Find the (a) x-intercept and (b) y-intercept of 2x − y = 4.

Answer

(2, 0); (0, −4)

Graphs of Circles

Circles are the set of all points, (x, y) a given distance, r, from the center point (h, k). The distance formula gives the equation of a circle.

$$ r = \sqrt{(x - h)^2 + (y - k)^2} $$

Square both sides and rearrange.

$$ (x - h)^2 + (y - k)^2 = r^2 $$

Graph a circle by plotting the center and then moving the radius away from the center in every direction.

Graph a circle by plotting the center and then moving the radius away from the center in every direction. See figure 12.

Example 4: Graph a Circle

Graph \((x - 2)^2 + y^2 = 9\).

Solution

Compare \((x - 2)^2 + y^2 = 9\) with \((x - h)^2 + (y - k)^2 = r^2\) to see that h = 2, k = 0 because \((y - 0)^2 = y^2\). Also, \(r^2 = 9\), so r = 3. Thus the center is (2, 0) and the radius is 3. Plot the center and then move the distance of the radius in every direction.

Practice Exercises

1. What is the y-intercept?
2. How do you find the x- and y-intercepts?

Find the x-intercept and the y-intercept without graphing. Write the coordinates of each intercept.

3. y = −2x + 2
4. \(y = \frac{2}{3}x + \frac{1}{3}\)
5. 2x + y = −5
6. y = x² − 1

Graph the equation by making a table.

7. y = −2x + 2
8. \(y = \frac{2}{3}x + \frac{1}{3}\)
9. 2x + y = −5
10. y = x² − 1

Use your graphing calculator to find the y-intercept. On the TI-84 do this by 1) entering the equation in the Y= menu, 2) pressing the GRAPH button, and 3) using the 2ND CALC button and choosing 1:value from the menu. At the lower part of the screen you will see “x=” and a blinking cursor. You may enter any number for x and it will display the y value for any x value you input. Use this and plug in x = 0 to find the y-intercept. On a NumWorks, 1) select Grapher from the home screen. 2) Enter the equation in the expressions tab. 3) Select the Graph tab to see the graph. Pressing any number followed by EXE will move the cursor to that x-value. The y-value can be read off the bottom of the screen.

11. y = −x + 3
12. y = x − 4
13. \(y = -\frac{3}{2}x + \frac{1}{2}\)

Use your graphing calculator to find the x-intercept. On the TI-84 do this by 1) entering the equation in the Y= menu, 2) pressing the GRAPH button, and 3) using the 2ND CALC button and choosing 2:zero from the menu. At the lower part of the screen you will see “Left Bound?” and a blinking cursor on the graph of the line. Move this cursor to the left of the x-intercept, hit ENTER. Now it says “Right Bound?” Move the cursor to the right of the x-intercept, hit ENTER. Now it says “Guess?” Move your cursor to the left somewhere in between the left and right bound near the x-intercept. Hit ENTER. At the bottom of your screen it will display the coordinates of the x-intercept or the “zero”. On the NumWorks, 1) select Grapher from the home screen. 2) Enter the equation in the expressions tab. 3) Select the Graph tab to see the graph. 4) Use the arrow pad to select Calculate at the top of the graph. 5) Select Find and then 6) select Zeros. The zeros can be read of the bottom of the graph. If there are multiple zeros, the left and right arrows will alternate between them.

14. y = −x + 3
15. y = x − 4
16. \(y = -\frac{3}{2}x + \frac{1}{2}\)

For the following exercises, (a) find the center and (b) radius and (c) graph the circle.

17. x² + y² = 9
18. (x + 2)² + (y − 1)² = 4
19. (x − 3)² + y² = 8
20. (x + 3)² + (y − 2)² = 36

Mixed Review

21. (1-01) Find the distance between (1, 2) and (4, −9)
22. (1-01) Find the midpoint between (1, 2) and (4, −9)

Solve the following equations for y:

23. x + 2y = 4
24. 3x − y = 10
25. \(\frac{x + y}{2} = 12\)

Answers

1. The point where the graph intersects the y-axis.
2. Let the other variable = 0 and solve
3. (1, 0); (0, 2)
4. \(\left(-\frac{1}{2}, 0\right)\); \(\left(0, \frac{1}{3}\right)\)
5. \(\left(-\frac{5}{2}, 0\right)\); (0, −5)
6. (−1, 0), (1, 0); (0, −1)
7. 
8. 
9. 
10. 
11. ; (0, 3)
12. ; (0, −4)
13. ; (0, .5)
14. ; (3, 0)
15. ; (4, 0)
16. ; (0.333, 0)
17. (0, 0); 3;
18. (−2, 1); 2;
19. (3, 0); \(2\sqrt{2}\);
20. (−3, 2); 6;
21. \(\sqrt{130}\)
22. \(\left(\frac{5}{2}, -\frac{7}{2}\right)\)
23. \(y = -\frac{1}{2}x + 2\)
24. y = 3x − 10
25. y = −x + 24