Problem 25 - Precalculus : Building Concepts and Connections

Question

Sketch the graph of each function. $$f(x)=2+3 e^{x}$$

Step-by-Step Solution

Answer

The graph of the function \(f(x)=2+3 e^{x}\) starts from the point (0, 2) on the y-axis and increases rapidly as \(x\) increases due to the exponential term.

Step 1: Identify the Base and Vertical Shift

The function \(f(x)=2+3 e^{x}\) is an exponential function where the base is \(e\), and the equation has a vertical shift of 2 units up. This means the graph will start from the point (0, 2) on the y-axis instead of the origin.

Step 2: Identify the Vertical Scaling Factor

The number 3 in front of the \(e^{x}\) term is a vertical scaling factor. This will make the graph steeper. An increase in x will result in the output of the function to increase by a factor of 3. This will make the graph rise more rapidly as x increases.

Step 3: Sketch the graph

Start by plotting the y-intercept at (0, 2). Then, note the steepness of the curve due to the scaling factor of 3. Draw the curve starting from the y-intercept, rising sharply to the right (indicative of the \(e^{x}\) portion of the function) and approaching, but never touching, the y-axis from the right (a horizontal asymptote introduced by the +2 shift). The graph will show a gradually increasing slope as \(x\) increases.

Key Concepts

Graph Sketching

When sketching the graph of an exponential function like \( f(x) = 2 + 3e^x \), it's important to understand the basic shape and behavior that different parts of the function dictate. The base function \( e^x \) is known for being continuously increasing, and its curve is gentle at smaller values of \( x \) and steep as \( x \) increases.

To start sketching:

• Identify key elements of the graph such as the y-intercept, which in this function is at (0, 2) due to the vertical shift.
• Recognize that exponential functions have a horizontal asymptote. Initially, the base function \( e^x \) has an asymptote at \( y = 0 \), but here it is shifted up to \( y = 2 \).
• Once these elements are plotted, draw the exponential curve rising sharply after the y-intercept, bearing in mind its rapid increase due to exponential growth.

Graph sketching is a visual way to interpret functions, giving a tangible representation of abstract mathematical concepts.

Vertical Shift

In the function \( f(x) = 2 + 3e^x \), the vertical shift can be easily spotted by the constant term outside the exponent. A vertical shift moves the entire graph up or down without affecting its shape.

Here, the graph is shifted 2 units upwards. This means that the line \( y = 0 \), which would have been the horizontal asymptote for \( e^x \), becomes \( y = 2 \). Thus, every point on the graph is moved 2 units higher than where it would be without this term.

Characteristics of vertical shifts:

• Simple Addition/Subtraction: A positive vertical shift adds to each output value of \( e^x \), moving the graph up.
• Impact on Intercepts and Asymptotes: It changes the intercepts and the horizontal line that the graph never crosses, known as the asymptote.

This creates a graph that still rises but does so starting from a different baseline.

Vertical Scaling

Vertical scaling in functions like \( f(x) = 2 + 3e^x \) involves multiplying the exponential term by a constant. The scaling factor here is 3, which affects how steeply the graph rises.

Understanding vertical scaling:

• Steepness: The larger the scaling factor, the steeper the exponential increase becomes. For every unit increase in \( x \), the output increases by this factor times the rate of \( e^x \).
• Graphically Obvious: A scaling factor makes the graph appear stretched vertically. More scaling makes the rise sharper and more dramatic.
• Relates to Multiplier: Over a region, output values are larger compared to the base function \( e^x \) because they're scaled up by the multiplier. This doesn't affect where the graph starts (that’s dictated by shifts, not scaling).

Vertical scaling is crucial in adjusting how a function behaves and looks visually, enabling tailored modeling of exponential growth based on real-world scenarios.