What are x- and y- intercepts?
- x-intercepts are where the graph crosses the x-axis
- y-intercepts are where the graph crosses the y-axis
Finding x- and y- intercepts
Line:
Examine the graph of the line to the left. This graph is missing a lot of things. It needs arrows on the axis and on both ends of the line. The graph is also missing x and y axis labels and, most importantly, axis interval labels. But, let's assume that each box is one interval:
The line crosses the x-axis at (-1,0), so -1 is the x-intercept.
The line crosses the y-axis at (0,1), so 1 is the y-intercept.
If you didn't have a graph to look at, you could draw one, or manipulate the equation. Remember y-intercept form? The equation of this graph is given to you: y = x+1. 1 is the y-intercept in this form of a line equation, so you would not even need to look at the graph. To find the x-axis intercept, you would need to solve the equation for x. Let y=0 because at the x-intercept, y will be 0:
y=x+1
0=x+1
-1=x so, even if you didn't have a graph, you could solve for the intercepts (you can apply the same substitute- 0-technique to find the y-intercept)
Examine the graph of the line to the left. This graph is missing a lot of things. It needs arrows on the axis and on both ends of the line. The graph is also missing x and y axis labels and, most importantly, axis interval labels. But, let's assume that each box is one interval:
The line crosses the x-axis at (-1,0), so -1 is the x-intercept.
The line crosses the y-axis at (0,1), so 1 is the y-intercept.
If you didn't have a graph to look at, you could draw one, or manipulate the equation. Remember y-intercept form? The equation of this graph is given to you: y = x+1. 1 is the y-intercept in this form of a line equation, so you would not even need to look at the graph. To find the x-axis intercept, you would need to solve the equation for x. Let y=0 because at the x-intercept, y will be 0:
y=x+1
0=x+1
-1=x so, even if you didn't have a graph, you could solve for the intercepts (you can apply the same substitute- 0-technique to find the y-intercept)
- lines will have no more than one x-intercept and no more than one y-intercept
Parabola:
Examine the parabola to the right. Just by glancing at the graph, we can see that there will be no x-intercept because the graph will never touch the x-axis (the vertex and lowest point the graph will ever reach is (1,1))
The x-intercept is therefore none/NA (remember that "0" is not the same as "none" in the math world).
The y-intercept is 2 because the graph touches the y-axis at (0,2).
We can see both intercepts, or lack thereof, just by looking at the graph, but we could also substitute 0 in for the variable x to find our y-intercept. However, we can't do that to find our x-intercept. Instead we would need to use the quadratic formula. See the video below for help memorizing the formula, and the parabola example below for more direction.
Examine the parabola to the right. Just by glancing at the graph, we can see that there will be no x-intercept because the graph will never touch the x-axis (the vertex and lowest point the graph will ever reach is (1,1))
The x-intercept is therefore none/NA (remember that "0" is not the same as "none" in the math world).
The y-intercept is 2 because the graph touches the y-axis at (0,2).
We can see both intercepts, or lack thereof, just by looking at the graph, but we could also substitute 0 in for the variable x to find our y-intercept. However, we can't do that to find our x-intercept. Instead we would need to use the quadratic formula. See the video below for help memorizing the formula, and the parabola example below for more direction.
In eighth grade, when I was first learning the quadratic function, my teacher made us watch this song several times everyday, and we had to stand up, sing, and clap. It was a little embarrassing and very annoying by the end, but I haven't forgotten the equation since, and my friends and I still mutter the song as we write the equation.
This parabola has one y-intercept and two x-intercepts.
We can see that the y-intercept is -1.
The x-intercepts are at -1 and 1.
But let's pretend that we couldn't see where the x-intercepts occurred. We can always use the quadratic formula:
x = (-b + SQRT(b)²-4ac)/2a
To use this formula we need to understand a little about how parabola formulas are created.
y = ax²+bx+c
Take the function to the left, all parabolas will have a term that is squared, in this case it is x². We don't have an x term, but we do have a shift down of 1. Our equation is y = x²-1. So, in the quadratic equation, our a = 1, our b = 0, and our c = -1. Now we need to plug in our terms and solve:
x = (0 + SQRT(0)²-4(1)(-1)) / 2(1)
x = (+ SQRT +4) / 2
x = + 2 / 2
x = + 1
Our x-intercepts are positive 1 and negative 1.
