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# Comparing linear functions: faster rate of change

Sal is given a table of values of a linear function and four linear graphs, and is asked to determine which graph increases faster than the function represented in the table. Created by Sal Khan.

## Want to join the conversation?

• I understand -7/3 < -9/4
but doesn't -7/3 represent the greater rate of change?
• Concur with @RasterFarlan.

Following the logic of -7/3 < -9/4 for a rate of change problem, a slope of 0 has a greater change than a slope of -200. A slope of 0 means no change, so that can't be correct.

Rate of change should be absolute value (distance from 0).

|-7/3| > |-9/4|
• On graph B why did he put down the point at x=1 and y=1, if we need to count the increase in y per 1 unit x, I would expect the "countdown" to start at zero y (aka on the x axis)

Following my trail of thought Graph B actually has a slope 6/1 and is another graph that satisfy the conditions. (Someone else also mentioned why does the exercise ask for "graphs" in plural.) Otherwise this seems really straight forward but this really confused me.
• The countdown doesn't necessarily have to start at 0. As in Graph B, the x-intercept is not an integer. The formula for slope only calls for any 2 points. Indeed, it could start at 0, but it would be harder to count.

For graph B, you could use the points (-1, -4) and (0, 1) to find the slope of the graph.
``(1 - (-4))/(0 - (-1)) = 5/1 = 5``

Since 5 is not greater than 5, it doesn't satisfy the question.

Hope this helps!
• I did the exercise and somehow I got it wrong and I don't understand why.

Since slope = Change in Y / Change in X, if 4/1 = 4, it means that when X moves 1 to the right, Y moves up 4 right? I even divided the numerator by the denominator, the value is smaller than the integer, and somehow the larger integer was the right answer (Sometimes). Please help! :(
• Your initial slope intuition is correct.

• Can anyone explain to me these please I'm having so much trouble on trying to comprehend
• Sal has to find the slope of the function first.

Slope is the change in x/change in y.
An example of this is when we have the points
(3,6) and (7,9).

First we would subtract the first x value from the second one.

7-3=4
Change in x is 4

Next we find the difference in y.It is the same process as before but we put the y values.

9-6=3
Change in y is Now we divide the change in x over change in y.

4/3=0.75
The slope is 0.75

If a line were to increase faster than this
its slope would have to be greater than 0.75

If it was increasing slower than the example it would require a slope less than 0.75
• I'm in Pre~Algebra and i'm going through this, but i still don't get y=mx+b. could i have help.
• y = mx + b is a form for writing the equation of a line (linear equation) as a function of y (as in, it shows what formula you need to use to find any "y" value on your line).

What this means is that if you know the slope (represented by the "m" variable) and the y-intercept (represented by the "b" variable), then you can plug in any value for x, and when you simplify the whole right side, you will have solved for y.

This means you can put in any value of x and get its corresponding y value, so you have a coordinate pair that you can plot on the graph. It makes finding y values and x,y pairs on your line very easy.

You can also manipulate this form to find out other information. For example, if you don't know the y-intercept, but you do know the slope and at least one point on the line, you could plug in those numbers and then solve for b, since b represents the y-intercept.

When you write the equation for a line in this form, it's called slope-intercept form.

It also helps to understand what a linear equation or linear function is, and how they can be useful. Slope-intercept form is just one way to write a linear function/linear equation. So if you don't know what a linear equation is and what it can be used for, try learning more about that, and you can better understand why y = mx + b is useful.
• At , there is a mistake. Graph B slope is 4, not 5.
• No, there is no error.
Sal went from the point (0, 1) to the point (1, 6). This makes the slope = 5/1 = 5
You may be thinking the 2nd point is (1, 5), but it isn't, The point is 1 above the 5 on the y-axis.
Hope this helps.
• I am not sure how can you define if it is decreasing and increasing on the graph.
so if x is minus than is it decreasing?
• When we say a function increases, we're talking about it's y value.
If the slope of a line is positive and you increase your x value, the y value will increase too, so we say it's increasing.
Similarly, if the slope is negative and you increase your x value, the y value will decrease, so we say it's decreasing.
• As one can see, the question asks for "which GRAPHS..." although the only answer possible is one graph. Why does the question want to fool us? Or is this just a mistake? O
• I don't think it's a mistake. But I think it would be better if he always said "...which graph, or graphs...".
• this diddnt help at all like what are we suposed to do if the slope is diffrent for instance -3.5 to -1.5 is just -3.5 -2.0=-1.5 but then -1.5 -(-0.5) is just -1 then it -1 - 1 which is 0 so how do i find my slope
• Why are you subtracting 3 times? The slope formula is:
m = (y2-y1) / (x2-x1)
There are just 2 subtractions in the formula.
What were the ordered pairs that you started with?
• What would happen if x increases by 2 on the graph, then go to an integer for f, how would that work?
• If x increases by 2, y will increase by the slope times 2.

## Video transcript

f is a linear function whose table of values is shown below. And they give us three different x-values and the corresponding f of x values. Which graphs show functions that are increasing faster than f? So when we're talking about increasing faster, we're really talking about a higher rate of change of y with respect to f, or a higher rate of change of the vertical axis with respect to the horizontal axis, which is another way of saying which of these have a steeper slope than the function f? So let's see what the change in our vertical axis is with respect to our change in our horizontal axis. Once again, the Greek letter-- this triangle is the Greek letter delta, which is shorthand for "change in." So this is the change in f over the change in x. So we see over here, when x changes by 1, the value of our function changes by positive 5. And it's linear, so that's true. Between any two points, the ratio between our change in f and our change in x is the same. If we go up 1 again, we have plus 1 in the x-direction, we are once again increasing by 5. If you start from this point and go all the way here, so if you go plus 2 along x, you're going to go plus 10 along f. So it would be 10 over 2, which is still 5. So either way, the slope, or the rate of change of the vertical axis with respect to the horizontal axis, is 5 for f. Now, let's see which of these increase faster. Well, a isn't even increasing. So A is decreasing. As x increases, y is decreasing. So that definitely can't be the case. If we look at this one right over here, it looks like-- let's see, if we start over here, if we increase 1 along the x-direction, if our change in x is 1, it looks like our change in y is exactly 5-- 1, 2, 3, 4, 5. So it looks like for choice B, our slope is exactly 5, or our change in y over change in x is exactly 5. So it's not increasing faster than f. It's increasing the same as f. Now let's look at C. So I'm going to try to find a point where it looks like I have an integer point right over here. So that's the point, negative 3, negative 3. And if I move 1 in the x-direction, it looks like I'm increasing by more than 5. I'm increasing 1, 2, 3, 4, 5, 6, 7, 8, it looks like. So this one looks like it has a slope of 8. So this one is increasing faster than f, so we'll circle that right over there. And now let's look at this choice. So if we start right over here-- and I just picked this point because that's at a nice integer coordinate. It's at the point 2, negative 4. If we increase x by 1, then we increase y by 1, 2, 3-- looks like about 3 and 1/2, definitely not 5. In order for it to increase as fast as f, it would have to increase by 5, so it would have to be up here. So it would have to go 1, 2, 3, 4, 5. It would have had to have been up here. The line would've looked something more like that just to match f, much less grow faster than f. So D does not meet the criteria. It is only C.