Problem 3 - Algebra and Trigonometry with Analytic Geometry

Question

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

Step-by-Step Solution

Answer

(a) The graph moves 4 units left; (b) The graph moves 4 units up.

Step 1: Understanding the Base Graph

The base graph is the function \(y = e^{x}\), which is an exponential function. It passes through the point (0,1), increases rapidly as \(x\) increases, and approaches the x-axis (y = 0) as \(x\) decreases.

Step 2: Transformation for f(x)=e^{x+4}

The function \(f(x) = e^{x+4}\) represents a horizontal shift. Rewriting the expression in terms of a shift, \(f(x) = e^{x+4} = e^{x} \cdot e^{4}\). This tells us that every point on \(y=e^{x}\) moves left by 4 units on the x-axis.

Step 3: Sketching f(x)=e^{x+4}

To sketch \(f(x) = e^{x+4}\), move the entire graph of \(y = e^{x}\) 4 units to the left. The new graph will still be an exponential curve, but it will cross the y-axis at \(e^{4}\), because when \(x = 0\), \(f(0) = e^{0+4} = e^{4}\).

Step 4: Transformation for f(x)=e^{x}+4

The function \(f(x) = e^{x} + 4\) represents a vertical shift. Add 4 to each output value of \(y = e^{x}\). This means every point on \(y = e^{x}\) is moved up by 4 units along the y-axis.

Step 5: Sketching f(x)=e^{x}+4

To sketch \(f(x) = e^{x}+4\), take the graph of \(y = e^{x}\) and move it up by 4 units. The horizontal asymptote changes from \(y = 0\) to \(y = 4\), and the y-intercept moves to the point (0,5), since \(f(0) = e^{0} + 4 = 1 + 4 = 5\).

Key Concepts

Horizontal Transformations

When dealing with exponential functions, a horizontal transformation involves shifting the graph along the x-axis. Let's consider the function \(f(x) = e^{x+4}\). The term \(x+4\) inside the exponent indicates a shift in the graph. If we simplify \(e^{x+4}\), it becomes \(e^{x} \cdot e^{4}\). What does this mean? Essentially, this shift results in the entire graph of \(y = e^{x}\) being moved to the left by 4 units.

To put it in simpler terms, if you picture the graph of \(y = e^{x}\) like a sheet of graph paper, you're sliding this entire sheet to the left 4 squares. The shape of the graph remains the same, but its position changes.

**Key Points:**

• The graph moves horizontally (left/right).
• For \(f(x) = e^{x+4}\), the movement is to the left by 4 units.
• This does not alter the y-values of the graph, just its position on the x-axis.

Vertical Transformations

Vertical transformations imply shifting the graph up or down along the y-axis. Now, look at the function \(f(x) = e^{x} + 4\). Here, the \(+4\) is positioned outside the exponential function, indicating a vertical shift. What effect does this have on the graph? It lifts the entire graph of \(y = e^{x}\) up by 4 units.

Imagine that the baseline of the graph of \(y = e^{x}\) is being raised vertically higher and every single point on the graph is elevated by 4 y-units. As a result, the graph leans completely upwards by 4 units while maintaining its shape.

**Important Details:**

• Graph moves vertically (up/down).
• In \(f(x) = e^{x} + 4\), the graph shifts up by 4 units.
• This affects the y-values, and also changes the y-intercept and the horizontal asymptote.
• The horizontal asymptote moves from \(y = 0\) to \(y = 4\).

Graph Sketching

Sketching graphs of transformed exponential functions can seem challenging at first. But breaking it down into transformations makes it manageable and systematic. Begin with the base graph of \(y = e^{x}\), which has a characteristic steep, rising curve and a horizontal asymptote at \(y = 0\).

**Understanding Adjustments:**

• For horizontal transformations, like in \(f(x) = e^{x+4}\), visualize the movement first: left by 4 units. Plot critical points accordingly.
• For vertical adjustments, as with \(f(x) = e^{x} + 4\), lift the entire curve up by 4 units. Ensure the horizontal asymptote moves from \(y = 0\) to \(y = 4\).

**Final Steps:**

• Ensure important features like intercepts and asymptotes are correctly adjusted.
• Draw smooth curves to represent the exponential nature of the function. Keep the rate of increase consistent with the base function.
• Check transformations visually; practice adjusting graphs by plotting several points.

With practice, sketching these transformed graphs becomes a simple process of shifting and adjusting key elements based on the transformations involved.