Problem 2

Question

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.

Step-by-Step Solution

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Answer

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

1Step 1: 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.

2Step 2: 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.

3Step 3: 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.

4Step 4: 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.

Key Concepts

Exponential GrowthLinear FunctionsGraphing Window Settings

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.

Problem 2

Problem 3

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