Problem 47 - Brief Applied Calculus

Question

For the functions \(y_{1}=\left(\frac{1}{3}\right)^{x}, \quad y_{2}=\left(\frac{1}{2}\right)^{x}\), \(y_{3}=2^{x}\), and \(y_{4}=3^{x}\) a. Predict which curve will be the highest for large values of \(x\). b. Predict which curve will be the lowest for large values of \(x\). c. Check your predictions by graphing the functions on the window \([-3,3]\) by \([0,5]\). d. From your graph, what is the common \(y\) -intercept? Why do all such exponential functions meet at this point?

Step-by-Step Solution

Answer

Highest: \(y_4 = 3^x\), Lowest: \(y_1 = \left(\frac{1}{3}\right)^x\). Common y-intercept is 1.

Step 1: Analyzing the growth rates

Compare the base values of the exponential functions. Since exponential functions grow according to their base values raised to the power of \(x\), higher base values indicate faster growth. For large \(x\), a larger base results in a higher value of the function.

Step 2: Predicting the highest curve

Among the functions, \(y_{4}=3^{x}\) has the largest base of 3. Therefore, as \(x\) becomes large, \(3^{x}\) will grow more rapidly than the others. Thus, \(y_4\) will be the highest curve for large values of \(x\).

Step 3: Predicting the lowest curve

\(y_{1}=(\frac{1}{3})^{x}\) has the smallest base value less than 1, which means it will decay (decrease) as \(x\) increases. Therefore, \(y_1\) will be the lowest curve for large values of \(x\).

Step 4: Graphing the functions

Plotting these functions in a given window of \([-3, 3]\) for \(x\) and \([0, 5]\) for \(y\) helps confirm the predictions visually. Each function will exhibit its growth or decay patterns according to its base.

Step 5: Determining the common y-intercept

For exponential functions in the form \(a^{x}\), setting \(x = 0\) gives \(a^{0} = 1\), hence all exponential functions \(y_{1}, y_{2}, y_{3},\) and \(y_{4}\) will intersect the y-axis at 1. This is why they share a common y-intercept at \(y = 1\).

Key Concepts

Growth Rates

In exponential functions, growth rates are determined by the base of the function. An exponential function is generally of the form \( y = a^x \), where the base \( a \) is a crucial factor in determining how quickly the function grows or decays. For values of \( x \) that are large and positive, functions with a larger base \( a \) grow more quickly.

In the given exercise, we compare \( y_1 = \left( \frac{1}{3} \right)^x \), \( y_2 = \left( \frac{1}{2} \right)^x \), \( y_3 = 2^x \), and \( y_4 = 3^x \).

• \( y_4 = 3^x \) grows the fastest because it has the largest base of 3.
• \( y_1 = \left( \frac{1}{3} \right)^x \) decreases as \( x \) increases because its base is less than 1, leading it to decay rather than grow.

Understanding growth rates helps in predicting and comparing how functions behave as \( x \) changes.

Graphing

Graphing the functions helps provide a visual representation of their growth or decay patterns over an interval. When graphing exponential functions, the way the curve behaves is influenced by the base.

For the exercise's graphing task within the window of \([-3, 3]\) for \( x \) and \([0, 5]\) for \( y \), each exponential function will display distinct characteristics.

• Functions with bases greater than 1, such as \( y_3 = 2^x \) and \( y_4 = 3^x \), show exponential growth and curve steeply upwards as \( x \) increases.
• Functions with bases between 0 and 1, like \( y_1 = \left( \frac{1}{3} \right)^x \) and \( y_2 = \left( \frac{1}{2} \right)^x \), decay and slope towards along the horizontal axis as \( x \) increases towards positive values.

Graphing is a crucial technique in confirming analytical predictions about behavior of exponential functions.

Y-Intercept

The y-intercept is the point at which a graph crosses the y-axis, providing a starting point for analyzing the overall behavior of the function. For exponential functions of the form \( a^x \), the y-intercept occurs at the point where \( x = 0 \).

To find the y-intercept, substitute \( x = 0 \) into each function:

• For \( y_1 = \left( \frac{1}{3} \right)^x \), \( y_2 = \left( \frac{1}{2} \right)^x \), \( y_3 = 2^x \), and \( y_4 = 3^x \), substituting \( x = 0 \) gives \( a^0 = 1 \) in every case.

Thus, all functions in the exercise intersect the y-axis at \( y = 1 \). This common y-intercept arises because any non-zero number raised to the power of zero equals 1, making exponential functions particularly predictable and easy to identify on a graph at their starting point.