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Chapter 5: Problem 1

Determine which of the following functions are linear, which are exponential, and which are neither. In each case identify the vertical intercept. a. \(C(t)=3 t+5\) b. \(f(x)=3(5)^{x}\) c. \(y=5 x^{2}+3\) d. \(Q=6\left(\frac{3}{2}\right)^{t}\) e. \(P=7(1.25)^{t}\) f. \(T=1.25 n\)

Short Answer

Expert verified
a. Linear, intercept = 5; b. Exponential, intercept = 3; c. Neither, intercept = 3; d. Exponential, intercept = 6; e. Exponential, intercept = 7; f. Linear, intercept = 0.

Step by step solution

01

Identify the Function Type for a

The function given is \(C(t) = 3t + 5\). A linear function is of the form \(y = mx + b\). Since this function fits that form, it is linear.
02

Identify the Vertical Intercept for a

The vertical intercept of a linear function \(y = mx + b\) is \(b\). Hence, the vertical intercept for \(C(t) = 3t + 5\) is 5.
03

Identify the Function Type for b

The function given is \(f(x) = 3(5)^{x}\). An exponential function is of the form \(y = a \times b^x\). Since this function fits that form, it is exponential.
04

Identify the Vertical Intercept for b

To find the vertical intercept of an exponential function \(y = a \times b^x\), set \(x = 0\). So, \(f(0) = 3(5)^0 = 3\). The vertical intercept is 3.
05

Identify the Function Type for c

The function given is \(y = 5x^2 + 3\). This is a quadratic function (not linear or exponential) since it includes a term with \(x^2\). Hence, it is neither.
06

Identify the Vertical Intercept for c

For any function, the vertical intercept is the value of the function when the input (\(x\)) is 0. For \(y = 5x^2 + 3\), \(y(0) = 5(0)^2 + 3 = 3\). So the vertical intercept is 3.
07

Identify the Function Type for d

The function given is \(Q = 6 \times \big(\frac{3}{2}\big)^{t}\). This fits the form of an exponential function \(y = a \times b^t\). Hence, it is exponential.
08

Identify the Vertical Intercept for d

To find the vertical intercept of \(Q = 6 \times \big(\frac{3}{2}\big)^{t}\), set \(t = 0\). So, \(Q(0) = 6 \times \big(\frac{3}{2}\big)^0 = 6\). The vertical intercept is 6.
09

Identify the Function Type for e

The function given is \(P = 7 (1.25)^t\). This fits the form of an exponential function \(y = a \times b^t\). Hence, it is exponential.
10

Identify the Vertical Intercept for e

To find the vertical intercept of \(P = 7 (1.25)^t\), set \(t = 0\). So, \(P(0) = 7 (1.25)^0 = 7\). The vertical intercept is 7.
11

Identify the Function Type for f

The function given is \(T = 1.25 n\). A linear function is of the form \(y = mx + b\) but without a constant term, it's still linear. Hence, it is linear.
12

Identify the Vertical Intercept for f

The intercept occurs where \(n = 0\). \(T(0) = 1.25 \times 0 = 0\). Thus, the vertical intercept is 0.

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Key Concepts

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

Linear Functions
Let's discuss linear functions. A linear function is a mathematical expression that forms a straight line when graphed. This type of function is characterized by the equation format: \(y = mx + b\), where:
  • \(m\) represents the slope (the rate of change).
  • \(b\) represents the vertical intercept (where the line crosses the y-axis).
If you see a term multiplied by a variable and a constant term, it's likely a linear function. For example, in the exercise given, we identified two linear functions:

  • a. \(C(t) = 3t + 5\)
  • f. \(T = 1.25n\)
Linear functions are straightforward to identify and work with due to their simplicity. The main characteristic is that the graph will always produce a straight line with a constant slope.
Exponential Functions
Next, let's dive into exponential functions. These functions look like \(y = a \times b^x\), where:
  • \(a\) is a constant (the initial value or amplitude).
  • \(b\) is the base, which is a positive real number.
  • \(x\) is the exponent and is usually represented by a variable.
Exponential functions grow much faster or decay much more rapidly than linear functions. Examples from the exercise include:

