Verified Answer :

  • The y-intercept of a function occurs at the point where \(x = 0\).
  • Substitute \(x = 0\) into the equation \(y = 5(2)^x\).
  • Use the rule that any non-zero base raised to the power of zero is 1: \(2^0 = 1\).
  • Calculate the result: \(y = 5(1) = 5\).
  • The final answer is \(\boxed{5}\).

Explanation

  1. Understanding the Y-Intercept

To find the y-intercept of the exponential function \(y = 5(2)^x\), we need to identify the value of \(y\) when the graph of the function crosses the y-axis. The y-axis is defined by the line where the x-coordinate is equal to 0. Therefore, to find the y-intercept, we substitute \(x = 0\) into the given equation.

  1. Calculating the Y-Intercept

We have the equation \(y = 5(2)^x\). Setting \(x = 0\), we get:
\[y = 5(2)^0\]
By the laws of exponents, any non-zero number raised to the power of 0 is equal to 1. Thus, \(2^0 = 1\). Substituting this back into the equation:
\[y = 5 \times 1 = 5\]
The y-intercept occurs at the point \((0, 5)\), which means the y-intercept value is 5.

Examples

Exponential functions are used in many real-world applications, such as modeling population growth, radioactive decay, or compound interest. For instance, if an initial investment of \$5,000 grows based on a specific interest rate, we can model the value of the investment over time. Understanding the y-intercept helps us identify the starting value of such investments or population sizes at time \(t = 0\). In this case, starting with an initial base multiplier of 5 and a growth factor of 2, we know that at time zero, the total value is exactly 5.

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