Book cover for Precalculus with Limits

Precalculus with Limits

Ron Larson

ISBN #9781439049099

2nd Edition

7,319 Questions

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724,371 Students Helped

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This section introduces the Cartesian plane as a powerful tool in mathematics for representing ordered pairs and visualizing relationships between two variables. Students learn how to plot points, use the Distance and Midpoint Formulas, and perform translations, which are essential for understanding more complex topics in coordinate geometry. Applications range from sports strategies, such as measuring a football pass, to business scenarios like sales estimation, emphasizing the practical relevance of these concepts.

Learning Objectives

1

Explain the structure of the Cartesian plane including axes, origin, and quadrants.

2

Demonstrate how to plot ordered pairs and interpret coordinates on the rectangular coordinate system.

3

Derive and apply the Distance Formula using the Pythagorean Theorem to find the distance between two points.

4

Utilize the Midpoint Formula to determine the midpoint of a line segment and apply translations to coordinate points.

5

Model real-life scenarios such as a football pass or annual revenue estimation using coordinate geometry.

Key Concepts

CONCEPT

DEFINITION

Cartesian Plane

A plane defined by two perpendicular real number lines (the x-axis and y-axis) that intersect at the origin, dividing the plane into four quadrants.

x-axis

The horizontal axis in the Cartesian plane that represents the set of all real numbers for the x-coordinate.

y-axis

The vertical axis in the Cartesian plane that represents the set of all real numbers for the y-coordinate.

Origin

The point of intersection of the x-axis and y-axis, designated as (0, 0).

Quadrants

The four regions into which the Cartesian plane is divided by the x- and y-axes.

Ordered Pair

A pair of numbers (x, y) that represents a point in the Cartesian plane, where x is the horizontal coordinate and y is the vertical coordinate.

Distance Formula

A formula derived from the Pythagorean Theorem used to calculate the distance between two points (x1, y1) and (x2, y2): d = √[(x2 - x1)² + (y2 - y1)²].

Midpoint Formula

A formula used to find the midpoint of a segment joining two points, given by: Midpoint = ((x1 + x2)/2, (y1 + y2)/2).

Translation

A type of transformation that shifts a point or figure a certain distance in a given direction by adding a constant to its coordinates.

Example Problems

Example 1

Match each term with its definition. (a) $ x $-axis (i) point of intersection of vertical axis and horizontal axis (b) $ y $-axis (ii) directed distance from the $ x $-axis (c) origin (iii) directed distance from the $ y $-axis (d) quadrants (iv) four regions of the coordinate plane (e) $ x $-coordinate (v) horizontal real number line (f) $ y $-coordinate (vi) vertical real number line

Example 2

In Exercises 2 - 4, fill in the blanks. An ordered pair of real numbers can be represented in a plane called the rectangular coordinate system or the ______________ plane.

Example 3

In Exercises 2 - 4, fill in the blanks. The __________ _____________ is a result derived from the Pythagorean Theorem.

Example 4

In Exercises 2 - 4, fill in the blanks. Finding the average values of the representative coordinates of the two endpoints of a line segment in a coordinate plane is also known as using the _____________ _______________.

Example 5

In Exercises 5 and 6, approximate the coordinates of the points.
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Step-by-Step Explanations

QUESTION

How do you plot the point (3, 4) on the Cartesian plane?

STEP-BY-STEP ANSWER:

Step 1: Identify the x-coordinate and y-coordinate from the ordered pair, which are 3 and 4 respectively.
Step 2: On the x-axis, move 3 units to the right of the origin.
Step 3: On the y-axis, move 4 units upward from the origin.
Step 4: The intersection of the vertical line through x=3 and the horizontal line through y=4 is the point (3, 4).
Final Answer: The point (3, 4) is plotted at that exact intersection in the Cartesian plane.

Plotting a Point in the Cartesian Plane

QUESTION

How do you find the distance between the points (2, 1) and (3, 4)?

STEP-BY-STEP ANSWER:

Step 1: Identify the coordinates: (x1, y1) = (2, 1) and (x2, y2) = (3, 4).
Step 2: Substitute these into the Distance Formula: d = √[(3 - 2)² + (4 - 1)²].
Step 3: Calculate the differences: 3 - 2 = 1 and 4 - 1 = 3.
Step 4: Square each difference: 1² = 1 and 3² = 9.
Step 5: Add the squared differences: 1 + 9 = 10.
Step 6: Take the square root: d = √10.
Final Answer: The distance between (2, 1) and (3, 4) is √10 units.

Using the Distance Formula

QUESTION

How do you find the midpoint of the segment connecting (5, 7) and (9, 3)?

STEP-BY-STEP ANSWER:

Step 1: Identify the coordinates: (x1, y1) = (5, 7) and (x2, y2) = (9, 3).
Step 2: Apply the Midpoint Formula: Midpoint = ((5 + 9)/2, (7 + 3)/2).
Step 3: Compute the averages: (5+9)/2 = 7 and (7+3)/2 = 5.
Final Answer: The midpoint is (7, 5).

Using the Midpoint Formula

QUESTION

How do you translate the point (-2, 3) three units to the right and two units upward?

STEP-BY-STEP ANSWER:

Step 1: Identify the translation amounts: add 3 to the x-coordinate and add 2 to the y-coordinate.
Step 2: Calculate the new x-coordinate: -2 + 3 = 1.
Step 3: Calculate the new y-coordinate: 3 + 2 = 5.
Final Answer: The translated point is (1, 5).

Translating Points

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Common Mistakes

  • Mixing up the x- and y-coordinates when plotting or calculating distances.
  • Neglecting the proper order of operations when applying the Distance Formula, especially with negative numbers.
  • Incorrectly averaging coordinates in the Midpoint Formula by not dividing the sum by 2.
  • Forgetting to adjust the coordinates appropriately during translations.