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Chapter 10: Problem 25

Sketch graphs of the exponential functions. Label your axes. $$ y=800(0.9)^{t} $$

Short Answer

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Question: Sketch the graph of the exponential function y = 800(0.9)^t. Answer: To sketch the graph of this exponential function, follow these steps: 1. Identify the function form and key features: The given function is y = 800(0.9)^t, which has an initial value (a) of 800 and a growth factor (b) of 0.9. Since the growth factor is between 0 and 1, the value of y decreases as t increases. 2. Label axes: Label the horizontal axis as t (time) and the vertical axis as y (the value of the function at time t). 3. Plot points: Calculate the values of y for a set of points t, such as -1, 0, 1, 2, 3, and plot them on the graph. 4. Sketch the graph: Connect the plotted points with a smooth curve. The graph should be asymptotic to the horizontal axis (y=0) as t goes to infinity. 5. Summary and review: Ensure that your graph represents the given exponential function, with a decreasing value as t increases and an asymptotic behavior as t approaches infinity.

Step by step solution

01

Identify the function form and key features

Write down the given function and identify its form. The given function is: $$ y=800(0.9)^t $$ This is an exponential function of the form: $$ y=a(b)^{t} $$ Here, a is the initial value (800) and b is the growth factor (0.9). As t increases, the value of y decreases since the growth factor is between 0 and 1.
02

Label axes

Label the axes as follows: On the horizontal axis, write the variable t for time, and on the vertical axis, write y to represent the value of the function at time t.
03

Plot points

Since this is an exponential function and we know its general shape, we only need to plot a few points to sketch the graph. Choose a set of points t as -1, 0, 1, 2, 3, and calculate the corresponding values of y using the given function: $$ y(-1)=800(0.9)^{-1} \\ y(0)=800(0.9)^0 \\ y(1)=800(0.9)^1 \\ y(2)=800(0.9)^2 \\ y(3)=800(0.9)^3 $$
04

Sketch the graph

Plot the calculated points from Step 3 on the graph. You will notice that as t increases, the value of y decreases. The graph is asymptotic to the horizontal axis (y=0), meaning it will not touch the axis but will approach it as t goes to infinity. With the points plotted, sketch the smooth curve that represents the function.
05

Summary and review

We have plotted the graph of the function: $$ y=800(0.9)^t $$ We identified the initial value and growth factor, labeled the axes, plotted points on the graph, and connected the points to sketch the exponential function. Review the graph to ensure that it meets all the criteria and properly represents the function.

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Key Concepts

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

Exponential Decay
Exponential decay in a function occurs when the value of the function decreases over time. This is seen in functions like \( y=800(0.9)^t \). Here, the base of the exponential is less than 1, specifically 0.9.

This means with each increase in \( t \), the function's value \( y \) becomes smaller, creating a downward sloping curve when graphed.
- Exponential decay is contrasted with exponential growth, where the base is greater than 1, causing the function to increase over time.
- The decay rate in this example is inherently tied to the factor of 0.9, indicating a 10% decrease with each time unit.Keep in mind that as \( t \) moves towards infinity in an exponential decay scenario, \( y \) approaches zero but never quite reaches it, making the x-axis a horizontal asymptote.
Graphing Functions
Graphing an exponential function such as \( y=800(0.9)^t \) involves step-by-step plotting of points calculated at different values of \( t \).
- Start by identifying key features like the initial value and the growth factor as well as the general shape of the curve鈥攄ecreasing in the case of decay.
- Ensure the axes are labeled correctly with \( t \) (time) on the horizontal and \( y \) (value) on the vertical axis.
To successfully plot the graph:
  • Select a range of \( t \) values (e.g., -1, 0, 1, 2, 3) and compute corresponding \( y \) values.
  • Plot the points on the graph paper with an appropriate scale.
  • Note that the curve will slope downwards, getting closer to but never touching the t-axis.
The graph should show a smooth curve connecting the plotted points, clearly exhibiting the characteristic decline of exponential decay.
Initial Value
The initial value in an exponential function provides the starting point of the graph at \( t=0 \).
The function \( y=800(0.9)^t \) has 800 as its initial value.
- This means that initially, when no time has passed, \( y \) equals 800.
- On the graph, this denotes the point at which the curve begins on the \( y \)-axis.Understanding the initial value is crucial as it sets the context of the function, indicating what the measurement starts with initially before any decay or growth occurs.
- For exponential decay functions, this value is typically larger than the subsequent values as the function decreases over time.
- It can help provide insight into the scale and scope of the function in a real-world application or context.
Growth Factor
The growth factor in an exponential function is the value by which the function is multiplied as \( t \) increases.
In the function \( y=800(0.9)^t \), 0.9 is the growth factor.
- Although it is termed 'growth factor', when this value is between 0 and 1, it actually represents a reduction or decay.
- Each time \( t \) increases by one unit, \( y \) is multiplied by 0.9, steadily decreasing its value.The growth factor determines the nature of the function:
  • If the factor is greater than 1, the function experiences exponential growth.
  • If the factor is less than 1, like in our function, it represents exponential decay.
Understanding the growth factor is key to predicting the long-term behavior of the function, helping determine how quickly or slowly the function values approach zero.

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Most popular questions from this chapter

The equations describe the value of investments after \(t\) years. For each investment, give the initial value, the continuous growth rate, the annual growth factor, and the annual growth rate. $$ V=20,000 e^{-0.44 t} $$

Prices are increasing at \(5 \%\) per year. What is wrong with the statements? Correct the formula in the statement. A. \$6 item costs \(\$(6 \cdot 1.05)^{7}\) in 7 years' time.

In year \(t,\) the population, \(L,\) of a colony of large ants is \(L=2000(1.05)^{t}\), and the population of a colony of small ants is \(S=1000(1.1)^{t}\). (a) Construct a table showing each colony's population in years \(t=5,10,15,20,25,30,35,40\). (b) The small ants go to war against the large ants; they destroy the large ant colony when there are twice as many small ants as large ants. Use your table to determine in which year this happens. (c) As long as the large ant population is greater than the small ant population, the large ants harvest fruit that falls on the ground between the two colonies. In which years in your table do the large ants harvest the fruit?

For the functions in the form \(P=a b^{t / T}\) describing population growth. (a) Give the values of the constants \(a, b\), and \(T\). What do these constants tell you about population growth? (b) Give the annual growth rate. \(P=50\left(\frac{1}{2}\right)^{t / 6}\)

Without solving them, say whether the equations in Problem had a positive solution, a negative solution, a zero solution, or no solution. Give a reason for your answer. \((3.2)^{2 y+1}(1+3.2)=(3.2)^{y}\)
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