Book cover for Calculus of a Single Variable

Calculus of a Single Variable

Ron Larson, Bruce Edwards

ISBN #9781285060286

10th Edition

6,214 Questions

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101,749 Students Helped

Homework Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This section introduces methods to graph equations through point plotting, intercept analysis, and symmetry testing. It emphasizes how combining graphical, analytic, and numerical approaches deepens understanding. The content also explains how to find points of intersection between graphs and applies these techniques to build and interpret mathematical models in realistic scenarios. Additionally, the concept of slope is elaborated as a measure of rate, essential both in theoretical and real-life applications.

Learning Objectives

1

Describe and graph equations using point plotting, intercepts, and symmetry tests.

2

Apply analytic, numerical, and graphical approaches to find intercepts, test symmetry, and determine points of intersection.

3

Analyze mathematical models that represent real-life data and predict outcomes using regression and linear models.

4

Determine the slope of a line from two points and interpret its meaning in real-world contexts.

Key Concepts

CONCEPT

DEFINITION

Graph of an Equation

The set of all solution points (ordered pairs) that satisfy a given equation, typically represented in the coordinate plane.

Intercept

A point where a graph crosses the x-axis (x-intercept) or y-axis (y-intercept), found by setting the other variable to zero.

Symmetry

A property of a graph where one part is a mirror image of another. Types include symmetry with respect to the x-axis, y-axis, and the origin.

Point Plotting

A technique for sketching graphs by constructing a table of values and plotting corresponding points on the coordinate plane.

Slope

A measure of the rate of change, representing the vertical change (rise) over the horizontal change (run) between two points on a line.

Mathematical Model

An equation or system of equations used to represent and predict real-life data while balancing accuracy and simplicity.

Example Problems

Example 1

Matching In Exercises $1-4,$ match the equation with its graph. [The graphs are labeled (a), (b), (c), and (d).] $$ y=-\frac{3}{2} x+3 $$

Example 2

Matching In Exercises $1-4,$ match the equation with its graph. [The graphs are labeled (a), (b), (c), and (d).] $$ y=\sqrt{9-x^{2}} $$

Example 3

Matching In Exercises $1-4,$ match the equation with its graph. [The graphs are labeled (a), (b), (c), and (d).] $$ y=3-x^{2} $$

Example 4

Matching In Exercises $1-4,$ match the equation with its graph. [The graphs are labeled (a), (b), (c), and (d).] $$ y=x^{3}-x $$

Example 5

Sketching a Graph by Point Plotting In Exercises $5-14$ , sketch the graph of the equation by point plotting. $$ y=\frac{1}{2} x+2 $$
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Step-by-Step Explanations

QUESTION

How do you sketch the graph of y = x² - 2 using point plotting?

STEP-BY-STEP ANSWER:

Step 1: Solve the equation for y by substituting a set of x-values into y = x² - 2.
Step 2: Create a table of values; for example, when x = -2, -1, 0, 1, 2, compute y.
Step 3: Plot the corresponding points (for instance, (-2, 2), (-1, -1), (0, -2), (1, -1), (2, 2)).
Step 4: Connect the points with a smooth curve to reveal the shape of the parabola.
Final Answer: The graph is a symmetric parabola opening upward with vertex at (0, -2).

Sketching a Graph by Point Plotting

QUESTION

How do you find the x-intercepts and y-intercepts of the equation 3x + y = 7?

STEP-BY-STEP ANSWER:

Step 1: To find the y-intercept, set x = 0. Substitute into the equation: 3(0) + y = 7, so y = 7. The y-intercept is (0, 7).
Step 2: To find the x-intercept, set y = 0. Substitute into the equation: 3x + 0 = 7, so x = 7/3. The x-intercept is (7/3, 0).
Final Answer: The intercepts are (0, 7) for the y-axis and (7/3, 0) for the x-axis.

Finding Intercepts

QUESTION

How do you test if the graph of y = 2x³ - x is symmetric with respect to the origin?

STEP-BY-STEP ANSWER:

Step 1: Replace x with -x to see what happens: y becomes 2(-x)³ - (-x) = -2x³ + x.
Step 2: Replace y with -y in the original equation to check for origin symmetry: -y = 2x³ - x, which implies y = -2x³ + x.
Step 3: Since the result is equivalent to the expression obtained in Step 1, the graph is symmetric with respect to the origin.
Final Answer: The graph of y = 2x³ - x is symmetric with respect to the origin.

Testing for Symmetry

QUESTION

How do you compute the slope of a line passing through the points (-2, 0) and (3, 1)?

STEP-BY-STEP ANSWER:

Step 1: Use the slope formula m = (y2 - y1) / (x2 - x1).
Step 2: Substitute the given values: m = (1 - 0) / (3 - (-2)) = 1 / 5.
Final Answer: The slope of the line is 1/5.

Calculating the Slope of a Line

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Common Mistakes

  • Relying on too few plotted points, which can misrepresent the true shape of the graph.
  • Mixing up the roles of x-intercepts and y-intercepts when setting variables to zero.
  • Inconsistently applying the symmetry tests by incorrectly substituting variables.
  • Misinterpreting the viewing window in graphing utilities, leading to an incomplete or inaccurate representation of a graph.
  • Not maintaining consistent subtraction order when computing slopes from pairs of points.