Understanding the Y-Intercept of the Exponential Function y 24 ? 2^x

Welcome to our comprehensive guide on the y-intercept of the exponential function y 24 ? 2^x. In this article, we will break down the concept of y-intercept, explain the given equation, and provide a detailed step-by-step solution to find the y-intercept. By the end of this guide, you will have a clear understanding of how to approach similar problems and apply this knowledge effectively.

What is the Y-Intercept?

The y-intercept of a function is the point where the graph of the function intersects the y-axis. At this point, the x-coordinate is 0. In other words, the y-intercept is the value of the function when x 0.

Exploring the Exponential Function y 24 ? 2^x

The given function is y 24 ? 2^x. This is an example of an exponential function, where 2^x represents the exponential term.

Finding the Y-Intercept

To find the y-intercept, we need to substitute x 0 into the equation and solve for y.

Step-by-Step Solution

Start with the given equation: y 24 ? 2^x. Substitute x 0 into the equation: y 24 ? 2^0 Simplify the equation: 2^0 1 (Any number raised to the power of 0 is 1) So, y 24 ? 1 y 24 The y-intercept is the point where x 0, so the coordinates are (0, 24).

No matter what the value of x is, when x 0, the function y 24 ? 2^x always equals 24. Therefore, the y-intercept is 24.

Graphical Interpretation

Let's consider a graphical interpretation of the function y 24 ? 2^x.

When x 0, y 24, which is the y-intercept. When x increases by 1, y doubles because 2^x is an exponential term. For example, when x 1, y 24 ? 2^1 48. When x 2, y 24 ? 2^2 96, and so on.

The graph of y 24 ? 2^x will start at the point (0, 24) and then grow exponentially as x increases.

Practice Problems

Now that you understand how to find the y-intercept of an exponential function, try solving a few practice problems:

Find the y-intercept of the function y 30 ? 4^x. Find the y-intercept of the function y 50 ? 5^x. Find the y-intercept of the function y 100 ? 3^x.

Solution:

For y 30 ? 4^x, when x 0: y 30 ? 4^0 30 ? 1 30. The y-intercept is (0, 30).

For y 50 ? 5^x, when x 0: y 50 ? 5^0 50 ? 1 50. The y-intercept is (0, 50).

For y 100 ? 3^x, when x 0: y 100 ? 3^0 100 ? 1 100. The y-intercept is (0, 100).

Conclusion

Understanding the y-intercept of an exponential function is crucial for analyzing and interpreting the behavior of such functions. By substituting x 0 into the equation and solving for y, we can easily find the y-intercept.

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