[MUSIC PLAYING] Let's look at our objectives for today. We'll start by defining the x and y-intercepts. We'll then do some examples finding the x and y-intercepts from a graph. And finally, we'll do some examples finding the x and y-intercepts from an equation.
Let's start by defining x and y-intercepts. The x-intercept is the location on a graph where a line or a curve intersects the x-axis. The coordinate pair of any x-intercept is x, 0 because the value of y is always 0 on the x-axis. The y-intercept is the location on the graph where a line or curve intersects the y-axis. The coordinate pair of any y-intercept is 0, y because the value of x is always 0 on the y-axis.
Now let's do an example finding the x and y-intercepts on a graph. The graph below shows the height above ground of a plane during its descent in relation to time in minutes. The y-intercept is represented by the pair 0, 30,000. This means that after 0 minutes of descent, the plane is 30,000 feet above the ground. The x-intercept of the graph is represented by the order paired 10, 0. This means that after 10 minutes, the height of the plane is 0 feet above ground.
Now let's look at an example of how to find the x and y-intercepts from an equation. An equation in two variables is an equation with terms involving two distinct variables. Most commonly, these variables are x and y. For example, the equation y equals negative 45x plus 1080 represents the balance of debt during a 24-month repayment period. The graph of this equation is also shown.
The x-intercept is found by substituting 0 for y in the equation and then solving the equation for x. Doing this, we have 0 equals negative 45x plus 1080. We start to solve this equation for x by subtracting 1080 on both sides, which gives us negative 1080 equals negative 45x. We then divide by negative 45 on both sides, which give us x equals 24. Therefore, the x-coordinate of the x-intercept is 24, and the ordered pair of the x-intercept is 24, 0. So after 24 months, the balance of debt is $0.
The y-intercept is found by substituting 0 for x in the equation and then solving the equation for y. So now, substituting 0 for x, our equation is y equals negative 45 times 0 plus 1080. Multiplying negative 45 times 0 gives us 0. We then add 0 plus 1080, which is 1080. So y equals 1080. Therefore, the y-coordinate of the y-intercept is 1080, and the ordered pair of the y-intercept is 0, 1080. So after 0 months, the balance of debt is $1,080.
Note that for equations written as y equals ax plus b, as in our example, the y-intercept can easily be defined by b, or the constant value at the end of the equation, since a times x will always be 0 when x is 0 for any value of a.
Here's another example. We have an equation in slope intercept form, y equals 5x minus 30. To find the x-intercept, we know that the value of y will be 0, so we substitute 0 for y into the equation and solve for x. This gives us 0 equals 5x minus 30. To solve for x, we start by adding 30 on both sides of the equation, which gives us 30 is equal to 5x.
We then divide by 5 on both sides, which gives us 6 is equal to x. The x-intercept can be written as the ordered pair 6 comma 0. Notice that the y value of the ordered pair is 0 because the value of y is always 0 at the x-intercept.
Next, to find the y-intercept, we substitute 0 in for x into the equation and solve for y. We know that the y-intercept will always be the value of b in our equation, because when x is 0, a times x will always equal 0 for any value of a. However, we can complete the steps to solve the equation for y to show that this is true.
Substituting 0 for x gives us y equals 5 times 0 minus 30. Simplifying the right side of the equation by multiplying 5 times 0 gives us 0 minus 30. This gives us y equals negative 30. The y-intercept can be written as the ordered pairs 0, negative 30. Notice that the x value of the ordered pair is 0 because the value of x is always 0 at the y-intercept.
Let's review our key points from today. Make sure you get these on so you can refer to them later. The x-intercept is the location on a graph where a line or curve intersects the x-axis. The coordinate pair of any x-intercept is x, 0. The y-intercept is a location on the graph where a line or a curve intersects the y-axis. The coordinate pair of any y-intercept is 0, y.
The x-intercept can be found by substituting 0 for y in an equation and solving the equation for x. And similarly, the y-intercept can be found by substituting 0 for x in an equation and solving for y. So I hope these key points and examples helped you understand a little bit more about identifying intercepts of a line. Keep using your notes and keep on practicing, and soon you'll be a pro. Thanks for watching.
An equation with terms involving two distinct variables.
The location on a graph where a line or curve intersects the x-axis: (x, 0).
The location on a graph where a line or curve intersects the y-axis: (0, y).