Problem 38 - Intermediate Algebra

Question

For Problems \(35-52\), graph each exponential function. $$ f(x)=5^{x} $$

Step-by-Step Solution

Answer

Plot the curve from negative to positive x using points like (-2, \(\frac{1}{25}\)), (0, 1), (2, 25). It rises steeply to the right.

Step 1: Understand the Function

The given function is an exponential function, which can be written as \( f(x) = 5^x \). This means that for any input \( x \), the output is \( 5 \) raised to the power of \( x \). Exponential functions showcase rapid growth as \( x \) increases.

Step 2: Identify Key Characteristics

The function \( f(x) = 5^x \) has a base greater than one (5), meaning it is an increasing exponential function. Key points to identify are the y-intercept at (0,1), since \( 5^0 = 1 \), and no x-intercept since exponential functions do not touch the x-axis.

Step 3: Select Data Points for Plotting

Choose several points to plot the function. Common choices are \( x = -2, -1, 0, 1, 2 \). Calculate the corresponding function values: - \( f(-2) = \frac{1}{25} \)- \( f(-1) = \frac{1}{5} \)- \( f(0) = 1 \)- \( f(1) = 5 \)- \( f(2) = 25 \)

Step 4: Plot and Draw the Graph

Using the calculated points \( (-2, \frac{1}{25}), (-1, \frac{1}{5}), (0, 1), (1, 5), (2, 25) \), plot these on a Cartesian plane. The points illustrate the rapid increase in the function's values as \( x \) increases. Next, draw a smooth curve through these points, noting the curve's behavior as it approaches zero along the negative x-direction and increases steeply as \( x \) becomes positive.

Step 5: Interpret the Graph

The graph of \( f(x) = 5^x \) should show a curve that starts very close to zero on the left, passes through (0,1), and sharply rises as \( x \) increases. The function has a horizontal asymptote at \( y = 0 \) suggesting the function never reaches or crosses the x-axis.

Key Concepts

Graphing Exponential Functions

Graphing exponential functions, like the function \( f(x) = 5^x \), provides a visual representation of how these functions behave as \( x \) varies. Exponential graphs have distinct characteristics that set them apart. Generally, they increase or decrease rapidly depending on the base. For the function \( f(x) = 5^x \), the base is greater than one, indicating the function will grow rapidly as \( x \) increases.

• The graph typically passes through the point (0,1) because \( 5^0 = 1 \). This is a fundamental starting reference point.
• There will not be an x-intercept since exponential functions never cross the x-axis.
• As you plot the graph by connecting points like \( (-2, \frac{1}{25}), (-1, \frac{1}{5}), (0, 1), (1, 5), (2, 25) \), notice the steep rise as \( x \)'s value increases.

By plotting these points on the Cartesian plane and connecting them with a smooth curve, you create a graph with a distinct upward swoosh. This graphical representation can help visualize exponential growth's quick increase.

Properties of Exponential Growth

Exponential growth reflects how functions like \( f(x) = 5^x \) increase rapidly for positive scenarios. It is one of the most important properties, especially in scenarios where things double or multiply quickly over time, such as populations or financial investments. Here are some critical things to understand about exponential growth:

• Rapid Growth: In the function \( f(x) = 5^x \), the output magnifies quickly as \( x \) goes upwards. This is why it rises sharply on the graph.
• Non-linear: Exponential growth is not a straight line. It curves upwards more and more steeply without leveling off or reversing back.
• Multiplicative Basis: The basis of this growth is multiplication by a constant factor, here '5', for every point \( x \) moves forward. This multiplicative nature makes it inherently faster growing than linear functions.

Understanding these properties helps clarify why exponential functions can be so impactful in real-world scenarios, as they often model rapid change or expansion.

Horizontal Asymptote

The concept of a horizontal asymptote is crucial when interpreting exponential graphs like \( f(x) = 5^x \). An asymptote is a line that the function approaches but never quite reaches or crosses. In the case of our function, the horizontal asymptote is located at \( y = 0 \).

• Closer to Zero: As \( x \) becomes more negative, the graph of \( f(x) = 5^x \) gets closer to the x-axis (\( y = 0 \)), but will never actually touch it. This means the function value is positive but tiny for very negative \( x \).
• Approaching Infinity: On the positive side, as \( x \) increases, \( f(x) \) grows larger indefinitely, leaving the horizontal asymptote behind.
• Significance: The horizontal asymptote illustrates the idea that although growth can be immense, some function behaviors such as approaching the x-axis just indefinitely approach without convergence.

Recognizing these asymptotes on a graph assists in understanding the full scope of exponential function behavior and predicting how the graph behaves towards extreme values of \( x \).