Book cover for Algebra and Trigonometry Real Mathematics, Real People

Algebra and Trigonometry Real Mathematics, Real People

Ron Larson

ISBN #9781305071735

7th Edition

6,909 Questions

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234,597 Students Helped

Homework Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This section introduces the method of graphing equations by identifying solution points, creating tables of values, and plotting these points on a coordinate system. It emphasizes both the traditional point-plotting method and the use of graphing utilities to create accurate graphs. Additionally, real-world applications, such as modeling a marathon run or calculating wages based on sales, underscore the importance of understanding the relationship between algebraic equations and their geometric representations.

Learning Objectives

1

Determine whether a given point is a solution to an equation by substitution.

2

Sketch graphs of equations using the point-plotting method by isolating variables and creating a table of values.

3

Utilize graphing utilities effectively to generate accurate graphs and adjust viewing windows.

4

Apply algebraic, graphical, and numerical approaches to solve real-world problems modeled by equations.

Key Concepts

CONCEPT

DEFINITION

Graph

The set of all solution points of an equation, representing the relationship between variables.

Solution Point

A point (a, b) that satisfies an equation when a is substituted for x and b for y.

Point-Plotting Method

A procedure to sketch the graph of an equation by isolating a variable, creating a table of values, plotting the points on a coordinate system, and connecting them.

Intercepts

Points where the graph crosses the axes; the x-intercept is where y = 0 and the y-intercept is where x = 0.

Graphing Utility

A computer or calculator tool that plots equations quickly by generating many solution points and allowing adjustments to viewing windows.

Example Problems

Example 1

For an equation in $x$ and $y,$ if substitution of $a$ for $x$ and $b$ for $y$ satisfics the equation, then the point $(a, b)$ is a _____.

Example 2

The set of all solution points of an equation is the _____ of the equation.

Example 3

Name three common approaches you can use to solve problems mathematically.

Example 4

List the steps for sketching the graph of an equation by point plotting.

Example 5

Determine whether each point lies on the graph of the equation. $y=\sqrt{x+4}$ (a) (0,2) (b) (12,4)
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Step-by-Step Explanations

QUESTION

Determine whether the point (2, 13) lies on the graph of y = 10x − 7.

STEP-BY-STEP ANSWER:

Step 1: Write down the equation y = 10x − 7.
Step 2: Substitute x = 2 into the equation to compute y.
Step 3: Calculate 10(2) − 7 = 20 − 7 = 13.
Step 4: Compare the computed value with the given y-coordinate (13).
Final Answer: Since 13 equals 13, the point (2, 13) is a solution and lies on the graph.

Verifying a Solution Point

QUESTION

Sketch the graph of the equation 3x + y = 6 by point plotting.

STEP-BY-STEP ANSWER:

Step 1: Isolate y in the equation: y = 6 − 3x.
Step 2: Choose several convenient values for x (e.g., −1, 0, 1, 2, 3) and compute the corresponding y-values.
Step 3: Create a table of values: for x = −1, y = 9; x = 0, y = 6; x = 1, y = 3; x = 2, y = 0; x = 3, y = −3.
Step 4: Plot these points on a coordinate plane.
Step 5: Connect the points with a smooth straight line, which represents the graph.
Final Answer: The graph is a straight line passing through the plotted points, clearly illustrating the relationship defined by y = 6 − 3x.

Graphing by Point Plotting

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Common Mistakes

  • Failing to substitute both x and y values correctly when verifying a solution point.
  • Not rewriting the equation to isolate y when using the point-plotting method, leading to confusion in creating a table of values.
  • Choosing an inappropriate viewing window on a graphing utility, which can distort the graph’s features.
  • Forgetting to plot enough points to accurately represent the shape of the graph, especially for nonlinear equations like parabolas or circles.
  • Neglecting the need to adjust scale settings (e.g., using a square setting) when graphing circles to avoid distortion.