When it comes to functions and lines, the y-intercept is the point at which that function crosses the y-axis.
Another way of thinking about this is that the y-intercept is the point on the function where x equals 0.
In this video, we are going to discuss some helpful methods for how to find the y-intercept when you are not given the function.
There are three methods that can be used to find the y-intercept of a linear function, and the right method depends on what information is already known.
The first method is used whenever you know the slope of the line and the coordinates of a point on the line.
Whenever this information is known, you can quickly solve for the value of the y-intercept.
Remember, linear functions can be written in slope-intercept form, y equals mx plus b, where m is the slope, and b is the y-intercept.
Let’s try an example of this method now.
Find the y-intercept of the linear function with a slope of negative 3 that passes through the point (1,5).
From this information, we know the slope, m, is equal to negative 3, and that when x equals 1, the output y is equal to 5.
Let’s plug those numbers into the slope-intercept form of a line.
So, we have our slope intercept form, y equals mx plus b, and then we're going to plug in our values for y, m, and x.
So, we said y is equal to 5, m is equal to negative 3, and x is equal to 1.
So we have, 5 equals, negative 3 times 1 plus b.
Now the only unknown is b, which is the y-intercept.
From here, we can solve for b by multiplying negative 3 and 1.
So, we'll get, 5 equals, negative 3 plus b, and then we add 3 to both sides, so we'll get, 8 is equal to b.
So, by plugging the slope and the known x- and y-coordinates into the slope-intercept form of a line, y equals mx plus b,
we were able to solve for the value of the y-intercept and see that the y-intercept is (0,8).
A second method for finding the y-intercept of a linear function is used when the coordinates of two points are given.
In this case, the slope is not known yet.
This second method is much like the first, but with an added step.
We first need to determine the slope so that we can then reuse the technique discussed in the first method.
To acquire the slope of the function, we can use the rise-over run formula, where m is equal to the change in y divided by the change in x.
Once the slope is determined we can quickly apply the first method once again to find b.
Let’s try this with an example.
Find the y-intercept of the linear function that passes through the points (4, negative 1) and (6,0).
Let’s start by using the slope formula to determine the slope of this function.
So, slope is equal to the change in y, so we have 0 minus negative 1, over the change in x, so 6 minus 4.
So, then we can simplify this and we'll get 1 over 2, so the slope is one-half.
Now that we have the slope of the function, let's plug m and one of the points into the slope-intercept form equation.
Either of the two points will work, so I'm going to use (6,0).
Let's write out the function, y equals mx plus b, and then we're going to plug in what we know.
So, we know y is equal to 0, m is equal to one-half, x is equal to 6, and then we're solving for b.
Now all we have to do is simplify.
So, we'll have 0 is equal to, one-half times 6 is 3, plus b, then to get b by itself we can subtract 3 from both sides and get, negative 3 is equal to b.
And just like that, we now know that the y-intercept of this function is (0, negative 3).
This second technique is much like the first but with the added step of finding the slope.
The third technique for finding the y-intercept of a linear function is used whenever an equation of the function is already known
but not necessarily in slope-intercept form.
For example, if we need to know the y-intercept of the function that follows the equation, 10 plus 5y equals negative 5x, we can find it with a simple trick.
Remember, the y-intercept is the point of a function that lies on the y-axis.
It is also the point you get when you plug in x equals 0.
To find the y-intercept from an equation, plug in x equals 0 and simplify.
So, for this equation, we're going to start by writing our equation out.
We have, 10 plus 5y equals negative 5x, and then we plug in 0 for x, so we have, 10 plus 5y equals, negative 5 times 0.
And all we have to do is simplify and solve for y.
So, we have, 10 plus 5y is equal to, negative 5 times 0, which is 0, and then we subtract 10 from both sides which gives us, 5y equals negative 10.
And we divide by 5 on both sides to get, y is equal to negative 2.
So, the y-intercept of this line is (0, negative 2).
One final method for finding y-intercepts can be applied for parabolic functions.
If the equation of a quadratic function can be arranged in standard form, y equals a x-squared plus bx plus c, then the y-intercept will occur at height c.
This is because when x equals 0, the first and second terms go to 0 and only the constant terms remain.
So, for the function, y equals, x minus 2, squared minus 3, we can determine the y-intercept by expanding, simplifying, and locating the constant term.
So, we're going to start by foiling out the squared portion.
So, we're going to start by rewriting it as, x minus 2, times x minus 2, minus 3
and then if we FOIL this, we'll get, y equals x-squared minus 2x, minus 2x, plus 4, minus 3.
And now we're going to combine like terms.
So, we'll have, y equals x-squared, minus 2x minus 2x is minus 4x, and plus 4 minus 3 is plus 1.
Now that we have the function in standard form, we can see that c equals 1, so the y-intercept is at (0,1).
Note that this could also be found by plugging in 0 for x and solving for y, like we did before.
To find the y-intercept of a linear function, you can either solve for b from, y equals mx plus b, if you are given one or more points
(where you must determine the slope sometimes),
or you can plug in x equals 0 and solve for y if you are given an equation of the function.
The y-intercept of a parabola can be found by writing the function in standard form and locating the constant term.
You’re now ready to try some examples on your own!
I hope this video was helpful.
Thanks for watching, and happy studying!