Problem 11 - Precalculus with Calculus Previews

Question

Sketch the graph of the given function \(f\). Find the \(y\) -intercept and the horizontal asymptote of the graph. State whether the function is increasing or decreasing. $$ f(x)=-1+e^{x-3} $$

Step-by-Step Solution

Answer

The y-intercept is \((0, -1 + e^{-3})\); the horizontal asymptote is \(y = -1\). The function is always increasing.

Step 1: Identify the y-intercept

The y-intercept is where the graph of the function intersects the y-axis. To find it, set \(x = 0\) in the function and solve for \(f(x)\):\[f(x) = -1 + e^{0-3} = -1 + e^{-3}\].Therefore, the y-intercept is at the point \((0, -1 + e^{-3})\).

Step 2: Find the horizontal asymptote

The horizontal asymptote is determined by looking at the behavioral trend of \(f(x)\) as \(x\) approaches infinity. As \(x\) approaches infinity, \(e^{x-3}\) approaches infinity, which suggests that \(f(x)\) has no horizontal asymptote going to infinity. As \(x\) approaches negative infinity, \(e^{x-3}\) approaches zero, hence \(f(x)=-1 + e^{x-3}\) approaches \(-1\). So, the horizontal asymptote is \(y = -1\).

Step 3: Determine if the function is increasing or decreasing

The function's increasing or decreasing behavior can be determined by examining its derivative. The derivative \(f'(x) = e^{x-3}\) is always positive because \(e^{x-3} > 0\) for all real \(x\). Therefore, the function \(f(x) = -1 + e^{x-3}\) is always increasing.

Step 4: Sketch the graph

To sketch the graph, start by plotting the y-intercept, which is slightly above \(-1\) on the y-axis, as \(-1 + e^{-3}\) is a small positive number. Draw the horizontal asymptote as a dotted line along \(y = -1\). Since the function is increasing for all x values, sketch a curve starting near the asymptote for large negative \(x\) and moving upwards past the y-intercept, continuing to rise as \(x\) increases.

Key Concepts

Understanding the Y-Intercept

The y-intercept of a function is a vital feature that helps to anchor the graph on the coordinate plane. It is the point where the graph intersects the y-axis. To find the y-intercept of the function, you simply set the value of the independent variable, which is usually denoted as \(x\), to zero. This helps in solving the function for the output value, often denoted as \(f(x)\) or \(y\).
For the function \(f(x) = -1 + e^{x-3}\), calculate \(f(0)\) to find the y-intercept. After substituting zero for \(x\), you get \(f(0) = -1 + e^{-3}\). This is a small positive number approximately equal to \(-1 + 0.0498\).
Thus, the point where the graph intersects the y-axis is around \((0, -0.9502)\). This is crucial for sketching the graph, as it informs you where to start drawing the curve on the y-axis.

Exploring the Horizontal Asymptote

Horizontal asymptotes describe the behavior of a function as the value of \(x\) moves toward infinity or negative infinity. They are horizontal lines that the graph of the function approaches but never actually reaches.
For exponential functions like \(f(x) = -1 + e^{x-3}\), it's especially important to determine the function's end behavior. As \(x\) approaches infinity, \(e^{x-3}\) becomes very large, which indicates no asymptote at positive infinity.
However, as \(x\) approaches negative infinity, \(e^{x-3}\) approaches zero. Therefore, the function \(-1 + e^{x-3}\) approaches \(-1\). Hence, the graph will smooth out towards the line \(y = -1\) but will not intersect it.
This line, \(y = -1\), serves as a boundary that guides how close the graph can get as \(x\) trends toward negative infinity. Remember to draw this line when sketching to illustrate the behavior of the graph.

Identifying an Increasing Function

Understanding whether a function is increasing or decreasing is a key aspect when graphing. An increasing function means that as \(x\) increases, the value of \(f(x)\) also increases.
To determine this for the function \(f(x) = -1 + e^{x-3}\), we need to explore its derivative, \(f'(x)\). The derivative \(f'(x) = e^{x-3}\) is always greater than zero since exponential functions with a base greater than one never produce negative values.
This means that for any value of \(x\), the rate of change is positive, confirming that \(f(x)\) is always increasing. As such, the graph will constantly go upwards as \(x\) moves from left to right.

• **Always Positive Derivative**: Confirms increasing nature.
• **Graphing Implication**: The curve rises without dropping anywhere.

Recognizing the increasing nature of this function will help in accurately predicting how the graph behaves over its domain.