Book cover for Precalculus: A Right Triangle Approach

Precalculus: A Right Triangle Approach

Judith A. Beecher, Marvin L. Bittinger, David J. Ellenbogen, Judith A. Penna

ISBN #9780321783967

5th Edition

5,905 Questions

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25,202 Students Helped

Homework Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This section explains the basics of graphing in the Cartesian coordinate system, detailing how to plot ordered pairs, determine intercepts, and graph equations such as lines and circles. It emphasizes using mathematical equations like the distance and midpoint formulas to analyze relationships between points, and shows the importance of proper scaling and using technology to accurately draw graphs. Mastery of these concepts bridges the gap between algebraic manipulation and geometric visualization.

Learning Objectives

1

Explain the role of graphs in displaying, interpreting, and analyzing data in a visual format.

2

Demonstrate how to plot points and graph equations using the Cartesian coordinate system, including determining intercepts.

3

Apply the distance and midpoint formulas to solve problems involving points, segments, and circles in the coordinate plane.

4

Graph different types of functions (linear, quadratic, circle) and understand the impact of scaling and window settings on graph appearance.

Key Concepts

CONCEPT

DEFINITION

Graph

A visual representation of data or equations on a coordinate plane, used to display relationships between variables.

Cartesian Coordinate System

A system using two perpendicular axes (x-axis and y-axis) that intersect at the origin (0, 0) to locate points via ordered pairs.

Ordered Pair

A pair of numbers (x, y) that represent a point's horizontal and vertical locations in the coordinate plane.

Intercepts

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

Distance Formula

A formula to compute the distance between two points: d = √[(x₂ - x₁)² + (y₂ - y₁)²].

Midpoint Formula

A formula to find the midpoint of a segment connecting two points: ((x₁ + x₂)/2, (y₁ + y₂)/2).

Standard Form of a Circle

An equation of a circle expressed as (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius.

Example Problems

Example 1

Find the coordinates of points $A, B, C, D, E,$ and $F$.

Example 2

Find the coordinates of points $G, H, I, J, K,$ and $L .$

Example 3

Find the coordinates of points $G, H, I, J, K,$ and $L$. $$ (4,0),(-3,-5),(-1,4),(0,2),(2,-2) $$

Example 4

Find the coordinates of points $G, H, I, J, K,$ and $L$. $$ (1,4),(-4,-2),(-5,0),(2,-4),(0,3) $$

Example 5

Find the coordinates of points $G, H, I, J, K,$ and $L$. $$ (1,4),(-4,-2),(-5,0),(2,-4),(0,3) $$
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Step-by-Step Explanations

QUESTION

Graph and label the intercepts of the line represented by the equation 2x + 3y = 18.

STEP-BY-STEP ANSWER:

Step 1: Find the y-intercept by setting x = 0. Substitute x = 0 into 2(0) + 3y = 18 to get 3y = 18, so y = 6. The y-intercept is (0, 6).
Step 2: Find the x-intercept by setting y = 0. Substitute y = 0 into 2x + 3(0) = 18 to get 2x = 18, so x = 9. The x-intercept is (9, 0).
Step 3: Optionally, choose an additional value for x (for example, x = 3) and solve for y: 2(3) + 3y = 18 → 6 + 3y = 18; subtract 6 to obtain 3y = 12, so y = 4. This gives the point (3, 4).
Step 4: Plot all three points on the coordinate plane and draw a straight line through them.
Final Answer: The line has intercepts (0, 6) and (9, 0) with an additional check point (3, 4) confirming the graph.

Graphing a Linear Equation Using Intercepts

QUESTION

Find the distance between the points A = (1, 2) and B = (13, -6).

STEP-BY-STEP ANSWER:

Step 1: Identify the coordinates: let (x₁, y₁) = (1, 2) and (x₂, y₂) = (13, -6).
Step 2: Substitute the values into the distance formula: d = √[(13 - 1)² + (-6 - 2)²].
Step 3: Compute the differences: 13 - 1 = 12 and -6 - 2 = -8.
Step 4: Square the differences: 12² = 144 and (-8)² = 64.
Step 5: Add the squared differences: 144 + 64 = 208.
Step 6: Take the square root: d = √208, which simplifies to approximately 14.4.
Final Answer: The distance between A and B is approximately 14.4 units.

Using the Distance Formula

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Common Mistakes

  • Mixing up the x and y coordinates when plotting points on the graph.
  • Failing to label axes or incorrectly scaling the graph, which leads to distorted representations.
  • Not substituting zero correctly when finding intercepts (setting x = 0 for y-intercept and y = 0 for x-intercept).
  • Forgetting to square the differences in the distance formula or misapplying absolute values.
  • Overlooking the need to check computed points on the graph, which can result in drawing an inaccurate line.