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Answer 1 · 2018-04-30T22:20:21

$y = - \frac{1}{2} x - 1$ $y=-1/2x-1$ if $- 6 \leq x \leq 2$ $-6<=x<=2$
$y = 2 x - 6$ $y=2x-6$ if $2 \leq x \leq 5$ $2<=x<=5$

Explanation:

There are two separate functions to find in this piecewise definition. If you look at the graph, there are also two lines on separate intervals. If we find the equations of those two lines and find out the x-value range for each, we can fill in the answer.

Start with the first line. (You'll need to know how to find the equation of a line using the slope-intercept form.)
You can see that the y-intercept is at $y = - 1$ $y=-1$ .
Also, the $\frac{r i s e}{r u n}$ $(rise)/(run)$ (the slope) is $- \frac{1}{2}$ $-1/2$ .
So the equation of the first line is: $y = - \frac{1}{2} x - 1$ $y=-1/2x-1$ .
Next, see where the line starts and ends. Looking at the graph, it starts at $x = - 6$ $x=-6$ and ends at $x = 2$ $x=2$ , so the interval is: $- 6 \leq x \leq 2$ $-6<=x<=2$ .

For the second line, use the same process. The slope is $\frac{2}{1}$ $2/1$ , and if you continue the line, it crosses the y-axis at $y = - 6$ $y=-6$ (y-intercept).
So the equation for the second line is: $y = 2 x - 6$ $y=2x-6$ .
Again, the second line starts at $x = 2$ $x=2$ and ends at $x = 5$ $x=5$ so the interval is: $2 \leq x \leq 5$ $2<=x<=5$ .

Put all this together and you get the answer:
$y = - \frac{1}{2} x - 1$ $y=-1/2x-1$ if $- 6 \leq x \leq 2$ $-6<=x<=2$
$y = 2 x - 6$ $y=2x-6$ if $2 \leq x \leq 5$ $2<=x<=5$

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