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Chapter 5: Problem 2

Technology Sketch the graphs of \(y=2^{t}\) and \(y=4+2.7725887(t-2)\) a. Using a window of \(0 \leq x \leq 2.5,0 \leq y \leq 6\). b. Using a window of \(1.5 \leq x \leq 2.5,0 \leq y \leq 6\). c. Using a window of \(1.8 \leq x \leq 2.2,3.3 \leq y \leq 4.6\). Mark the point (2,4) on each graph.

Short Answer

Expert verified
Plot each function in the specified windows and mark (2,4).

Step by step solution

01

Understand Equations and Point to Plot

First, identify the equations to be plotted, which are \( y=2^t \) and \( y=4+2.7725887(t-2) \). Additionally, you will need to mark the point \((2,4)\) on both graphs.
02

Graph Settings and Plotting for (a)

Set up your graphing tool to use the window \(0 \leq x \leq 2.5\) and \(0 \leq y \leq 6\). Plot the graph of \( y=2^t \), which is an exponential function increasing rapidly from left to right. Then plot \( y=4+2.7725887(t-2) \), a linear function with a slope, intersecting the y-axis at a certain point when extended. Mark the point \((2,4)\) on both graphs.
03

Graph Settings and Plotting for (b)

Adjust the graphing window to \(1.5 \leq x \leq 2.5\) and \(0 \leq y \leq 6\). Re-plot both functions: \( y=2^t \), which should still show exponential growth albeit slightly restrained by the reduced x-range, and \( y=4+2.7725887(t-2) \), which may look more like a straight diagonal line. Again, ensure the point \((2,4)\) is marked on both graphs.
04

Graph Settings and Plotting for (c)

This time, you need to change the graph window to \(1.8 \leq x \leq 2.2\) and \(3.3 \leq y \leq 4.6\). With this narrow window focusing around \(x = 2\) and \(y = 4\), plot \( y=2^t \) and note how the exponential graph is comparatively flat within this range. Plot \( y=4+2.7725887(t-2) \), which should intersect or run parallel very close to \(y=2^t\), given the small range of x-values. Ensure the point \((2,4)\) is clearly indicated on both graphs.

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Key Concepts

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

Exponential Growth
Exponential growth describes a process where a quantity, such as a population or value, increases at a rate proportional to its current size. This means that the more you have, the faster it grows. Mathematically, exponential growth can be represented by a function of the form \(y = a \, b^t\), where \(a\) is the initial quantity, \(b\) is the base or growth factor, and \(t\) represents time or another independent variable.
In the exercise, the function \(y = 2^t\) exemplifies exponential growth. As \(t\) increases, the value of \(y = 2^t\) increases rapidly. Exponential functions are characterized by their
  • rapidly increasing curve that rises steeply
  • sensitive dependence on the initial value \(a\)
  • consistent multiplicative rate of change determined by \(b\)
Through graphing experiments, you can see how changing the base \(b\) or shifting the function can drastically affect its behavior.
Linear Functions
Linear functions are among the simplest types of functions and are of the form \(y = mx + c\), where \(m\) is the slope and \(c\) is the y-intercept. These functions graph as straight lines that have a constant rate of change, represented by the slope \(m\).
In our case, the function \(y = 4 + 2.7725887(t - 2)\) represents a linear function. Here, we can identify:
  • The slope \(m = 2.7725887\), indicating how much \(y\) changes for each unit change in \(t\).
  • The function shifts vertically and horizontally, connecting it to the coordinate point \((t, y)\) offset by \((2,4)\).
Linear functions are straightforward to graph, and they reveal how two quantities are directly proportional or related through a simple ratio.
Graphing Window Settings
Graphing window settings are essential for accurately visualizing functions on a graph. These settings define the range of values for the x-axis and y-axis that your graphing tool will display. Understanding how to adjust these parameters allows you to focus on areas of interest or analyze specific segments of a graph.
In the original exercise, multiple graphing window settings were proposed:
  • Settings like \(0 \leq x \leq 2.5\) and \(0 \leq y \leq 6\) provide a broad look at both functions, capturing their behaviors over a sizable range.
  • Tighter settings, such as \(1.8 \leq x \leq 2.2\) and \(3.3 \leq y \leq 4.6\), zoom in on particular sections of the graph, allowing detailed analysis near the point \((2,4)\).
Balancing between a wider over-view and a detailed close-up gives deeper insights into the interactions between exponential and linear graphs.

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Most popular questions from this chapter

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