Problem 10 - 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. $$ f(x)=9-e^{x} $$

Step-by-Step Solution

Answer

Y-intercept: (0, 8), Horizontal asymptote: \( y = 9 \), Function is decreasing.

Step 1: Identify the Function Type

The function is in the form of a transformation of the exponential function. The base function is \( e^x \), and this function is shifted and flipped due to transformation rules.

Step 2: Find the Y-intercept

To find the y-intercept, set \( x = 0 \) in the function \( f(x) = 9 - e^x \). Calculate \( f(0) = 9 - e^0 = 9 - 1 = 8 \). Hence, the y-intercept is \( (0, 8) \).

Step 3: Determine the Horizontal Asymptote

Exponential functions of the form \( f(x) = a - e^x \) have horizontal asymptotes. As \( x \to \infty \), \( e^x \to \infty \) hence \( f(x) = 9 - e^x \to -\infty \). As \( x \to -\infty \), \( e^x \to 0 \) hence \( f(x) = 9 - 0 = 9 \). Thus, the horizontal asymptote is \( y = 9 \).

Step 4: Analyze the Function's Behavior (Increasing/Decreasing)

The derivative of \( f(x) \) is \( f'(x) = -e^x \). Since \( e^x \) is always positive for all real \( x \), \( f'(x) \) is always negative. Therefore, \( f(x) = 9 - e^x \) is a decreasing function.

Step 5: Sketch the Graph

Based on the previous steps, sketch the graph of \( f(x) = 9 - e^x \):- Start at (0, 8) for the y-intercept.- As \( x \to \infty \), the graph goes downwards infinitely due to the decreasing nature of the function and nears -\infty.- As \( x \to -\infty \), the graph approaches the horizontal asymptote \( y = 9 \).

Key Concepts

Horizontal Asymptote

When dealing with exponential functions like \( f(x) = 9 - e^x \), understanding horizontal asymptotes is crucial. A horizontal asymptote is a horizontal line that the graph of a function approaches as \( x \) heads towards infinity or negative infinity.
The function \( f(x) = 9 - e^x \) tends towards a stable value, which occurs because the exponential part \( e^x \) changes drastically as \( x \) changes.
For example:

• As \( x \to -\infty \): \( e^x \to 0 \), thus \( f(x) = 9 - 0 = 9 \). Hence, the function approaches a horizontal line at \( y = 9 \) on the left.


• As \( x \to \infty \): \( e^x \to \infty \), thus \( f(x) \to -\infty \). Therefore, there is no horizontal asymptote on the right as \( f(x) \) diminishes indefinitely downwards.

This asymptotic behavior helps in sketching the graph accurately, showing how it behaves in extremes.

Decreasing Function

Exponential functions can either increase or decrease, and for \( f(x) = 9 - e^x \), the function is decreasing. This means as \( x \) increases, the value of \( f(x) \) decreases.

The key to understanding why this happens lies in the derivative. The derivative of \( f(x) \) is \( f'(x) = -e^x \), which is always negative since \( e^x \) is always positive. A negative derivative indicates function values are getting smaller as \( x \) moves towards larger values:

• When \( x \to 0 \), \( f'(x) = -e^0 = -1 \) (a small but consistent decrease).
• As \( x \to \infty \), \( f'(x) \) becomes more negative, accelerating the decrease of \( f(x) \).

This behavior confirms that the function decreases without bound as \( x \to \infty \), impacting how its graph looks and behaves.

Y-intercept

The y-intercept of a function is where the graph crosses the y-axis, i.e., the value of the function when \( x = 0 \). For the function \( f(x) = 9 - e^x \), let's learn how you find this point.

Simply substitute \( x = 0 \) into the function:
\( f(0) = 9 - e^0 = 9 - 1 = 8 \).
Thus, the y-intercept is the point \((0, 8)\).
This calculation means that the graph will intersect the y-axis at this point.

• It's a useful reference point for sketching the graph since it shows the starting height when \( x \) is zero.
• This point reflects that at zero input (where x is zero), the output (y-value) is 8.

The y-intercept is essential in giving context to what happens as the graph moves horizontally along the axis.