Problem 3 - College Algebra

Question

The graph of \(f(x)=3^{x}\) is refl cted about the \(y\) -axis and stretched vertically by a factor of \(4 .\) What is the equation of the new function, \(g(x) ?\) State its \(y\) -intercept, domain, and range.

Step-by-Step Solution

Answer

The equation is \( g(x) = 4 \cdot 3^{-x} \). The y-intercept is 4, the domain is all real numbers, and the range is \( (0, \infty) \).

Step 1: Reflect the Graph about the y-axis

Reflecting the graph of the function about the y-axis involves changing the sign of the x-variable in the function. Thus, if the original function is \( f(x) = 3^x \), the reflected function is \( f(-x) = 3^{-x} \).

Step 2: Stretch the Graph Vertically

A vertical stretch by a factor of 4 means multiplying the function by 4. Thus, applying this to our reflected function \( f(-x) = 3^{-x} \), the new function becomes \( g(x) = 4 \cdot 3^{-x} \).

Step 3: Identify the y-intercept

The y-intercept of a function occurs where \( x = 0 \). Substituting \( x = 0 \) into \( g(x) = 4 \cdot 3^{-x} \), we get \( g(0) = 4 \cdot 3^{0} = 4 \cdot 1 = 4 \). Thus, the y-intercept is 4.

Step 4: Determine the Domain of the New Function

The domain of the function \( g(x) = 4 \cdot 3^{-x} \) is all real numbers since there are no restrictions on x for exponential functions.

Step 5: Determine the Range of the New Function

The range of \( g(x) = 4 \cdot 3^{-x} \) is all positive real numbers because an exponential function like \( 3^{-x} \) is always positive, and multiplying it by 4 does not change this property. Thus, the range is \( (0, \infty) \).

Key Concepts

Reflection about the y-axis

To reflect a graph about the y-axis, you need to change the sign of the x-variable. Imagine flipping the curve over the vertical axis, like looking at a mirror. If our original function is \( f(x) = 3^x \), after reflection, it becomes \( f(-x) = 3^{-x} \). This transformation flips the directional growth of the curve. Instead of increasing, it now decreases. The dominant idea is to reverse the input direction while maintaining the same exponential base. It’s like reversing the tape of a movie without changing the story itself.

Vertical Stretch

A vertical stretch involves multiplying the function by a constant factor, which affects how quickly or slowly the graph changes along the y-axis. For the function \( f(-x) = 3^{-x} \), multiplying it by 4 gives us \( g(x) = 4 \cdot 3^{-x} \). The effect is an elongation vertically, meaning each point on the graph moves further away from the x-axis. This does not change the x-values but scales the height. Visually, if you were stretching a rubber band, each point rises to four times its original height. This transformation affects how steep or flat the graph appears.

Exponential Function Domain and Range

Understanding the domain and range of an exponential function helps us know where it can operate and what values it produces. The domain of \( g(x) = 4 \cdot 3^{-x} \) is all real numbers. This happens because you can raise 3 to any real number, be it fraction, zero, or negative, without restrictions.
The range of the function is different. Since \( 3^{-x} \) is always positive, multiplying by 4 doesn’t change this fact. The function never touches or goes below zero. Thus, the range is \( (0, \infty) \), meaning it only includes positive numbers. This forms an essential feature of exponential functions, making them widely applicable in modeling growth scenarios.