  • b. \(f(x) = 3(5)^x\)
  • d. \(Q = 6\big(\frac{3}{2}\big)^t\)
  • e. \(P = 7 (1.25)^t\)
Exponential functions are recognizable by their rapid increase or decrease and are often used in contexts like population growth or radioactive decay. In these examples, when you plot the function, it creates a curve that either swoops upward or downward, depending on the base.
Vertical Intercepts
Finally, let's talk about vertical intercepts. The vertical intercept is the point where the graph of a function crosses the y-axis. This is the value of the function when the independent variable (x or t) is zero. To find the vertical intercept, simply set your variable to zero:
  • For a linear function \(y = mx + b\), the intercept is \(b\).
  • For an exponential function \(y = a \times b^x\), the intercept is \(a\) because \(b^0 = 1\).
Let's look at the examples:

  • a. \(C(t) = 3t + 5\) -> vertical intercept is 5.
  • b. \(f(x) = 3(5)^x\) -> vertical intercept is 3.
  • d. \(Q = 6 \big(\frac{3}{2}\big)^{t}\) -> vertical intercept is 6.
  • e. \(P = 7 (1.25)^t\) -> vertical intercept is 7.
  • f. \(T = 1.25n\) -> vertical intercept is 0.
Vertical intercepts help understand where the graph begins in the context of real-world applications. For instance, in population growth, it might represent the initial population before growth starts.

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Most popular questions from this chapter

According to the Arkansas Democrat Gazette (February \(27,\) 1994): Jonathan Holdeen thought up a way to end taxes forever. It was disarmingly simple. He would merely set aside some money in trust for the government and leave it there for 500 or 1000 years. Just a penny, Holdeen calculated, could grow to trillions of dollars in that time. But the stash he had in mind would grow much bigger-to quadrillions or quintillions-so big that the government, one day, could pay for all its operations simply from the income. Then taxes could be abolished. And everyone would be better off. a. Holdeen died in 1967 , leaving a trust of \(\$ 2.8\) million that is being managed by his daughter, Janet Adams. In 1994 , the trust was worth \(\$ 21.6\) million. The trust was debated in Philadelphia Orphans' Court. Some lawyers who were trying to break the trust said that it is dangerous to let it go on, because "it would sponge up all the money in the world." Is this possible? b. After 500 years, how much would the trust be worth? Would this be enough to pay off the current national debt (over \(\$ 7\) trillion in 2004\() ?\) What about after 1000 years? Describe the model you used to make your predictions.

Plutonium- 238 is used in bombs and power plants but is dangerously radioactive. It decays very slowly into nonradioactive materials. If you started with 100 grams today, a year from now you would have 99.2 grams. a. Construct an exponential function to describe the decay of plutonium- 238 over time. b. How much of the original 100 grams of plutonium- 238 would be left after 50 years? After 500 years?

Each of the following three exponential functions is in the $$ \begin{array}{l} \text { standard form } y=C \cdot a^{x} . \\ y=2^{x} \quad y=5^{x} \quad y=10^{x} \end{array} $$ a. In each case identify \(C\) and \(a\). b. Specify whether each function represents growth or decay. In particular, for each unit increase in \(x,\) what happens to \(y ?\) c. Do all three curves intersect? If so, where? d. In the first quadrant, which curve should be on top? Which in the middle? Which on the bottom? e. Describe any horizontal asymptotes. f. For each function, generate a small table of values. g. Graph the three functions on the same grid and verify that your predictions in part (d) are correct.

Identify the doubling time or half-life of each of the following exponential functions. Assume \(t\) is in years. [Hint: What value of \(t\) would give you a growth (or decay) factor of 2 (or \(1 / 2\) )?] a. \(Q=70(2)^{t}\) b. \(Q=1000(2)^{t / 50}\) c. \(Q=300\left(\frac{1}{2}\right)^{t}\) d. \(Q=100\left(\frac{1}{2}\right)^{t / 250}\) e. \(N=550(2)^{t / 10}\) f. \(N=50\left(\frac{1}{2}\right)^{t / 20}\)

Construct both a linear and an exponential function that go through the points (0,6) and (1,9) .
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