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### Course: Algebra (all content) > Unit 3

Lesson 12: Comparing linear functions- Comparing linear functions: equation vs. graph
- Comparing linear functions: same rate of change
- Comparing linear functions: faster rate of change
- Compare linear functions
- Comparing linear functions word problem: climb
- Comparing linear functions word problem: walk
- Comparing linear functions word problem: work
- Comparing linear functions word problems

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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?(7 votes)- 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|(10 votes)

- 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.(7 votes)- 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!(7 votes)

- Can anyone explain to me these please I'm having so much trouble on trying to comprehend(7 votes)
- 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(6 votes)

- 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! :((4 votes)- Your initial slope intuition is correct.

As for your confusion over the result, I need more information on the actual problem given.(10 votes)

- I'm in Pre~Algebra and i'm going through this, but i still don't get y=mx+b. could i have help.(4 votes)
- 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.(6 votes)

- At2:05, there is a mistake. Graph B slope is 4, not 5.(2 votes)
- 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.(9 votes)

- 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?(2 votes)- 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.(5 votes)

- 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(3 votes)
- I don't think it's a mistake. But I think it would be better if he always said "...which graph, or graphs...".(4 votes)

- What would happen if x increases by 2 on the graph, then go to an integer for f, how would that work?(3 votes)
- If x increases by 2, y will increase by the slope times 2.(3 votes)

- what do you mean by respect(3 votes)
- When you hear the phrase “with respect to f” you can substitute it with “in relation to f” or “relative to f”. Sorry it took one year for anyone to answer you. You probably figured it out by now since it is everywhere in algebra and trigonometry.(3 votes)

## 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.