Linear Function

Question

Write an equation for a linear function whose graph has the given characteristics.

Passes through (−2, 4) and (2, −14)

Answer 1

The word "linear function" is specifically referring to a line. Something you should commit to memory is that all lines have the following form: y = mx + b, where m is the slope, and b is the y-intercept. Thus, if you know the slope and y-intercept of a line, then you can write down its equation. The slope is defined by m = Δy/Δx (the "Delta" symbol Δ means "change in," so the slope is simply the change in y divided by the change in x), and the y-intercept b is defined by the value of y when x = 0 (you can see this by looking at the equation I provided. Plug in x = 0, and we're left with y = b).

So let's get back to the problem, and let's start with what we know. The line passes through two points, (-2, 4) and (2, -14). Something to know about lines is that the slope is the same everywhere on the line, so no matter what two points on the line we choose, we will always have m = Δy/Δx. Therefore, all we need to do is find the change in y and the change in x, then we have our slope, and we're halfway there! In this case, the change in y is: Δy = -14 - 4 = -18, and the change in x is: Δx = 2 - (-2) = 2 + 2 = 4, so the slope is m = Δy/Δx = -18/4 = -9/2, where in the last equality we simply cancel the common factor of 2 in the numerator and denominator. It's worth mentioning that to calculate m, I chose the second point, (2, -14) as the first coordinate and (-2, 4) as the second, but if you calculated Δy and Δx from the point (-2, 4) to (2, -14), you obtain the same slope.

So what about b? Well, again, let's look at what we know. We have two points that each are on the line, so they must satisfy the equation: y = mx + b. Furthermore, we just found that m=-9/2. Therefore, if we plug in the x and y values of one of the points into our linear equation, then we should be able to solve for b! Let's see how this works with the point (-2, 4). The y value is 4, and the x value is -2, so we have: 4 = -2(-9/2) + b. -2(-9/2) = 9, so we have 4 = 9 + b. Finally, we subtract 9 from both sides to isolate b to obtain b = -5. You should again check for yourself that had we chosen the point (2, -14), we would have obtained the same value of the y-intercept, since both points exist on the same line!

So, putting it all together, our final answer for the equation of the line is: y = -9/2 x - 5