Problem 2 Plot in a single diagram the gra... [FREE SOLUTION]

Chapter 10: Problem 2

Plot in a single diagram the graphs of the exponential functions \(y=4^{t}\) and \(y=3\left(4^{t}\right)\) (a) Do the two graphs display the general positional relationship suggested in Fig. \(10.2 b ?\) (b) Do the two curves have the same \(y\) intercept? Why? (c) Sketch the graph of the function \(y=\frac{3}{2}\left(4^{t}\right)\) in the same diagram.

Short Answer

Expert verified

(a) Yes, aligned with Fig. 10.2b. (b) No, different due to stretch. (c) Lies between the other two functions.

Step by step solution

Understand the Functions

We are given two exponential functions: 1. \( y = 4^t \) 2. \( y = 3(4^t) \) Both functions have a base of 4, meaning they will grow exponentially as \( t \) increases. The second function is a vertically stretched version of the first one by a factor of 3. This means \( y = 3(4^t) \) is three times higher at each point along the \( t \)-axis than the function \( y = 4^t \).

Plot the Functions

Create a graph with \( t \) on the horizontal axis and \( y \) on the vertical axis. Plot the function \( y = 4^t \) first. It is an exponential curve starting from \( y = 1 \) when \( t = 0 \) and increases rapidly as \( t \) increases. Now, plot \( y = 3(4^t) \). Begin at \( y = 3 \) when \( t = 0 \), and note that this curve will increase at the same rate as \( y = 4^t \) but will be three times as high.

Compare Positional Relationship

Observing the plotted graphs, check if \( y = 3(4^t) \) is consistently above \( y = 4^t \) across all values of \( t \). This relationship aligns with the typical positional relationship exhibited by exponential functions when one is a vertical stretch of another, as shown in the provided figure reference, Fig. 10.2b.

Check for Same Y-Intercepts

In both functions, substitute \( t = 0 \) to find the \( y \)-intercepts: 1. For \( y = 4^t \), \( y = 4^0 = 1 \). 2. For \( y = 3(4^t) \), \( y = 3(4^0) = 3 \). We find that the \( y \)-intercepts are different because the factor 3 in the second function only multiplies the output of the base function.

Sketch Third Function

Plot the function \( y = \frac{3}{2}(4^t) \) on the same diagram. This function is also a vertical stretch of \( y = 4^t \) by a factor of \( \frac{3}{2} \). It will have a \( y \)-intercept at \( y = \frac{3}{2} \) when \( t = 0 \) and will lie between the other two functions on the graph, starting higher than \( y = 4^t \) but lower than \( y = 3(4^t) \).

Summary of Findings

(a) Yes, the graphs show a consistent vertical stretch relationship. (b) No, they have different \( y \)-intercepts because the vertical stretch causes different starting values. (c) \( y = \frac{3}{2}(4^t) \) is between the two other graphs, showing another vertical stretch.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Graphing Exponential Functions

To graph exponential functions like \( y = 4^t \), it's important to recognize their basic form. An exponential function typically has the structure \( y = a b^t \), where \( a \) and \( b \) are constants, with \( b > 0 \). The base here is 4, which indicates how the function grows as \( t \) increases. These functions start slowly and increase rapidly, rising sharply as \( t \) grows larger. When graphing, plot a few points by choosing values for \( t \) to understand the curve's shape. This function, \( y = 4^t \), will start at \( (0, 1) \) because any number raised to the power of zero is 1.

Vertical Stretch

A vertical stretch occurs when an exponential function is multiplied by a constant factor greater than 1, altering its graph without changing its exponential base. Consider the function \( y = 3(4^t) \). The 3 multiplies each output from the base function \( y = 4^t \), stretching it vertically. This transformation makes the curve steeper and shifts it upwards. At \( t = 0 \), where the basic function starts at 1, this stretched version starts at 3. The entire graph is lifted while maintaining the same growth trend as the original. Similarly, for \( y = \frac{3}{2}(4^t) \), the constant \( \frac{3}{2} \) results in a less pronounced vertical stretch compared to \( y = 3(4^t) \). Understanding this effect is crucial for analyzing changes in exponential graphs.

Y-Intercept

The y-intercept of a function is the point where its graph crosses the y-axis, occurring when \( t = 0 \). For the exponential function \( y = 4^t \), substituting \( t = 0 \) gives \( y = 1 \). In contrast, for the stretched version \( y = 3(4^t) \), substituting \( t = 0 \) results in \( y = 3 \). Thus, the y-intercept changes according to the vertical stretch factor. For \( y = \frac{3}{2}(4^t) \), the intercept is \( \frac{3}{2} \). The y-intercept provides a foundational point from which the rest of the graph builds, crucial for drawing and comparing the graphs of different exponential functions.

Comparing Graphs

When comparing graphs of exponential functions, pay attention to their growth rate, vertical shifts, and y-intercepts. Take \( y = 4^t \), \( y = 3(4^t) \), and \( y = \frac{3}{2}(4^t) \) as examples. While all share the same base, their vertical stretches differentiate them. This affects their relative positions on a graph: \( y = 3(4^t) \) is always above \( y = \frac{3}{2}(4^t) \), which, in turn, is above \( y = 4^t \). Each graph's starting point, or y-intercept, also distinguishes them, with each curve positioned higher or lower based on its stretch factor. Evaluating these aspects helps in visualizing the relationships and drawing the graphs accurately.

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