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Construct a table and graph the equation by plotting points: y = 1 2 x + 2.

x y = 1 2 x + 2 ( x , y )
−2 y = 1 2 ( −2 ) + 2 = 1 ( −2 , 1 )
−1 y = 1 2 ( −1 ) + 2 = 3 2 ( 1 , 3 2 )
0 y = 1 2 ( 0 ) + 2 = 2 ( 0 , 2 )
1 y = 1 2 ( 1 ) + 2 = 5 2 ( 1 , 5 2 )
2 y = 1 2 ( 2 ) + 2 = 3 ( 2 , 3 )
This is an image of a graph on an x, y coordinate plane. The x and y-axis range from negative 5 to 5.  A line passes through the points (-2, 1); (-1, 3/2); (0, 2); (1, 5/2); and (2, 3).
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Graphing equations with a graphing utility

Most graphing calculators require similar techniques to graph an equation. The equations sometimes have to be manipulated so they are written in the style y = _____ . The TI-84 Plus, and many other calculator makes and models, have a mode function, which allows the window (the screen for viewing the graph) to be altered so the pertinent parts of a graph can be seen.

For example, the equation y = 2 x 20 has been entered in the TI-84 Plus shown in [link] a. In [link] b, the resulting graph is shown. Notice that we cannot see on the screen where the graph crosses the axes. The standard window screen on the TI-84 Plus shows −10 x 10 , and −10 y 10. See [link] c .

This is an image of three side-by-side calculator screen captures.  The first screen is the plot screen with the function y sub 1 equals two times x minus twenty.  The second screen shows the plotted line on the coordinate plane.  The third screen shows the window edit screen with the following settings: Xmin = -10; Xmax = 10; Xscl = 1; Ymin = -10; Ymax = 10; Yscl = 1; Xres = 1.
a. Enter the equation. b. This is the graph in the original window. c. These are the original settings.

By changing the window to show more of the positive x- axis and more of the negative y- axis, we have a much better view of the graph and the x- and y- intercepts. See [link] a and [link] b.

This is an image of two side-by-side calculator screen captures.  The first screen is the window edit screen with the following settings: Xmin = negative 5; Xmax = 15; Xscl = 1; Ymin = -30; Ymax = 10; Yscl = 1; Xres =1.  The second screen shows the plot of the previous graph, but is more centered on the line.
a. This screen shows the new window settings. b. We can clearly view the intercepts in the new window.

Using a graphing utility to graph an equation

Use a graphing utility to graph the equation: y = 2 3 x 4 3 .

Enter the equation in the y= function of the calculator. Set the window settings so that both the x- and y- intercepts are showing in the window. See [link] .

This image is of a line graph on an x, y coordinate plane. The x-axis has numbers that range from negative 3 to 4. The y-axis has numbers that range from negative 3 to 3.  The function y = -2x/3 + 4/3 is plotted.
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Finding x- Intercepts and y- Intercepts

The intercepts    of a graph are points at which the graph crosses the axes. The x- intercept    is the point at which the graph crosses the x- axis. At this point, the y- coordinate is zero. The y- intercept is the point at which the graph crosses the y- axis. At this point, the x- coordinate is zero.

To determine the x- intercept, we set y equal to zero and solve for x . Similarly, to determine the y- intercept, we set x equal to zero and solve for y . For example, lets find the intercepts of the equation y = 3 x 1.

To find the x- intercept, set y = 0.

y = 3 x 1 0 = 3 x 1 1 = 3 x 1 3 = x ( 1 3 , 0 ) x −intercept

To find the y- intercept, set x = 0.

y = 3 x 1 y = 3 ( 0 ) 1 y = −1 ( 0 , −1 ) y −intercept

We can confirm that our results make sense by observing a graph of the equation as in [link] . Notice that the graph crosses the axes where we predicted it would.

This is an image of a line graph on an x, y coordinate plane. The x and y-axis range from negative 4 to 4.  The function y = 3x – 1 is plotted on the coordinate plane

Given an equation, find the intercepts.

  1. Find the x -intercept by setting y = 0 and solving for x .
  2. Find the y- intercept by setting x = 0 and solving for y .

Finding the intercepts of the given equation

Find the intercepts of the equation y = −3 x 4. Then sketch the graph using only the intercepts.

Set y = 0 to find the x- intercept.

y = −3 x 4 0 = −3 x 4 4 = −3 x 4 3 = x ( 4 3 , 0 ) x −intercept

Set x = 0 to find the y- intercept.

y = −3 x 4 y = −3 ( 0 ) 4 y = −4 ( 0 , −4 ) y −intercept

Plot both points, and draw a line passing through them as in [link] .

This is an image of a line graph on an x, y coordinate plane. The x-axis ranges from negative 5 to 5. The y-axis ranges from negative 6 to 3.  The line passes through the points (-4/3, 0) and (0, -4).
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Find the intercepts of the equation and sketch the graph: y = 3 4 x + 3.

x -intercept is ( 4 , 0 ) ; y- intercept is ( 0 , 3 ) .

This is an image of a line graph on an x, y coordinate plane. The x and y axes range from negative 4 to 6.  The function y = -3x/4 + 3 is plotted.
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Using the distance formula

Derived from the Pythagorean Theorem , the distance formula    is used to find the distance between two points in the plane. The Pythagorean Theorem, a 2 + b 2 = c 2 , is based on a right triangle where a and b are the lengths of the legs adjacent to the right angle, and c is the length of the hypotenuse. See [link] .

Questions & Answers

A laser rangefinder is locked on a comet approaching Earth. The distance g(x), in kilometers, of the comet after x days, for x in the interval 0 to 30 days, is given by g(x)=250,000csc(π30x). Graph g(x) on the interval [0, 35]. Evaluate g(5)  and interpret the information. What is the minimum distance between the comet and Earth? When does this occur? To which constant in the equation does this correspond? Find and discuss the meaning of any vertical asymptotes.
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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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