How Do You Use X-Intercepts to Find Y-Values?

Understanding x-intercepts and how to use them to find y-values is a fundamental skill in algebra. The x-intercept of a function is a point where the graph of the function crosses the x-axis, meaning the y-value of that point is 0. This guide will walk you through the process of finding x-intercepts and using those points to determine y-values for various x-values.

What Are X-Intercepts?

An x-intercept is a point (x, y) on the graph of a function where the y-value is 0. To find the x-intercept, set the y-value equal to 0 and solve for x. This process is essential for understanding the behavior of a function and where it crosses the x-axis.

Steps to Find X-Intercepts and Corresponding Y-Values

Set y to 0: Begin by setting the y-value of the function equal to 0. This is because at the x-intercept, the y-value is 0. Solve for x: Solve the resulting equation for x to find the x-values of the x-intercepts. Substitute x-values back into the original function: Once you have the x-values, you can substitute them back into the original equation to find the corresponding y-values.

Example: Finding X-Intercepts of a Quadratic Function

Let's consider the function y 2x2 - 4x - 3. To find the x-intercepts, we set y equal to 0:

2x2 - 4x - 3 0

Simplifying this, we get:

x2 - 2x - 1.5 0

To solve this quadratic equation, we can use the quadratic formula:

x [-b plusmn; sqrt(b2 - 4ac)] / (2a)

In this case, a 1, b -2, and c -1.5. Plugging these values into the formula:

x [2 plusmn; sqrt((-2)2 - 4(1)(-1.5))] / (2(1))

x [2 plusmn; sqrt(4 6)] / 2

x [2 plusmn; sqrt(10)] / 2

We get two x-values:

x (2 sqrt(10)) / 2 and x (2 - sqrt(10)) / 2

Using X-Intercepts to Find Other Y-Values

Once you have the x-intercepts, you can use them to find other y-values. For example, if you want to find the y-value when x 2:

y 2(22) - 4(2) - 3

Calculating this gives:

y 2(4) - 8 - 3

y 8 - 8 - 3

y -3

Thus, the point (2, -3) is on the graph of the function.

Conclusion

Mastering the use of x-intercepts to find y-values is crucial for understanding the behavior of various functions. By solving for x-intercepts and using those points to find other y-values, you can gain deeper insights into the functions you are working with. This method is particularly useful in algebra and calculus, where understanding the behavior of functions is essential.