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## 8th grade

### Course: 8th grade > Unit 3

Lesson 10: 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: same 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 has the same rate of change as the function represented in the table. Created by Sal Khan.

## Want to join the conversation?

- Why is f and y used interchangeably?(19 votes)
- Functions are like that. When you're describing functions, you write them as f(x).

F(x) is basically the same thing as y.

Hope this helps.(46 votes)

- Let's look at a concrete example. Suppose Alice spends $2,333 a day on cheese. I don't know why, she must really like cheese. Suppose Bob spends $2,000 a day on cheese. Who is spending the most?

The answer is Alice. But notice that I didn't tell you how much money Alice or Bob started with? Would it make a difference to your answer if Alice started with $1,000,000, or if she started with $4,000? No - we don't care about where we start or even where we end up, we just care about how fast the number is changing, and so we don't have to worry about that extra constant (the +1).(7 votes) - What is the difference of a linear function, and a regular function?(2 votes)
- A linear function is a function that is first order -- in other words, its graph is a straight line. All the other functions do not graph as a straight line.(2 votes)

- Isn't there a typo in this video?

It says that the function is increasing at a rate of 7/4, and yet the constant of 8/14 doesn't follow that rule.

Am I misunderstanding something or was it just a small mistake?(3 votes)- Where do you see a constant of 8/14 in the video - could you give a time stamp? Are you talking about the point (8,13)? We do not do slope by points, but by changes in x and y, so 13-6=7 and 8-4=4, slope of 7/4.(3 votes)

- At2:08what are the triangles used for?(2 votes)
- The triangles are the greek symbol for delta, which in this case represents the change(5 votes)

- At1:30, what is the exact rate of change? I can't tell precicely from the graph! Please help!(4 votes)
- find any 2 points where you know the x and y coordinates(0 votes)

- what is the diffrence between linear function and normal funtions(2 votes)
- This question has no meaning because there is no such thing as a "normal function." A linear function is one where both variables have a maximum exponent of 1. A quadratic function has the independent variable with an exponent of 2 and the dependent variable exponent 1. A exponential function has the independent variable in the exponent. An absolute value function has the independent variable within the absolute value bars. A rational function has the independent variable in the denominator. All are just functions, what do you mean by a normal function?(4 votes)

- At1:35, how can we know that line A is increasing faster than line F and line C is increasing slower than line F?(2 votes)
- the slope is a indication of how fast a line is increasing, so A is steeper than F and C is flatter than F.(3 votes)

- Couldn't you just find the slope of each line and check if it is equal to your f and x values?(2 votes)
- Doesn't F(x) equal y? But in this video f is y.(2 votes)
- @JustKidding

Great Question! Sometimes the equation is written with function notation, f(x), instead of y. It means the same thing, but shows what input value was used to find the output.

Hope this helps.(1 vote)

## Video 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.