Problem 1 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 1-4: Use the graph of \(y=e^{x}\) to help sketch the graph of \(f\). (a) \(f(x)=e^{-x}\) (b) \(f(x)=-e^{x}\)

Step-by-Step Solution

Answer

(a) Reflect \(y=e^x\) over y-axis; (b) Reflect \(y=e^x\) over x-axis.

Step 1: Understanding the Baseline Graph

The function \(y = e^x\) is an exponential function with a base greater than one. Its graph passes through the point \( (0, 1) \) and is asymptotic to the x-axis on the left (as \(x \to -\infty\)). It increases rapidly and is never negative.

Step 2: Transforming for (a)

For \(f(x) = e^{-x}\), substitute \(-x\) for \(x\) in \(y = e^x\). This results in reflecting the graph of \(y = e^x\) over the y-axis. Therefore, the graph of \(f(x) = e^{-x}\) still passes through \( (0, 1) \), but it becomes asymptotic to the x-axis on the right (as \(x \to \infty\)) and decreases as \(x\) increases.

Step 3: Transforming for (b)

For \(f(x) = -e^x\), multiply the original function by \(-1\). This reflects the graph of \(y = e^x\) across the x-axis. The new graph passes through the point \( (0, -1) \), is asymptotic to the x-axis and decreases as \(x\) increases to positive infinity.

Key Concepts

Graph Transformations

Graph transformations involve changing the position, shape, or size of a graph without altering the essence of the function itself. In exponential functions like \( y = e^x \), you may encounter transformations such as reflections, shifts, or stretches. The original graph of \( y = e^x \) is a smooth curve that starts low near the y-axis and rises steeply as \( x \) increases.

• Horizontal Shift: Moving the graph left or right.
• Vertical Shift: Increasing or decreasing the graph's height.
• Vertical Stretch/Compression: Stretching the graph away from the x-axis or compressing it towards the x-axis.
• Reflections: Flipping the graph over a given axis, which is crucial in understanding the specific transformations involved in exponential functions.

Each transformation modifies the graph's appearance, making it essential to understand these changes to sketch the accurate graph. By seeing how an exponential graph is affected, students grasp how algebraic changes influence graph behavior.

Reflection Across Axes

Reflections result in flipping the graph over a line, such as the x-axis or y-axis, creating a mirror image. Understanding reflections can help you sketch new variations of familiar functions.

For part (a), \(f(x) = e^{-x}\): This involves reflecting \( y = e^x \) across the y-axis. Imagine holding a mirror at the y-axis; the graph facing right now faces left. This keeps the point \( (0,1) \) stationary while altering the direction the graph tends towards infinity, making it asymptotic to the x-axis as \(x\) approaches infinity while decreasing.

For part (b), \(f(x) = -e^x\): This involves reflecting the graph over the x-axis. Here, the graph of \( y = e^x \) inverts vertically. This transformation takes the curve from above the x-axis and mirrors it below. The point \( (0,1) \) shifts to \( (0,-1) \), emphasizing the reflection.

Asymptotes

An asymptote is a line that a graph approaches but never quite touches. In exponential functions, asymptotes describe the behavior of the graph at extreme values of \(x\).

For the base function \( y = e^x \), the graph has a horizontal asymptote at \( y = 0 \), or the x-axis. As \( x \) becomes more negative, the graph gets closer to this line, but never intersects it.

When considering transformations:

• For \( f(x) = e^{-x} \): The asymptote remains the x-axis. However, the graph approaches this asymptote from the right, as \( x \to \infty \), due to the reflection across the y-axis.
• For \( f(x) = -e^x \): Although reflected, the asymptote remains \( x = 0 \). The graph approaches the asymptote from below as it flips across the x-axis.

Asymptotes are crucial in understanding the limits and long-term trends of functions. They help predict graph behavior at large or small values of \(x\) and form a key part of graph sketching and analysis.