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Chapter 6: Problem 95

Find an exponential function whose graph has the horizontal asymptote \(y=2\) and contains the points (0,3) and \((1,5) .\)

Short Answer

Expert verified
The exponential function is \( y = 3^x + 2 \).

Step by step solution

01

Identify the general form of the exponential function

An exponential function can generally be written as \( y = a \times b^x + c \), where \(c\) is the horizontal asymptote. Given the horizontal asymptote \( y = 2 \), we can refine this to \( y = a \times b^x + 2 \).
02

Use the point (0,3) in the function

Substitute \(x = 0\) and \(y = 3\) into the refined function. This gives us: \[ 3 = a \times b^0 + 2 \] Since \( b^0 = 1 \), this simplifies to: \[ 3 = a + 2 \] Solving for \(a\): \[ a = 1 \]
03

Substitute \(a\) back into the function

Now, using \( a = 1 \), the equation becomes: \[ y = 1 \times b^x + 2 \] or more simply: \[ y = b^x + 2 \]
04

Use the point (1,5) to find \(b\)

Substitute \(x = 1\) and \(y = 5\) into the equation: \[ 5 = b^1 + 2 \] Simplifying: \[ 5 = b + 2 \] Solving for \(b\): \[ b = 3 \]
05

Write the final exponential function

Substitute \(b = 3\) back into the equation to get the final form of the exponential function: \[ y = 3^x + 2 \]

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Key Concepts

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

horizontal asymptote
The horizontal asymptote of a function is a horizontal line that the graph of the function approaches but never actually touches. For exponential functions, the horizontal asymptote helps determine the end behavior of the function. In our given problem, the horizontal asymptote is at \(y=2\). This tells us that as \(x\) becomes very large or very small, the value of \(y\) will get closer and closer to 2, but it will never quite reach it. So, the function values gravitate around \(y = 2\) for extreme values of \(x\).
exponential function form
An exponential function can generally be written in the form \(y = a \times b^x + c\). Here, \(a\) is a coefficient that stretches or compresses the graph vertically. \(b\) is the base of the exponential and it determines the rate of growth or decay. \(c\) is the horizontal asymptote value.
In this exercise, the provided horizontal asymptote \(y = 2\) tells us that \(c = 2\). When substituting the given points and solving for \(a\) and \(b\), we found the function to be \(y = 3^x + 2\).
solving exponential equations
Solving exponential equations involves manipulating the equation to isolate the variable, often using properties of exponents. Let's break down the solution from our original exercise:

First, we use the point (0, 3) to find \(a\):
The equation becomes \(3 = a \times 1 + 2\) so \(a = 1\).
With \(a = 1\), substituting in the second point (1, 5):
The modified equation becomes \(5 = b + 2\), which simplifies to \(b = 3\).
Thus, substituting these findings back into the form, our final function is \(y = 3^x + 2\).
substitution method in algebra
The substitution method in algebra is a technique used to solve systems of equations or verify points against a given function.
Here, we used this method to plug in known values into the general exponential function form to find unknowns like \(a\) and \(b\).
Substituting \(x = 0\) and \(y = 3\) helped us determine \(a\), while using \(x = 1\) and \(y = 5\) enabled us to find \(b\).
This step-by-step substitution ensured our values of \(a\) and \(b\) precisely fit the points provided, resulting in the accurate exponential function \(y = 3^x + 2\).

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Most popular questions from this chapter

Environmentalists often capture an endangered species and transport the species to a controlled environment where the species can produce offspring and regenerate its population. Suppose that six American bald eagles are captured, transported to Montana, and set free. Based on experience, the environmentalists expect the population to grow according to the model $$P(t)=\frac{500}{1+83.33 e^{-0.162 t}}$$ $$\text { where } t \text { is measured in years. }$$ (a) Determine the carrying capacity of the environment. (b) What is the growth rate of the bald eagle? (c) What is the population after 3 years? (d) When will the population be 300 eagles? (e) How long does it take for the population to reach one-half of the carrying capacity?

Uninhibited growth can be modeled by exponential functions other than \(A(t)=A_{0} e^{k t} .\) For example, if an initial population \(P_{0}\) requires \(n\) units of time to double, then the function \(P(t)=P_{0} \cdot 2^{t / n}\) models the size of the population at time t. Likewise, a population requiring \(n\) units of time to triple can be modeled by \(P(t)=P_{0} \cdot 3^{t / n}\). The population of a town is growing exponentially. (a) If its population doubled in size over an 8 -year period and the current population is 25,000 , write an exponential function of the form \(P(t)=P_{0} \cdot 2^{t / n}\) that models the population. (b) What will the population be in 3 years? (c) When will the population reach \(80,000 ?\) (d) Express the model from part (a) in the form \(A(t)=A_{0} e^{k t}\).

The concentration of alcohol in a person's bloodstream is measurable. Suppose that the relative risk \(R\) of having an accident while driving a car can be modeled by an equation of the form$$R=e^{k x}$$ where \(x\) is the percent concentration of alcohol in the bloodstream and \(k\) is a constant. (a) Suppose that a concentration of alcohol in the bloodstream of 0.03 percent results in a relative risk of an accident of \(1.4 .\) Find the constant \(k\) in the equation. (b) Using this value of \(k,\) what is the relative risk if the concentration is 0.17 percent? (c) Using the same value of \(k,\) what concentration of alcohol corresponds to a relative risk of \(100 ?\) (d) If the law asserts that anyone with a relative risk of having an accident of 5 or more should not have driving privileges, at what concentration of alcohol in the bloodstream should a driver be arrested and charged with a DUI? (e) Compare this situation with that of Example \(10 .\) If you were a lawmaker, which situation would you support? Give your reasons.

Strontium-90 is a radioactive material that decays according to the function \(A(t)=A_{0} e^{-0.0244 t}\) where \(A_{0}\) is the initial amount present and \(A\) is the amount present at time \(t\) (in years). Assume that a scientist has a sample of 500 grams of strontium-90. (a) What is the decay rate of strontium-90? (b) How much strontium-90 is left after 10 years? (c) When will 400 grams of strontium-90 be left? (d) What is the half-life of strontium-90?

Write each expression as a sum and/or difference of logarithms. Express powers as factors. \(\log _{7} x^{5}\)
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