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Chapter 9: Problem 4

Which of the following defines \(y\) as a linear function of \(x ?\) A. \(y=\frac{1}{4} x-\frac{5}{4}\) B. \(y=\frac{1}{x}\) C. \(y=x^{2}\) D. \(y=x^{3}\)

Short Answer

Expert verified
Option A, \(y = \frac{1}{4} x - \frac{5}{4}\), defines \(y\) as a linear function of \(x\).

Step by step solution

01

Identify the form of a linear function

A linear function is generally written in the form of \[y = mx + b\] , where \(m\) d is the slope and \(b\) d is the y-intercept. The defining characteristic of a linear function is that the variable \(x\) appears to the first power.
02

Examine option A

Given the equation \(y = \frac{1}{4} x - \frac{5}{4}\) , notice that \(x\) appears to the first power. It matches the form \(y = mx + b\) , so it is a linear function.
03

Examine option B

Given the equation \(y = \frac{1}{x}\) , notice that \(x\) is in the denominator. This is not in the form \(y = mx + b\) and is not a linear function.
04

Examine option C

Given the equation \(y = x^2\) , notice that \(x\) is raised to the second power. This does not match the form \(y = mx + b\) and is not a linear function.
05

Examine option D

Given the equation \(y = x^3\) , notice that \(x\) is raised to the third power. This does not match the form \(y = mx + b\) and is not a linear function.
06

Determine the linear function

After examining all options, it is concluded that option A, \(y = \frac{1}{4} x - \frac{5}{4}\), defines \(y\) as a linear function of \(x\).

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Key Concepts

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

headline of the respective core concept
In this section, we will dive into understanding Algebra, its role in analyzing linear functions, and how to recognize a linear function among various equations.
Understanding Algebra in Linear Functions
Algebra is a branch of mathematics that allows us to express and solve equations using variables.
It serves as a tool to find unknown values and understand relationships between quantities.
Linear functions, a core component of algebra, are defined by the equation \(y = mx + b\).

Here,
  • \(m\) represents the slope, which indicates how steep the line is.
  • \(b\) is the y-intercept, the point where the line crosses the y-axis.
A linear function graphs as a straight line.
This is crucial in algebra because the simplicity of linear functions makes them easier to work with compared to non-linear ones.
Slope-Intercept Form and its Importance
The slope-intercept form of a linear equation is \(y = mx + b\).
In this form:
  • The slope \(m\) shows the rate of change of \(y\) with respect to \(x\).If \(m\) is positive, the line rises; if negative, it falls.
  • The y-intercept \(b\) indicates the value of \(y\) when \(x = 0\).It's where the line starts on the y-axis.
This form is very intuitive and helpful for quickly identifying key characteristics of the linear function and for graphing its line.
In the given exercise, option A \(y = \frac{1}{4} x - \frac{5}{4}\) is in slope-intercept form, with a slope of \(\frac{1}{4}\) and a y-intercept of \(-\frac{5}{4}\). Usually, recognizing these will assist in graphing and interpreting linear functions accurately.
Function Properties of Linear Functions
Linear functions have several important properties:
  • Constant Rate of Change: The slope \(m\) remains constant throughout the function.
    This makes linear functions predictable and easy to work with.
  • One-to-One Function: Each \(x\) value has a unique \(y\) value, and vice versa.
  • Graphical Representation: They graph as straight lines.
  • Simple to Solve: Solving a linear function typically involves straightforward algebraic manipulations.
    You can easily find unknowns and intercepts.
Recognizing a linear function means identifying these properties.
In our exercise, option A maintains a constant slope and intercept, thereby confirming it as a linear function.
Options B, C, and D do not, as they involve transformations that disqualify them from being linear.

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Most popular questions from this chapter

Solve each problem. The amount of light (measured in foot-candles) produced by a light source varies inversely as the square of the distance from the source. If the illumination produced \(1 \mathrm{~m}\) from a light source is 768 foot- candles, find the illumination produced \(6 \mathrm{~m}\) from the same source.

The cost in dollars to produce \(x\) youth baseball caps is \(C(x)=4.3 x+75 .\) The revenue in dollars from sales of \(x\) caps is \(R(x)=25 x\) (a) Write and simplify a function \(P\) that gives profit in terms of \(x\). (b) Find the profit if 50 caps are produced and sold.

Solve each problem. The force with which Earth attracts an object above Earth's surface varies inversely as the square of the distance of the object from the center of Earth. If an object \(4000 \mathrm{mi}\) from the center of Earth is attracted with a force of \(160 \mathrm{lb},\) find the force of attraction if the object were \(6000 \mathrm{mi}\) from the center of Earth.

Solve each problem. The force needed to keep a car from skidding on a curve varies inversely as the radius of the curve and jointly as the weight of the car and the square of the speed. If 242 lb of force keeps a 2000 - \(\mathrm{lb}\) car from skidding on a curve of radius \(500 \mathrm{ft}\) at \(30 \mathrm{mph},\) what force (to the nearest tenth of a pound) would keep the same car from skidding on a curve of radius \(750 \mathrm{ft}\) at \(50 \mathrm{mph} ?\)

Determine whether each relation defines y as a function of \(x .\) (Solve for y first if necessary.) Give the domain. $$ y=x $$
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