Problem 3 - Calculus Early Transcendentals: Pearson New International Edition

Question

Sketch a graph of the given exponential function. $$ f(x)=2^{2 x} $$

Step-by-Step Solution

Answer

The graph is an exponential curve rising sharply with an asymptote at \(y=0\).

Step 1: Identify Function Type

The given function is an exponential function, where the base is 2 and it is raised to the power of \(2x\). This affects the rate at which the graph grows.

Step 2: Determine Key Points

Identify key points by substituting values of \(x\) into the equation to find corresponding \(y\) values. Use \(x = -1, 0, 1, 2\):- \(f(-1) = 2^{2(-1)} = 2^{-2} = \frac{1}{4}\)- \(f(0) = 2^{2(0)} = 2^0 = 1\)- \(f(1) = 2^{2(1)} = 2^2 = 4\)- \(f(2) = 2^{2(2)} = 2^4 = 16\)

Step 3: Plot the Points

On a coordinate plane, plot the points found in Step 2: \((-1, \frac{1}{4})\), \((0, 1)\), \((1, 4)\), \((2, 16)\). These points will help in visualizing how the function behaves.

Step 4: Draw the Exponential Curve

Using the points plotted, draw a smooth curve that starts near the x-axis (but never touches it, as the function never actually reaches zero) and rises steeply as \(x\) increases. Exponential functions like this one grow very quickly after the initial transition.

Step 5: Analyze the Graph

The graph of \(f(x) = 2^{2x}\) is a rapidly increasing curve that starts near zero when \(x\) is negative and rises sharply as \(x\) becomes positive. There is a horizontal asymptote at \(y = 0\) and no x-intercept.

Key Concepts

Graph Sketching

Graph sketching involves visualizing mathematical equations in a graphical form. It helps in understanding how functions behave. For exponential functions, this includes quickly identifying how the curve changes as the input value—or the x-coordinate—changes. In the given function, \(f(x) = 2^{2x}\), the curve starts close to the x-axis for negative x values, then rises steeply for positive x values. To sketch the graph, follow a few basic steps:

• Identify the function's properties, such as its base and exponent. Here, the base is 2, and the exponent is \(2x\).
• Determine critical points by calculating the output for different x values. This provides a framework for the shape of the graph.
• Draw a smooth curve through these points, ensuring that the slope reflects the function's exponential nature—a rapid increase.
• Keep in mind horizontal asymptotes, where the curve approaches the line y=0 but never touches or crosses it.

By understanding these steps, one can accurately create a visual representation of exponential functions.

Coordinate Points

Coordinate points are fundamental in graph sketching, providing exact locations on a coordinate plane. They consist of an x-coordinate (horizontal value) and a y-coordinate (vertical value). In graphing \(f(x) = 2^{2x}\), determine coordinate points by selecting x-values and computing the corresponding y-values:

• For \(x = 0\), \(y = f(0) = 1\), leading to the point \((0, 1)\).
• For \(x = 1\), \(y = f(1) = 4\), resulting in the point \((1, 4)\).
• For \(x = 2\), \(y = f(2) = 16\), giving the point \((2, 16)\).
• For negative x-values like \(x = -1\), \(y = f(-1) = \frac{1}{4}\), producing the point \((-1, \frac{1}{4})\).

Plotting these points on graph paper or a digital tool provides a framework for the sketch. This technique helps to visualize the rapid increase characteristic of exponential growth.

Exponential Growth

Exponential growth refers to a rate of increase that becomes more rapid in proportion to the growing total. In mathematical terms, this occurs when an exponential function such as \(f(x) = 2^{2x}\) is used. Here's how it manifests in the function:

• The base, 2, signifies a doubling process; every time x increases by 1, the output value (y) grows exponentially.
• The exponent, \(2x\), means this doubling process happens twice for each increase in x, leading to even faster growth.
• Initial values are close to zero, creating a curve that expands upward rapidly.

This rapid expansion is typical in natural processes, such as population growth or radioactive decay. Graphing such functions provides insight into how quickly values can increase, illustrating a profound concept in both mathematics and real-life applications.