Book cover for College Algebra

College Algebra

Michael Sullivan

ISBN #9780321979476

10th Edition

5,609 Questions

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320,854 Students Helped

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This section emphasizes the fundamental aspects of linear functions, including graphing using the slope-intercept form, interpreting the constant average rate of change as the slope, and determining the behavior of linear functions based on the sign of the slope. It also extends these ideas to real-world applications like depreciation, supply and demand equilibrium, and costs, and introduces scatter diagrams as a tool to visually assess relationships in data. Mastery of these concepts is crucial for constructing and interpreting linear models effectively.

Learning Objectives

1

Graph linear functions accurately using the slope-intercept form y = mx + b.

2

Utilize the concept of average rate of change to identify linear functions and interpret the slope.

3

Determine if a linear function is increasing, decreasing, or constant based on the sign of its slope.

4

Construct linear models from verbal descriptions, including applications such as depreciation, supply and demand, and cost functions.

5

Interpret scatter diagrams and understand the process of fitting a line of best fit to data.

Key Concepts

CONCEPT

DEFINITION

Linear Function

A function of the form f(x) = mx + b, whose graph is a straight line with slope m and y-intercept b.

Slope (m)

The rate at which the function changes; it represents the average rate of change, calculated as Δy/Δx. A positive slope indicates an increasing function, a negative slope indicates a decreasing function, and a zero slope indicates a constant function.

Y-intercept (b)

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

Average Rate of Change

For a function f(x), it is calculated as (f(x2) - f(x1)) / (x2 - x1). For linear functions, this value is constant and equals the slope m.

Nonlinear Function

Any function that does not have a constant rate of change; its graph is not a straight line.

Scatter Diagram

A graph of ordered pairs (x, y) used to visually assess the relationship between two variables, often the first step in building a linear model from data.

Equilibrium (in Supply and Demand)

The point where the quantity supplied equals the quantity demanded, often found by solving two linear equations.

Example Problems

Example 1

Graph $y=x^{2}-1 .(\mathrm{pp} .157-164)$

Example 2

Find the slope of the line joining the points $(2,5)$ and $(-1,3) .(\mathrm{pp.} .167-175)$

Example 3

Find the average rate of change of $f(x)=3 x^{2}-2,$ from 2 to $4 .(\mathrm{pp} .223-231)$

Example 4

Solve: $60 x-900=-15 x+2850 .(\mathrm{pp.} 82-87)$

Example 5

If $f(x)=x^{2}-4,$ find $f(-2)$
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Step-by-Step Explanations

QUESTION

Graph the linear function f(x) = -3x + 7 and identify its domain and range.

STEP-BY-STEP ANSWER:

Step 1: Identify the slope and y-intercept. Here, m = -3 and b = 7.
Step 2: Plot the y-intercept at (0, 7) on the Cartesian plane.
Step 3: Use the slope to find another point. Starting at (0, 7), move right 1 unit (x increases by 1) and down 3 units (y decreases by 3) to get (1, 4).
Step 4: Draw a straight line through the points (0, 7) and (1, 4).
Step 5: Since the domain and range of any linear function are all real numbers, state: Domain = ℝ and Range = ℝ.
Final Answer: The graph of f(x) = -3x + 7 is a straight line through (0, 7) and (1, 4) with domain and range as all real numbers.

Graphing a Linear Function

QUESTION

Show that the average rate of change for the function f(x) = -3x + 7 is equal to the slope.

STEP-BY-STEP ANSWER:

Step 1: Choose two distinct x-values, x1 and x2.
Step 2: Compute the function values: f(x1) = -3x1 + 7 and f(x2) = -3x2 + 7.
Step 3: Calculate the average rate of change: (f(x2) - f(x1)) / (x2 - x1).
Step 4: Substitute the values: [(-3x2 + 7) - (-3x1 + 7)] / (x2 - x1) = (-3x2 + 3x1) / (x2 - x1).
Step 5: Factor out -3: -3(x2 - x1) / (x2 - x1) = -3.
Final Answer: The average rate of change is -3, which is exactly the slope m of the function.

Using Average Rate of Change

QUESTION

Determine if the linear function g(x) = 5x - 2 is increasing, decreasing, or constant.

STEP-BY-STEP ANSWER:

Step 1: Identify the slope, m, from the equation. Here, m = 5.
Step 2: Since m is positive, the function increases as x increases.
Final Answer: g(x) is an increasing function across its entire domain.

Determining Function Behavior

QUESTION

A company buys cars for $31,500 each and depreciates them by $4,500 per year using straight-line depreciation. Write the linear function for the book value V(x) as a function of the car's age x (in years) and find the book value after 3 years.

STEP-BY-STEP ANSWER:

Step 1: The initial value is the y-intercept, b = 31,500.
Step 2: The annual depreciation is the slope, m = -4,500.
Step 3: Write the function: V(x) = -4500x + 31500.
Step 4: Evaluate the function at x = 3: V(3) = -4500(3) + 31500 = -13500 + 31500 = 18000.
Final Answer: The book value after 3 years is $18,000.

Building a Linear Model from a Verbal Description (Depreciation)

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Common Mistakes

  • Mistaking the y-intercept for the slope or failing to correctly identify the values of m and b in the equation.
  • Confusing average rate of change with instantaneous rate of change, especially in nonlinear contexts.
  • Forgetting that the domain and range of a linear function are all real numbers, unless additional context is provided.
  • In scatter diagrams, incorrectly plotting points or connecting them, which may misrepresent the underlying relationship.
  • Misinterpreting the sign of the slope leading to wrong conclusions about whether a function is increasing or decreasing.