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Construct a table and graph the equation by plotting points: $\text{\hspace{0.17em}}y=\frac{1}{2}x+2.$
$x$ | $y=\frac{1}{2}x+2$ | $\left(x,y\right)$ |
$\mathrm{-2}$ | $y=\frac{1}{2}\left(\mathrm{-2}\right)+2=1$ | $\left(\mathrm{-2},1\right)$ |
$\mathrm{-1}$ | $y=\frac{1}{2}\left(\mathrm{-1}\right)+2=\frac{3}{2}$ | $\left(-1,\frac{3}{2}\right)$ |
$0$ | $y=\frac{1}{2}\left(0\right)+2=2$ | $\left(0,2\right)$ |
$1$ | $y=\frac{1}{2}\left(1\right)+2=\frac{5}{2}$ | $\left(1,\frac{5}{2}\right)$ |
$2$ | $y=\frac{1}{2}\left(2\right)+2=3$ | $\left(2,3\right)$ |
Most graphing calculators require similar techniques to graph an equation. The equations sometimes have to be manipulated so they are written in the style $\text{\hspace{0.17em}}y=\_\_\_\_\_.\text{\hspace{0.17em}}$ 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 $\text{\hspace{0.17em}}y=2x-20\text{\hspace{0.17em}}$ 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 $\text{\hspace{0.17em}}\mathrm{-10}\le x\le 10,$ and $\text{\hspace{0.17em}}\mathrm{-10}\le y\le 10.\text{\hspace{0.17em}}$ See [link] c .
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.
Use a graphing utility to graph the equation: $\text{\hspace{0.17em}}y=-\frac{2}{3}x-\frac{4}{3}.$
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] .
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 $\text{\hspace{0.17em}}y=3x-1.$
To find the x- intercept, set $\text{\hspace{0.17em}}y=0.$
To find the y- intercept, set $\text{\hspace{0.17em}}x=0.$
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.
Find the intercepts of the equation $\text{\hspace{0.17em}}y=\mathrm{-3}x-4.\text{\hspace{0.17em}}$ Then sketch the graph using only the intercepts.
Set $\text{\hspace{0.17em}}y=0\text{\hspace{0.17em}}$ to find the x- intercept.
Set $\text{\hspace{0.17em}}x=0\text{\hspace{0.17em}}$ to find the y- intercept.
Plot both points, and draw a line passing through them as in [link] .
Find the intercepts of the equation and sketch the graph: $\text{\hspace{0.17em}}y=-\frac{3}{4}x+3.$
x -intercept is $\text{\hspace{0.17em}}\left(4,0\right);$ y- intercept is $\text{\hspace{0.17em}}\left(0,3\right).$
Derived from the Pythagorean Theorem , the distance formula is used to find the distance between two points in the plane. The Pythagorean Theorem, $\text{\hspace{0.17em}}{a}^{2}+{b}^{2}={c}^{2},$ 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] .
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