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Chapter 13: Problem 2

What is the \(y\) -intercept of the graph of \(f(x)=a^{x}\) where \(a>0\) and \(a \neq 1 ?\)

Short Answer

Expert verified
The y-intercept of the graph of the function \(f(x) = a^x\) is the point (0, 1).

Step by step solution

01

Set x to 0

We are looking for the y-intercept which means we want to find the value of the function when x = 0. So, we will plug in x = 0 into the given function: \(f(x) = a^x\).
02

Calculate f(0)

Now that we have set x to 0, our function becomes \(f(0) = a^0\). By the rule for exponents, any non-zero number raised to the power of 0 is equal to 1. Therefore, \(a^0 = 1\). So, we get \(f(0) = 1\).
03

Find the coordinates of the y-intercept

Since we have found the function value f(0) = 1, the y-intercept will be the point where x is 0 and y is the function's value at that point. Therefore, the y-intercept of the graph of the function \(f(x) = a^x\) is the point (0, 1).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding the Y-Intercept
When we talk about functions, especially exponential functions like \( f(x) = a^x \), an important feature is the \( y \)-intercept. This is the point where the graph of the function intersects the \( y \)-axis. To find this point, we set \( x \) to 0.
In our function, \( f(x) = a^x \), plugging in \( x = 0 \) means we calculate \( f(0) = a^0 \). Regardless of the value of \( a \), as long as it's not zero, \( a^0 = 1 \). This is a basic rule of exponents.
So, for any exponential function of the form \( f(x) = a^x \), the \( y \)-intercept will always be \( (0, 1) \). This is a convenient constant point on the graph, providing a reliable starting place for graphing.
Essential Properties of Exponents
Exponents can seem tricky, but there are a few simple rules that make them easier to work with. These basic properties help us solve problems like finding the \( y \)-intercept of exponential functions.
  • Zero Exponent Rule: This rule states that any nonzero number raised to the power of zero is 1. For instance, \( a^0 = 1 \) as long as \( a eq 0 \).
  • Product of Powers Rule: When you multiply two powers with the same base, you add the exponents: \( a^m \times a^n = a^{m+n} \).
  • Power of a Power Rule: To raise a power to another power, you multiply the exponents together: \( (a^m)^n = a^{m \times n} \).
  • Power of a Product Rule: When you have a product raised to an exponent, you apply the exponent to each factor in the product: \( (ab)^n = a^n \times b^n \).
These rules simplify expressions and help us evaluate functions efficiently.
Graphing Exponential Functions
Graphing exponential functions like \( f(x) = a^x \) is a great way to visualize their behavior. These graphs have unique features that distinguish them from other types of functions.
When \( a \) is greater than 1, the function \( a^x \) shows exponential growth. The graph rises sharply as \( x \) increases. Conversely, if \( 0 < a < 1 \), the function indicates exponential decay, and the graph falls as \( x \) increases.
  • Y-Intercept: As we've seen, the \( y \)-intercept is always at \( (0, 1) \) for \( f(x) = a^x \).
  • Horizontal Asymptote: The graph will approach the \( x \)-axis but never touch it, making \( y = 0 \) the horizontal asymptote.
  • Domain and Range: The domain of \( f(x) = a^x \) is all real numbers, but its range is \( y > 0 \), because the function never produces negative or zero values.
Understanding these characteristics can help when plotting and interpreting the graph of any exponential function.

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