Book cover for College Algebra: Real Mathematics, Real People

College Algebra: Real Mathematics, Real People

Ron Larson

ISBN #9781305778917

7th Edition

5,230 Questions

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38,634 Students Helped

Homework Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This section emphasizes the interconnectedness of algebraic, graphical, and numerical representations in solving equations. It covers how to identify and verify solution points, determine intercepts, and apply fundamental algebraic properties like the distributive and associative properties. Mastery of these skills facilitates accurate graphing and modeling of functions, enhancing problem-solving abilities in various mathematical contexts.

Learning Objectives

1

Identify and interpret solution points from tables and graphs.

2

Determine intercepts (x-intercepts and y-intercepts) of equations and understand their significance.

3

Understand and apply algebraic properties (distributive and associative properties) in simplifying expressions.

4

Translate between numerical, graphical, and algebraic representations of functions and equations.

Key Concepts

CONCEPT

DEFINITION

Solution Point

A coordinate pair (x, y) that satisfies the equation or represents a valid point on a graph.

Intercept

A point where a graph crosses an axis. The x-intercept occurs when y = 0, and the y-intercept occurs when x = 0.

Algebraic Representation

Expressing mathematical relationships using symbols and equations.

Graphical Representation

A visual depicted form of data or equations on coordinate axes.

Distributive Property

An algebraic property stating that a*(b + c) = a*b + a*c, allowing multiplication to be distributed over an addition inside parentheses.

Associative Property of Multiplication

A property stating that the way in which factors are grouped does not affect the product, i.e., (a*b)*c = a*(b*c).

Example Problems

Example 1

Name three types of rigid transformations.

Example 2

Match the rigid transformation of $y=f(x)$ with the correct representation, where $c>0$ a. $h(x)=f(x)+c$ b. $h(x)=f(x)-c$ c. $$\text { c. } h(x)=f(x-c)$$ d. $h(x)=f(x+c)$ i. horizontal shift $c$ units to the left ii. vertical shift $c$ units upward iii. horizontal shift $c$ units to the right iv. vertical shift $c$ units downward

Example 3

Fill in the blanks. A reflection in the $x$ -axis of $y=f(x)$ is represented by $h(x)=$ ________ while a reflection in the $y$ axis of $y=f(x)$ is represented by $h(x)=$ ____.

Example 4

Fill in the blanks. A nonrigid transformation of $y=f(x)$ represented by $c f(x)$ is a vertical stretch when ______ and a vertical shrink when ____.

Example 5

Sketch the graphs of the three functions by hand on the same rectangular coordinate system. Verify your results with a graphing utility. $$\begin{aligned} &f(x)=x\\ &g(x)=x-4\\ &h(x)=3 x \end{aligned}$$
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Step-by-Step Explanations

QUESTION

How do you verify a solution point from a table of values?

STEP-BY-STEP ANSWER:

Step 1: Identify the coordinate pair from the table (e.g., (1, 1/2)).
Step 2: Substitute the x-value of the point into the given equation to see if it returns the y-value.
Step 3: Check the equality to ensure the coordinate satisfies the equation.
Final Answer: If the substituted values match the coordinate pair, it is a correct solution point.

Finding a Solution Point

QUESTION

How can you find the x-intercept of an equation?

STEP-BY-STEP ANSWER:

Step 1: Set y = 0 in the given equation.
Step 2: Solve the equation for x.
Step 3: The solution is the x-coordinate of the x-intercept, with y=0.
Final Answer: The x-intercept is the point (x, 0).

Determining Intercepts

QUESTION

How do you apply the distributive property in simplifying an expression?

STEP-BY-STEP ANSWER:

Step 1: Identify the expression in the form a*(b + c).
Step 2: Multiply 'a' by 'b' and 'a' by 'c' separately.
Step 3: Write the simplified expression as a*b + a*c.
Final Answer: The expression is simplified correctly using the distributive property.

Using the Distributive Property

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Common Mistakes

  • Misinterpreting solution points by not validating that the given coordinate satisfies the equation.
  • Confusing x-intercepts with y-intercepts or neglecting to set the correct variable to zero.
  • Overlooking the proper distribution of multiplication across terms inside parentheses.
  • Mixing up the distributive property with the associative property, leading to incorrect simplifications.