Comparing linear functions: same rate of change
About this video
Sal finds the rate of change of a linear function f from a table: when x increases by 4, f increases by 7, so the rate is 7/4. He plots two points of f, (0, -1) and (4, 6), to compare visually with four graphs, and picks graph D, which he checks rises 7 when x goes from 4 to 8.
You'll be able to
- find the rate of change of a linear function from a table.
- compare the rate of change of a table to the slope of a graph.
- check a graph's slope by counting the rise for a given run.
Key words
- rate of change
- How much the output changes for a given change in the input; here, 7/4.
- slope
- The steepness of a line: vertical change divided by horizontal change.
- linear function
- A function whose graph is a straight line, changing at a constant rate.
Transcript
f is a linear function whose table of values is shown below. So they give us different values of x and what the function is for each of those x's. Which graphs show functions which are increasing at the same rate as f? So what is the rate at which f is increasing?
When x increases by 4, we have our function increasing by 7. So we could just look for which of these lines are increasing at a rate of 7/4, 7 in the vertical direction every time we move 4 in the horizontal direction. And an easy way to eyeball that would actually be just to plot two points for f, and then see what that rate looks like visually.
So if we see here when x is 0, f is negative 1. When x is 0, f is negative 1. So when x is 0, f is negative 1. And when x is 4, f is 6, so 1, 2, 3, 4, 5, 6, so just like that. And two points specify a line. We know that it is a linear function. You can even verify it here.
When we increase by 4 again, we increase our function by 7 again. We know that these two points are on f and so we get a sense of the rate of change of f. Now, when you draw it like that, it immediately becomes pretty clear which of these has the same rate of change of f.
A is increasing faster than f. C is increasing slower. A is increasing much faster than f. C is increasing slower than f. B is decreasing, so that's not even close. But D seems to have the exact same inclination, the exact same slope, as f. So D is what we would go with.
And we could even verify it, even if we didn't draw it in this way. Our change in f for a given change in x is equal to-- when x changed plus 4, our function changed plus 7. It is equal to 7/4. And we can verify that on D, if we increase in the x-direction by 4, so we go from 4 to 8, then in the vertical direction we should increase by 7, so 1, 2, 3, 4, 5, 6, 7.
And it, indeed, does increase at the exact same rate.
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