Problem 37 - Algebra for College Students

Question

Graph each of the exponential functions. $$ f(x)=-2^{x} $$

Step-by-Step Solution

Answer

The graph is a downward-sloping curve passing through points like (0, -1) and approaching the x-axis from below as x increases.

Step 1: Understand the Function

The given function is an exponential function of the form \( f(x) = -2^x \). This means that for any value of \( x \), the function will produce a value calculated by taking \( 2^x \) and then multiplying the result by \(-1\).

Step 2: Identify Key Characteristics

Identify important points for the graph such as the y-intercept. Since \( f(x) = -2^x \), when \( x = 0 \), \( f(0) = -2^0 = -1 \). Also note that this function will never cross the x-axis and will approach zero as \( x \to \infty \) from below the x-axis.

Step 3: Generate Points

Create a table of values by substituting several values for \( x \). For example, some values could be \( x = -2, -1, 0, 1, 2 \). Calculate \( f(x) = -2^x \) for these points:- \( x = -2 \), \( f(-2) = -2^{-2} = -\frac{1}{4} \)- \( x = -1 \), \( f(-1) = -2^{-1} = -\frac{1}{2} \)- \( x = 0 \), \( f(0) = -1 \)- \( x = 1 \), \( f(1) = -2^1 = -2 \)- \( x = 2 \), \( f(2) = -2^2 = -4 \)

Step 4: Plot the Points

Plot the points on a coordinate plane. The points from Step 3 give us the coordinates \((-2, -\frac{1}{4})\), \((-1, -\frac{1}{2})\), \((0, -1)\), \((1, -2)\), \((2, -4)\).

Step 5: Draw the Exponential Curve

Connect the plotted points with a smooth curve that passes through them. Ensure the curve approaches the x-axis (but never touches it) for large values of \( x \) and curves downward as \( x \) decreases. The overall shape is a downward-sloping curve.

Step 6: Verify Characteristics

Check that the graph starts at the point \((0, -1)\), continuously decreases, and approaches but does not touch the x-axis to the right. It is also important to see the curve reflecting these exponential characteristics, showing the exponential growth close to zero on the left and decreasing growth as \( x \) increases.

Key Concepts

Graphing Exponential Functions

Graphing exponential functions can initially feel tricky, but with a few steps, you can graph them confidently. An exponential function is one where the variable appears in the exponent. In the case of the function \( f(x) = -2^x \), the base is 2, a common base in exponential functions. The key to graphing these functions is understanding how the values change as you substitute different values of \( x \). Each point on the graph represents a unique output for a given input \( x \).

• Begin by substituting a range of \( x \) values (negative, zero, and positive) into the function.
• Calculate the corresponding \( y \) values to plot these points on the coordinate plane.
• Connect the dots smoothly in the direction the function shows growth or decay.

This gives you a visual representation of the function's behavior across its domain.

Exponential Decay

Exponential decay describes a process where quantities diminish over time. For the function \( f(x) = -2^x \), the graph demonstrates decay because as \( x \) increases, \( f(x) \) becomes more negative. This happens because the value of \( 2^x \) is always positive, but it's multiplied by \(-1\), flipping the function below the x-axis. Here are some important characteristics of exponential decay:

• The function never crosses the x-axis, only approaches it, making the x-axis an asymptote.
• As \( x \) becomes more negative, the values approach zero from the negative side.
• This reflects a decrease that continues towards an asymptote, without ever reaching or crossing it.

Understanding decay helps in predicting how the function behaves over an extended \( x \) range.

Coordinate Plane

The coordinate plane is your graphing playground and consists of a horizontal x-axis and vertical y-axis. For graphing functions like \( f(x) = -2^x \), you’ll plot the calculated points from the function on this plane.

• Each point from the function corresponds to an \( (x, y) \) coordinate on the graph.
• Use negative and positive values of \( x \) to capture the behavior of the function around critical points like \( (0, -1) \).

Important characteristics such as slope and direction become evident by observing how points relate to each other on the plane. Knowing how to accurately plot these will give you a snapshot of the function's behavior.

Function Characteristics

Each exponential function has specific characteristics. In \( f(x) = -2^x \), these traits define its unique shape on the graph.

• The y-intercept, which is the point where the function crosses the y-axis, is \( (0, -1) \).
• As \( x \) goes to positive infinity, \( f(x) \) keeps approaching zero, sloping downwards.
• The function is strictly decreasing due to its negative growth coefficient, creating a downward curve.

By recognizing these characteristics, you’re able to predict and draw consistent and accurate graphs of exponential functions, capturing their behaviors vertically and horizontally across the plane.