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Chapter 1: Problem 1

Determine whether or not the function is a power function. If it is a power function, write it in the form \(y=k x^{p}\) and give the values of \(k\) and \(p\) $$y=\frac{x}{5}$$

Short Answer

Expert verified
Yes, it is a power function with \(k=\frac{1}{5}\) and \(p=1\).

Step by step solution

01

Identify the given function

The given function is \( y = \frac{x}{5} \). We need to determine if this can be expressed as a power function of the form \( y = kx^p \).
02

Simplify the function to check for power function form

Rewrite the given function \( y = \frac{x}{5} \) in a way that fits the form \( y = kx^p \). Specifically, we can write it as \( y = \frac{1}{5}x^1 \), which resembles the form of a power function where \( k = \frac{1}{5} \) and \( p = 1 \).
03

Verify the power function

Verify that the rewritten function \( y = \frac{1}{5}x^1 \) indeed represents a power function. In this case, \( k \) is a constant \( \frac{1}{5} \) and \( p \) is the exponent \( 1 \), confirming it is a power function.

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Key Concepts

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

Power Function Form
A power function has the form \( y = kx^p \), making it a crucial component in understanding different types of mathematical equations and their behavior. When we say something is a power function, we're referring to any function where a single term is raised to a constant exponent.
  • Form Elements: It must have one term in the form of a variable raised to the power of an exponent.
  • Variable: Typically, the variable involved is \( x \), but it could be any other variable.
  • Simplicity: Unlike other polynomials that may have a sum of multiple terms, a power function only focuses on one term with an exponent.
When given a function, the task is to see if it fits this format. For example, transforming \( y = \frac{x}{5} \) into \( y = \frac{1}{5}x^1 \) proves it can be rewritten in power function form. This step helps in analyzing and classifying different functions and also indicates how they might behave or be manipulated mathematically.
Constant Coefficient
In the context of power functions, the constant coefficient \( k \) plays a vital role in scaling the effect of the function. It is the numerical part that stays constant regardless of the value of the variable \( x \).
  • Value of \( k \): It's the number that multiplies the variable raised to a power. For example, in \( y = \frac{1}{5}x^1 \), \( k = \frac{1}{5} \).
  • Influence: The coefficient determines the function's vertical stretch or shrink. It impacts the steepness of the graph, affecting how quickly the output increases or decreases.
Understanding \( k \) helps predict how the function behaves as \( x \) changes. If \( k \) is larger than 1, the function's growth multiplies rapidly; if between 0 and 1, it's less aggressive.
Exponent
The exponent \( p \) in a power function determines the degree and direction of the function. This impacts the curvature and overall shape of the graph.
  • Role of \( p \): In \( y = kx^p \), \( p \) signifies the power to which the variable is raised. In \( y = \frac{1}{5}x^1 \), \( p = 1 \).
  • Impact on Graph: A positive exponent leads to a standard power function shape, whereas a negative \( p \) would flip the curve, introducing new patterns like reciprocal functions.
  • Nature of p:**: Higher exponents mean curves will rise or drop faster as \( x \) moves away from zero. Linear: For \( p=1 \), the function is linear. The graph is a straight line.
Exploring the exponent allows you to comprehend how rapidly the outputs may change, adding depth to your understanding of function behavior in different mathematical contexts.

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Most popular questions from this chapter

The population of the world can be represented by \(P=\) \(7(1.0115)^{t},\) where \(P\) is in billions of people and \(t\) is years since \(2012 .\) Find a formula for the population of the world using a continuous growth rate.

With time, \(t,\) in years since the start of \(1980,\) textbook prices have increased at \(6.7 \%\) per year while inflation has been \(3.3 \%\) per year. \(^{68}\) Assume both rates are continuous growth rates. (a) Find a formula for \(B(t),\) the price of a textbook in year \(t\) if it \(\operatorname{cost} \$ B_{0}\) in 1980 (b) Find a formula for \(P(t),\) the price of an item in year \(t\) if it cost \(\$ P_{0}\) in 1980 and its price rose according to inflation. (c) A textbook cost \(\$ 50\) in \(1980 .\) When is its price predicted to be double the price that would have resulted from inflation alone?

In 2010 , there were about 246 million vehicles (cars and trucks) and about 308.7 million people in the US. \(^{70}\) The number of vehicles grew \(15.5 \%\) over the previous decade, while the population has been growing at \(9.7 \%\) per decade. If the growth rates remain constant, when will there be, on average, one vehicle per person?

Kleiber's Law states that the metabolic needs (such as calorie requirements) of a mammal are proportional to its body weight raised to the 0.75 power. \(^{86}\) Surprisingly, the daily diets of mammals conform to this relation well. Assuming Kleiber's Law holds: (a) Write a formula for \(C,\) daily calorie consumption, as a function of body weight, \(W\) (b) Sketch a graph of this function. (You do not need scales on the axes.) (c) If a human weighing 150 pounds needs to consume 1800 calories a day, estimate the daily calorie requirement of a horse weighing 700 lbs and of a rabbit weighing 9 lbs. (d) On a per-pound basis, which animal requires more calories: a mouse or an elephant?

The following table shows values of a periodic function \(f(x) .\) The maximum value attained by the function is 5 (a) What is the amplitude of this function? (b) What is the period of this function? (c) Find a formula for this periodic function. $$\begin{array}{|c|c|c|c|c|c|c|c|}\hline x & 0 & 2 & 4 & 6 & 8 & 10 & 12 \\\\\hline f(x) & 5 & 0 & -5 & 0 & 5 & 0 & -5 \\\\\hline\end{array}$$
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