Problem 44 - Intermediate Algebra

Question

For Problems \(35-52\), graph each exponential function. $$ f(x)=2^{x+1} $$

Step-by-Step Solution

Answer

Graph the function by plotting points and noting it has a horizontal asymptote at \(y = 0\).

Step 1: Identify the base function

The function given is \(f(x) = 2^{x+1}\). This can be rewritten as \(f(x) = 2 \times 2^x\), indicating it is an exponential function with base \(b = 2\) and a vertical stretch by a factor of 2.

Step 2: Determine the horizontal shift

Since the function is written as \(2^{x+1} = 2 \times 2^x\), this represents a base transformation of \(2^x\) shifted 1 unit left. This is because the exponent \(x+1\) affects the horizontal position by an opposite shift.

Step 3: Identify key points

Calculate key points to plot the graph. For \(x = -2, -1, 0, 1, 2\), calculate \(f(x)\):\[\begin{align*}f(-2) &= 2^{(-2)+1} = 2^{-1} = \frac{1}{2}, \f(-1) &= 2^{(-1)+1} = 2^0 = 1, \f(0) &= 2^{0+1} = 2^1 = 2, \f(1) &= 2^{1+1} = 2^2 = 4, \f(2) &= 2^{2+1} = 2^3 = 8.\end{align*}\]

Step 4: Graph the function

Plot the points \((-2, \frac{1}{2}), (-1, 1), (0, 2), (1, 4), (2, 8)\) on the coordinate plane. Draw a smooth curve through these points, noting the exponential growth as \(x\) increases, and approach zero as \(x\) becomes more negative.

Step 5: Analyze asymptotic behavior

Examine the asymptotic behavior of the function. There is a horizontal asymptote at \(y = 0\) because \(f(x) = 2^{x+1}\) approaches 0 as \(x\) approaches negative infinity. This helps in sketching the graph accurately.

Key Concepts

Graphing Transformations

Graphing transformations involve altering a function to shift, stretch, or reflect its graph. In our example, the function given is an exponential function, specifically \( f(x) = 2^{x+1} \). The transformation starts by identifying the base function \( 2^x \), where the base \( b \) is 2. This exponential function is shifted due to the term \((x+1)\), accounting for a horizontal shift to the left by 1 unit. It is important to remember that the expression \( x + c \) within the exponent causes a shift of \( c \) units in the opposite direction. Thus, \( x + 1 \) means a shift to the left. Additionally, the multiplication by 2 indicates a vertical stretch, meaning every point on the base graph is pulled away from the x-axis by a factor of 2. In summary, transformations adjust the graph's appearance while maintaining its overall shape, helping in repositioning its key features easily on a coordinate plane.

Key Points in Graphing

When graphing functions, identifying key points is crucial as these points serve as a guide for sketching the curve. For \( f(x) = 2^{x+1} \), calculating key points involves selecting values for \( x \) and computing the corresponding \( y \)-values. This provides a clear picture of the function's growth. In this case, let's calculate for \( x = -2, -1, 0, 1, 2 \).

• For \( x = -2 \), \( f(-2) = 2^{-2+1} = \frac{1}{2} \)
• For \( x = -1 \), \( f(-1) = 2^{-1+1} = 1 \)
• For \( x = 0 \), \( f(0) = 2^{0+1} = 2 \)
• For \( x = 1 \), \( f(1) = 2^{1+1} = 4 \)
• For \( x = 2 \), \( f(2) = 2^{2+1} = 8 \)

These points \((-2, \frac{1}{2}), (-1, 1), (0, 2), (1, 4), (2, 8)\) should be plotted on the graph. As you plot these, notice the exponential growth as \( x \) increases to the right, revealing the characteristic "J-shaped" curve of exponential functions.

Asymptotic Behavior

Understanding the asymptotic behavior of a function is essential while graphing, particularly with exponential functions. An asymptote is a line that a graph approaches but never actually touches. For \( f(x) = 2^{x+1} \), identifying the horizontal asymptote is key. This function approaches the line \( y = 0 \) as \( x \) moves toward negative infinity. That's because as \( x \) decreases, \( 2^{x+1} \) falls closer and closer to zero. The horizontal asymptote forms a boundary, demonstrating how the function behaves as \( x \) continues in either direction. Knowing this helps in sketching the curve accurately, providing a context for predicting the exponential decay of the graph as \( x \) decreases. Recognizing asymptotic behavior is a significant aspect of understanding exponential functions comprehensively.