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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?
• 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.
• At , how did Sal plot a "g" out of nowhere?
• 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.
• my teacher wants me to (sooner or later) describe a linear function. how should I do that?
• 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!
• why is math like this?
• is y=1/x - 7 a linear function
• No, it isn't. Anything that has the formula y=m/x + q is a hyperbola function.
• At , if both are increasing then don't the negatives cancel out and become positive?
• 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.
• 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.
• Graph the linear function with the equation y=1/2x+3
• 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:
• i dont care
• then why are you here?
• 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)
• 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.

## 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.