So, by looking at the graph or using the quadratic equation and setting x to 0, we can find the x- and y- intercepts. Although working it out is a lot more work, both ways should give you the same answer if your math is correct, and the two ways can also be used to check your solution.
We can see that the y-intercept is -1.
The x-intercepts are at -1 and 1.
But let's pretend that we couldn't see where the x-intercepts occurred. We can always use the quadratic formula:
x = (-b + SQRT(b)²-4ac)/2a
To use this formula we need to understand a little about how parabola formulas are created.
y = ax²+bx+c
Take the function to the left, all parabolas will have a term that is squared, in this case it is x². We don't have an x term, but we do have a shift down of 1. Our equation is y = x²-1. So, in the quadratic equation, our a = 1, our b = 0, and our c = -1. Now we need to plug in our terms and solve:
x = (0 + SQRT(0)²-4(1)(-1)) / 2(1)
x = (+ SQRT +4) / 2
x = + 2 / 2
x = + 1
Our x-intercepts are positive 1 and negative 1.
So, by looking at the graph or using the quadratic equation and setting x to 0, we can find the x- and y- intercepts. Although working it out is a lot more work, both ways should give you the same answer if your math is correct, and the two ways can also be used to check your solution.
- parabolas will have no more than two x-intercepts and no more than one y-intercept
Absolute Value:
Absolute value equations look a lot like a pointy parabola. To the left is the graph of the parent absolute value. The x- and y- intercepts are both 0.
Absolute value equations look a lot like a pointy parabola. To the left is the graph of the parent absolute value. The x- and y- intercepts are both 0.
- much like a parabola, absolute value equations will have no more than two x-intercepts and no more than one y- intercept
By glancing at the function to the right, we can see that there is one y-intercept and two x-intercepts. The y- intercept is 2, and the x-intercepts are at -4 and -2.
If we were unable to determine the intercepts by looking at the graph, we can plug 0 in for the variable that we are not trying to find:
Find y-intercept:
y=I0+3I-1
y=I3I-1
y=2
Find x-intercepts:
0=Ix+3I-1
0=x+3-1 and 0=x+3+1
x = -2 and x = -4
Notice that we treated the 1 like + 1.
If we were unable to determine the intercepts by looking at the graph, we can plug 0 in for the variable that we are not trying to find:
Find y-intercept:
y=I0+3I-1
y=I3I-1
y=2
Find x-intercepts:
0=Ix+3I-1
0=x+3-1 and 0=x+3+1
x = -2 and x = -4
Notice that we treated the 1 like + 1.
Square Root:
There are three types of square root equations as far as x- and y-intercepts:
Find the y-intercept:
y = SQRT(0) +3
y = 3
Find the x-intercept:
0 = SQRT(x) + 3
-3 = SQRT(x)
There is no x-intercept because the graph will never touch x=-3. The y-intercept is 3 and that is the lowest range that this graph ever reaches. We can confirm this by looking at the graph and seeing that the function never touches the x-axis. Graphs are a great way to check your work and save time.
There are three types of square root equations as far as x- and y-intercepts:
- 1 x-intercept, 1 y-intercept
- 1 x-intercept, no y-intercept/ no x-intercept, 1 y-intercept
- 0 x-intercept, 0 y-intercept
- Look at the first square root function. The equation will never touch the x-axis or the y-axis, so the function does not have an x- or y- intercept.
- Now look at the second square root function, this has 1 x-intercept and 1 y-intercept, both at (0,0). Therefore, the x- and y- intercepts are 0.
- The third square root function has only a y-intercept and no x-intercept. The y-intercept is 3, but we can see that the graph will never touch the x-axis.
Find the y-intercept:
y = SQRT(0) +3
y = 3
Find the x-intercept:
0 = SQRT(x) + 3
-3 = SQRT(x)
There is no x-intercept because the graph will never touch x=-3. The y-intercept is 3 and that is the lowest range that this graph ever reaches. We can confirm this by looking at the graph and seeing that the function never touches the x-axis. Graphs are a great way to check your work and save time.