Book cover for College Algebra

College Algebra

Michael Sullivan

ISBN #9780321716811

9th Edition

5,201 Questions

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40,131 Students Helped

Homework Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This section focuses on linear functions and their properties. Key concepts include the slope-intercept form, the meaning of slope and y-intercept, and how to graph linear functions. The section also emphasizes the constant average rate of change for linear functions and extends these ideas to real-world applications, such as depreciation and economic models. Understanding these fundamentals is essential for creating and analyzing linear models effectively.

Learning Objectives

1

Describe the structure and characteristics of a linear function, including slope and y-intercept.

2

Graph linear functions using the slope-intercept form and interpret the results.

3

Explain and compute the average rate of change and understand its relationship to the slope.

4

Determine whether a linear function is increasing, decreasing, or constant based on its slope.

5

Construct linear models from verbal descriptions and real-world data, such as depreciation and supply-demand scenarios.

Key Concepts

CONCEPT

DEFINITION

Linear Function

A function that can be written in the form f(x) = mx + b, where m is the slope and b is the y-intercept. Its graph is a straight line and its domain is all real numbers.

Slope (m)

The rate of change of the function, indicating how much y increases or decreases for a unit increase in x. A positive slope means the function is increasing and a negative slope means it is decreasing.

Y-intercept (b)

The point where the graph of the function crosses the y-axis, which is the value of the function when x = 0.

Average Rate of Change

The ratio of the change in the output to the change in the input over an interval; for a linear function, this average rate is constant and equal to the slope.

Straight-line Depreciation

A method of modeling the decrease in an asset’s value over time using a linear function, where the asset depreciates by a fixed amount each period.

Equilibrium Price

In supply and demand models, it is the price at which the quantity supplied equals the quantity demanded.

Example Problems

Example 1

Answers are given at the end of these exercises. If you get a wrong answer, read the pages listed in red. Graph $y=2 x-3 .(\mathrm{pp} .157-164)$

Example 2

Answers are given at the end of these exercises. If you get a wrong answer, read the pages listed in red. Find the slope of the line joining the points $(2,5)$ and $(-1,3) .(\mathrm{pp} .167-175)$

Example 3

Answers are given at the end of these exercises. If you get a wrong answer, read the pages listed in red. Find the average rate of change of $f(x)=3 x^{2}-2,$ from 2 to $4 .(\mathrm{pp} .222-230)$

Example 4

Answers are given at the end of these exercises. If you get a wrong answer, read the pages listed in red. Solve: $60 x-900=-15 x+2850 .(\mathrm{pp} .82-87)$

Example 5

Answers are given at the end of these exercises. If you get a wrong answer, read the pages listed in red. If $f(x)=x^{2}-4,$ find $f(-2) \cdot(\mathrm{pp} .200-208)$
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Step-by-Step Explanations

QUESTION

Graph the linear function f(x) = -3x + 7.

STEP-BY-STEP ANSWER:

Step 1: Identify the y-intercept (b). For f(x) = -3x + 7, b = 7. Plot the point (0, 7) on the Cartesian plane.
Step 2: Identify the slope (m). Here, m = -3, meaning that for every unit increase in x, y decreases by 3 units.
Step 3: From the y-intercept (0,7), move right 1 unit (to x = 1) and down 3 units (to y = 4) to plot an additional point (1, 4).
Step 4: Draw a straight line through these points extending in both directions. This is the graph of f(x) = -3x + 7.
Final Answer: The graph is a straight line with y-intercept at (0,7) and passing through (1,4), having a slope of -3.

Graphing a Linear Function

QUESTION

Given two points on a function, (x1, y1) and (x2, y2), explain how to calculate the average rate of change.

STEP-BY-STEP ANSWER:

Step 1: Identify the two points. Let the points be (x1, y1) and (x2, y2).
Step 2: Apply the formula: Average Rate of Change = (y2 - y1) / (x2 - x1).
Step 3: Substitute the coordinates of the points into the formula.
Step 4: Simplify the fraction to obtain the constant rate, which, in the case of a linear function, equals the slope.
Final Answer: The average rate of change is calculated using (y2 - y1)/(x2 - x1) and equals the slope m for a linear function.

Determining the Average Rate of Change

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Common Mistakes

  • Confusing the y-intercept with the point where the line crosses the x-axis.
  • Misinterpreting the slope’s sign, leading to errors in determining whether the function is increasing or decreasing.
  • Failing to correctly compute the average rate of change by mixing up the order of subtraction in the formula.
  • Assuming that only nonlinear functions have variable rates of change and overlooking the constant nature of linear function slopes.
  • Plotting points inaccurately or using the wrong step increments based on the slope.