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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: equation vs. graph

Sal is given the formula of a linear function and the graph of another, and is asked to determine which function increases faster. Created by Sal Khan.

## Want to join the conversation?

- If ax + by + c = 0 is a linear equation in two variables. Is x + 0y + 7 = 0 a linear equation in 2 variables?(16 votes)
- Yes, the second example is also a linear equation in two variables. Similarly,
`0x + 0y + 0z = 0`

is a linear equation in three variables, albeit one with an infinite number of solutions.

Having a coefficient of 0 doesn't mean that the variable goes away. It still exists, it is simply irrelevant and can take on any value without affecting the validity of the equation.(7 votes)

- my teacher wants me to (sooner or later) describe a linear function. how should I do that?(7 votes)
- Zachary,

Lets break up the word. Linear means something that makes a line (pretty simple,) and a function is a set of coordinates or points on a graph that have only one x value to every y value. So basically, a linear function does not curve, but goes in a strait line on a graph. If it curves, this is called a non-linear function.

I hope this helped!(19 votes)

- At0:30, how did Sal plot a "g" out of nowhere?(9 votes)
- He wasn't plotting a point named "g". He was showing that the line on the graph is the function "g(x)"; this is stated in the text of the exercise.(11 votes)

- At2:12, if both are increasing then don't the negatives cancel out and become positive?(4 votes)
- You are comparing the slopes, not multiplying the slopes.

Both lines are negative, so both lines slant down from left to right. The slope of line F is decreasing faster because its slope is more negative than the slope of line G.

Hope this helps.(4 votes)

- The way this was laid out confused me. I thought that the equation was referring to the graph as well, and I just assumed that they were representing the same thing until we actually broke down the slope of the graph.. The whole 'g is a function of x' thing is a bit confusing, and I wish that we went over that more in the functions lesson.

Hopefully we will touch on that in the next few videos.(6 votes) - is y=1/x - 7 a linear function(1 vote)
- No, it isn't. Anything that has the formula y=m/x + q is a hyperbola function.(10 votes)

- Graph the linear function with the equation y=1/2x+3(3 votes)
- Your equation is in slope-intercept form already so graphing it is pretty easy.

Just looking at the equation, you can tell it crosses the y-axis at (0,3) so plot that point.

Now use the slope (1/2) to find another point. Your slope tells you to go up one and over to the right 2 to get to the next point. Since it's a positive number, your line will slope upwards.

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|f you aren't comfortable with the slop-intercept form yet, I suggest:

https://www.khanacademy.org/math/algebra-home/alg-linear-eq-func/alg-slope-intercept-form/v/slope-intercept-form(5 votes)

- then why are you here?(2 votes)

- I dont understand where is line f. How do we know that it is decreasing? And when calculating the slope of f ,why is just -7/3x considered ,and not +1 (f = -7/3x + 1)(4 votes)
- f is only shown as an equation. Its slope is -7/3. The negative means that it is a downward slope. [-7/3 is a downward slope. 7/3 would be an upwards slope]. The slope of the line, g is -2/1. When calculating the slope we want to figure out the numbers before x

example: -7/3 x . The +1 is the point where the line crosses the y axis. It isn't used to calculate slope unless we are using it as one of two points to use the y2-y1 / x2-x1 process. We are not using this process because we already know the slope is -7/3.(2 votes)

## Video transcript

Two functions, f and
g, are described below. Which of these statements
about f and g is true? So they defined
function f as kind of a traditional linear
equation right over here. And this right over here is g. So this right over
here is g of x. And that also looks
like a linear function. We see it's a kind of a
downward sloping line. So let's look at our choices
and see which of these are true. f and g are both increasing, and
f is increasing faster than g. Well, when I look at g--
Well, first of all, g is definitely decreasing. So we already know
that that's false. And f is also decreasing. We see here it has
a negative slope. Every time we move forward
3 in the x direction, we're going to move down 7
in the vertical direction. So neither of these
are increasing so that's definitely not right. f and g are both increasing. Well, that's
definitely not right. So we know that both f
and g are decreasing. So this first choice says
they're both decreasing, and g is decreasing faster than f. So let's see what
the slope on g is. So the slope on g is every time
we move 1 in the x direction, positive 1 in the
x direction, we move down 2 in the y direction. So for g of x, if we were
to write our change in y over our change in x-- which
is our slope-- our change in y over change in x, when we
move one in the x direction, positive 1 in the
x direction, we move down 2 in the y direction. So our change in y over
change in x is negative 2. So g has a slope of negative 2. f has a slope of negative 7/3. Negative 7/3 is the same
thing as negative 2 and 1/3. So f's slope is more negative. So it is decreasing faster. So g is not decreasing faster
than f. f is decreasing faster than g. So this is not right. And then we have this choice--
f and g are both decreasing, and f is decreasing
faster than g. This is right, right over here. We have this last choice-- g is
increasing but f is decreasing. We know that's not true.
g is actually decreasing.