Problem 22 - Intermediate Algebra

Question

Graph each function. See Examples 1 and 2 . $$ f(x)=7^{x} $$

Step-by-Step Solution

Answer

The graph of \( f(x) = 7^x \) shows exponential growth with a horizontal asymptote at \( y = 0 \).

Step 1: Identify the Function

The function given is an exponential function: \( f(x) = 7^x \). This means the base of the exponential function is 7, and the exponent is the variable \( x \).

Step 2: Choose Key Points

Select key values of \( x \) to plot points on the graph. Common choices are \( x = -2, -1, 0, 1, 2 \). These will give us a sense of the shape and growth of the exponential function.

Step 3: Calculate Function Values

Calculate \( f(x) \) for each chosen value of \( x \):- For \( x = -2 \), \( f(x) = 7^{-2} = \frac{1}{49} \)- For \( x = -1 \), \( f(x) = 7^{-1} = \frac{1}{7} \)- For \( x = 0 \), \( f(x) = 7^0 = 1 \)- For \( x = 1 \), \( f(x) = 7^1 = 7 \)- For \( x = 2 \), \( f(x) = 7^2 = 49 \)

Step 4: Plot the Points

Using the calculated values, plot the points on a coordinate plane: - \( (-2, \frac{1}{49}) \)- \( (-1, \frac{1}{7}) \)- \( (0, 1) \)- \( (1, 7) \)- \( (2, 49) \)These points will show the exponential growth of the function.

Step 5: Draw the Graph of the Function

Draw a smooth curve through the plotted points. The curve should pass through each point and continue in both the negative and positive directions of \( x \), illustrating the rapid increase of values as \( x \) becomes more positive and the function approaching zero as \( x \) becomes more negative. The graph will have a horizontal asymptote at \( y = 0 \).

Key Concepts

Function Graphing

Graphing functions, especially exponential ones like \( f(x) = 7^x \), helps visualize how a function changes and behaves. The process involves evaluating the function at different \( x \)-values and plotting these points on the coordinate plane. By connecting these points smoothly, you can see the characteristics of the function's graph more clearly. Aim to choose \( x \)-values that neatly illustrate the symmetry or growth pattern of the function. For exponential graphs, this often includes negative, zero, and positive values of \( x \). Each plotted point represents a coordinate \((x, f(x))\), giving a visual representation of the function.

Exponential Growth

Exponential growth is a distinctive pattern where values increase rapidly over time. For the function \( f(x) = 7^x \), every time \( x \) increases by one unit, the value of the function multiplies by 7. This causes a sharp upward curve known as exponential growth.

• The point \( (0,1) \) is a critical feature, known as the y-intercept. Here, regardless of the base, \( 7^0 = 1 \).
• When \( x = 1 \), \( f(x) = 7 \), further reinforcing that the values are rising exponentially.
• Negative \( x \)-values lead to fractions smaller than 1, such as \( 7^{-2} = \frac{1}{49} \), illustrating the decrease prior to the rapid rise.

Understanding exponential growth is pivotal in predicting future values and interpreting real-world problems involving similar growth rates.

Coordinate Plane Plotting

The coordinate plane is a two-dimensional space that lets you track how functions behave visually. It consists of a horizontal axis (\( x \)-axis) and a vertical axis (\( y \)-axis). Each point on the plane is defined by a pair of coordinates \((x, y)\). For the function \( f(x) = 7^x \), you first calculate \( y \) for each \( x \) value, then plot these points.
Steps involved include:

• Decide which \( x \) values to use.
• Calculate the corresponding \( y \) (or \( f(x) \)) values.
• Plot points such as \( (-2, \frac{1}{49}) \) or \( (1, 7) \) on the plane.

Plotting accurately is critical for creating a reliable graph. Once this setup is complete, drawing a curve through your points completes the graph, highlighting the shape of the function and its features like asymptotes.