Book cover for College Algebra

College Algebra

Jay Abramson

ISBN #9781680920376

1st Edition

3,813 Questions

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418,521 Students Helped

Homework Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This section on linear functions covers the foundational concept of representing relationships with a constant rate of change. Key takeaways include understanding the slope and y-intercept, the slope-intercept form of a linear equation, and its applications in real-world scenarios. By mastering these concepts, students can analyze and graph linear relationships accurately.

Learning Objectives

1

Define and describe the characteristics of a linear function.

2

Identify and calculate the slope and y-intercept from a linear equation.

3

Graph linear functions using the slope-intercept form.

4

Apply linear functions to solve real-world problems.

5

Analyze how changes in slope and intercept affect the graph of a linear function.

Key Concepts

CONCEPT

DEFINITION

Linear Function

A function that can be graphed as a straight line, typically written in the form y = mx + b, where m is the slope and b is the y-intercept.

Slope

A measure of the steepness of the line, calculated as the ratio of the change in y to the change in x between any two points on the line.

Y-Intercept

The point where the line crosses the y-axis, represented by the value b in the equation y = mx + b.

Slope-Intercept Form

A common way of writing the equation of a line: y = mx + b, which makes it easy to identify the slope and y-intercept.

Example Problems

Example 1

Terry is kking down a steep hill. Terry's elevation, $E(t)$ in feet after $t$ seconds is given by $E(t)=3000-70 t .$ Write a complete sentence describing Terrys starting elevation and how it is changing over time.

Example 2

Jessica is walking home from a friend's house. After 2 minutes she is 1.4 miles from home. Twelve minutes after leaving, she is 0.9 milesf rom home. What is her rate in miles per hour?

Example 3

A boat is 100 miles away from the marina, sailing directly toward it at 10 miles per hour Write an equation for the distance of the boat from the marina after $t$ hours.

Example 4

If the graphs of two linear functions are perpendicular, describe the relationship between the slopes and the $y$ -intercepts.

Example 5

If a horizontal line has the equation $f(x)=a$ and a vertical line has the equation $x=a$ , what is the point of intersection? Explain why what you found is the point of intersection.
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Step-by-Step Explanations

QUESTION

Given two points on a line, (x₁, y₁) and (x₂, y₂), how do you calculate the slope?

STEP-BY-STEP ANSWER:

Step 1: Identify the coordinates of the two points.
Step 2: Subtract the y-coordinate of the first point from the y-coordinate of the second point: (y₂ - y₁).
Step 3: Subtract the x-coordinate of the first point from the x-coordinate of the second point: (x₂ - x₁).
Step 4: Divide the difference in y-values by the difference in x-values: Slope = (y₂ - y₁) / (x₂ - x₁).
Final Answer: The slope of the line is (y₂ - y₁) / (x₂ - x₁).

Finding the Slope of a Line

QUESTION

How do you write the equation of a line in the form y = mx + b if you know the slope and a point on the line?

STEP-BY-STEP ANSWER:

Step 1: Start with the slope-intercept equation: y = mx + b.
Step 2: Substitute the known slope (m) into the equation.
Step 3: Use the coordinates of the given point (x₁, y₁) by substituting x₁ for x and y₁ for y.
Step 4: Solve the equation for b, the y-intercept.
Step 5: Write the final equation with the identified slope and y-intercept.
Final Answer: The equation of the line is y = mx + b with the calculated values.

Writing the Equation in Slope-Intercept Form

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Common Mistakes

  • Mixing up the roles of the slope and the y-intercept in an equation.
  • Incorrectly computing the difference in coordinates when calculating the slope.
  • Assuming all functions that pass through two points are linear without verifying constant rate of change.
  • Failing to correctly substitute values into the slope-intercept form, leading to arithmetic mistakes.