Problem 8 - College Algebra with Corequisite Support

Question

For the following exercises, graph the function and its reflection about the \(y\) -axis on the same axes, and give the \(y\) -intercept. \(f(x)=3\left(\frac{1}{2}\right)^{x}\)

Step-by-Step Solution

Answer

The y-intercept is 3.

Step 1: Understand the Function

The function given is \( f(x) = 3 \left( \frac{1}{2} \right)^x \). This is an exponential decreasing function because the base \( \frac{1}{2} \) is between 0 and 1. It represents a transformation of the basic exponential function.

Step 2: Plot the Original Function

To graph \( f(x) = 3 \left( \frac{1}{2} \right)^x \), choose a set of \(x\) values, such as \(-2, -1, 0, 1, 2\), and calculate \( f(x) \) for each. - \( f(-2) = 3 \times 4 = 12 \)- \( f(-1) = 3 \times 2 = 6 \)- \( f(0) = 3 \times 1 = 3 \)- \( f(1) = 3 \times \frac{1}{2} = 1.5 \)- \( f(2) = 3 \times \frac{1}{4} = 0.75 \)Plot these points and draw a smooth curve through them.

Step 3: Reflect the Function Across the y-axis

The reflection of the function across the \(y\)-axis is found by replacing \(x\) with \(-x\) in the function. The reflected function is \( f(-x) = 3 \left( \frac{1}{2} \right)^{-x} = 3 \times 2^x \). Calculate and plot this function as well:- \( f(-2) = 3 \times \frac{1}{4} = 0.75 \)- \( f(-1) = 3 \times \frac{1}{2} = 1.5 \)- \( f(0) = 3 \times 1 = 3 \)- \( f(1) = 3 \times 2 = 6 \)- \( f(2) = 3 \times 4 = 12 \)Draw a smooth curve through these points. This curve should be a reflection of the original curve over the \(y\)-axis.

Step 4: Determine the y-intercept

The \(y\)-intercept is the value of the function when \(x = 0\). For both \( f(x) = 3 \left( \frac{1}{2} \right)^x \) and its reflection, \( f(-x) \), the \(y\)-intercept is \( f(0) = 3 \). Therefore, the \(y\)-intercept for both functions is the point \((0, 3)\).

Key Concepts

Exponential Functions

Exponential functions are characterized by a constant base raised to a variable exponent, such as \( f(x) = 3 \left( \frac{1}{2} \right)^x \). In this function, the base is \( \frac{1}{2} \) and it is less than 1, which indicates that the function is decreasing, or decaying, as \( x \) increases. This behavior is characteristic of exponential decay.
In general:

• If the base \( b \) is greater than 1, the function represents exponential growth.
• If the base \( b \) is between 0 and 1, the function represents exponential decay.

Exponential functions have many applications in real-world scenarios such as compound interest, population growth, and radioactive decay.

Function Reflection

Function reflection involves flipping a graph across a specific axis, much like looking at its mirror image. For a horizontal reflection across the \( y \)-axis, you substitute \( x \) with \( -x \) in the function. The original function \( f(x) = 3 \left( \frac{1}{2} \right)^x \) becomes \( f(-x) = 3 \cdot 2^x \), representing its reflection.
This transformation flips the curve from the right side of the \( y \)-axis to the left side, and vice versa. It's crucial to practice reflecting functions, as it helps in understanding symmetrical properties and graphical transformations. When graphing both the original and reflected functions together, their symmetry becomes evident.

Y-Intercept

The \( y \)-intercept of a function is the point where the graph crosses the \( y \)-axis. It corresponds to the output value when \( x = 0 \). For both the original function \( f(x) = 3 \left( \frac{1}{2} \right)^x \) and its reflection \( f(-x) = 3 \cdot 2^x \), the \( y \)-intercept is calculated as follows:When \( x = 0 \), both functions evaluate to \( f(0) = 3 \cdot 1 = 3 \). Therefore, the \( y \)-intercept for both is \((0, 3)\).
Recognizing the \( y \)-intercept is crucial for sketching accurate graphs. It provides a starting point and helps in drawing the function curve.

Transformation of Functions

Transformation of functions involves changing a function's position or shape on a graph. Common transformations include translations, reflections, stretches, and compressions.
In the function \( f(x) = 3 \left( \frac{1}{2} \right)^x \), the constant 3 represents a vertical stretch. This multiplies every output by 3, pulling the graph away from the \( x \)-axis. Meanwhile, substituting \( x \) with \( -x \) reflects the graph across the \( y \)-axis, as seen in \( f(-x) = 3 \cdot 2^x \).

• Vertical shifts involve adding or subtracting a constant to the whole function.
• Horizontal shifts involve adding or subtracting from the \( x \) variable.

Understanding these transformations is key in graphing complex functions, as they affect the graph's orientation and shape, leading to a deeper comprehension of the function's behavior.