The Cartesian Coordinate Plane - Master High-Frequency Concepts and Skills for Algebra Proficiency—FAST - Easy Algebra Step-by-Step

Easy Algebra Step-by-Step: Master High-Frequency Concepts and Skills for Algebra Proficiency—FAST! (2012)

Chapter 16. The Cartesian Coordinate Plane

In this chapter, you learn about the Cartesian coordinate plane.

Definitions for the Plane

The Cartesian coordinate plane is defined by two real number lines, one horizontal and one vertical, intersecting at right angles at their zero points (see Figure 16.1). The two real number lines are the coordinate axes. The horizontal axis, commonly called the x-axis, has positive direction to the right, and the vertical axis, commonly referred to as the y-axis, has positive direction upward. The two axes determine a plane. Their point of intersection is called the origin.

Image

Figure 16.1 Cartesian coordinate plane

Ordered Pairs in the Plane

In the (Cartesian) coordinate plane, you identify each point P in the plane by an ordered pair (x, y) of real numbers x and y, called its coordinates. The ordered pair (0, 0) names the origin. An ordered pair of numbers is written in a definite order so that one number is first and the other second. The first number is the x-coordinate, and the second number is the y-coordinate (see Figure 16.2). The order in the ordered pair (x, y) that corresponds to a point P is important. The absolute value of the first coordinate, x, is the perpendicular horizontal distance (right or left) of the point P from the y-axis. If x is positive, P is to the right of the y-axis; if x is negative, it is to the left of the y-axis. The absolute value of the second coordinate, y, is the perpendicular vertical distance (up or down) of the point P from the x-axis. If y is positive, P is above the x-axis; if y is negative, it is below the x-axis.

Image

Figure 16.2 Point P in a Cartesian coordinate plane

Problem Name the ordered pair of integers corresponding to point A in the coordinate plane shown.

Image

Solution

Image Step 1. Determine the x-coordinate of A.

The point A is 7 units to the left of the y-axis, so it has x-coordinate –7.

Step 2. Determine the y-coordinate of A.

The point A is 4 units above the x-axis, so it has y-coordinate 4.

Step 3. Name the ordered pair corresponding to point A.

(–7, 4) is the ordered pair corresponding to point A.

Two ordered pairs are equal if and only if their corresponding coordinates are equal; that is, (a, b) = (c, d) if and only if a = c and b = d.

Problem State whether the two ordered pairs are equal. Explain your answer.

Image

Image

Image

Image

Solution

a. (2, 7), (7, 2)

Image Step 1. Check whether the corresponding coordinates are equal.

(2, 7) ≠ (7, 2) because 2 and 7 are not equal.

Image

Image Step 1. Check whether the corresponding coordinates are equal.

Image

Image Step 1. Image either because – 4 ≠ 4 or because –1 ≠ 1.

d. Image because 6 = 6 and Image

Quadrants of the Plane

The axes divide the Cartesian coordinate plane into four quadrants. The quadrants are numbered with Roman numerals–I, II, III, and IV–beginning in the upper right and going around counterclockwise, as shown in Figure 16.3.

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Figure 16.3 Quadrants in the coordinate plane


Don’t forget that the quadrants are numbered counterclockwise.


In quadrant I, both coordinates are positive; in quadrant II, the x-coordinate is negative, and the y-coordinate is positive; in quadrant III, both coordinates are negative; and in quadrant IV, the x-coordinate is positive, and the y-coordinate is negative. Points that have 0 as one or both of the coordinates are on the axes. If the x-coordinate is 0, the point lies on the y-axis. If the y-coordinate is 0, the point lies on the x-axis. If both coordinates of a point are 0, the point is at the origin.

Problem Identify the quadrant in which the point lies.

a. (4, – 8)

b. (1, 6)

c. (– 8, – 3)

d. (– 4, 2)

Solution

a. (4, – 8)

Image Step 1. Note the signs of x and y.

x is positive and y is negative.

Step 2. Identify the quadrant in which the point lies.

Because x is positive and y is negative, (4, – 8) lies in quadrant IV.

b. (1, 6)

Image Step 1. Note the signs of x and y.

x is positive and y is positive.

Step 2. Identify the quadrant in which the point lies.

Because x is positive and y is positive, (1, 6) lies in quadrant I.

c. (– 8, – 3)

Image Step 1. Note the signs of x and y.

x is negative and y is negative.

Step 2. Identify the quadrant in which the point lies.

Because x is negative and y is negative, (– 8, – 3) lies in quadrant III.

d. (– 4, 2)

Image Step 1. Note the signs of x and y.

x is negative and y is positive.

Step 2. Identify the quadrant in which the point lies.

Because x is negative and y is positive, (– 4, 2) lies in quadrant II.

Finding the Distance Between Two Points in the Plane

If you have two points in a coordinate plane, you can find the distance between them using the formula given here.

Image Distance Between Two Points

The distance d between two points (x1, y1) and (x2, y2) in a coordinate plane is given by

Image


To avoid careless errors when using the distance formula, enclose substituted negative values in parentheses.


Problem Find the distance between the points (– 1, 4) and (5, – 3).

Solution

Image Step 1. Specify (x1, y1) and (x2, y2) and identify values for x1, y1, x2, and y2.

Let Image and Image Then Image and y2 = – 3.

Step 2. Evaluate the formula for the values from step 1.

Image

Step 3. State the distance.

The distance between (– 1, 4) and (5, – 3) is Image units.

Finding the Midpoint Between Two Points in the Plane

You can find the midpoint between two points using the following formula.

Image Midpoint Between Two Points

The midpoint between two points (x1, y1) and (x2, y2) in a coordinate plane is the point with coordinates

Image


When you use the midpoint formula, be sure to put plus signs, not minus signs, between the two x values and the two y values.


Problem Find the midpoint between (– 1, 4) and (5, – 3).

Solution

Image Step 1. Specify (x1, y1) and (x2, y2) and identify values for x1, y1, x2, and y2. Let Image and Image Then Image and y2 = – 3.

Step 2. Evaluate the formula for the values from step 1.

Image

Step 3. State the midpoint.

The midpoint between (– 1, 4) and (5, – 3) is Image.

Finding the Slope of a Line Through Two Points in the Plane

When you have two distinct points in a coordinate plane, you can construct the line through the two points. The slope describes the steepness or slant (if any) of the line. To calculate the slope of a line, use the following formula.

Image Slope of a Line Through Two Points

The slope m of a line through two distinct points, (x1, y1) and (x2, y2), is given by

Image


When you use the slope formula, be sure to subtract the coordinates in the same order in both the numerator and the denominator. That is, if y2 is the first term in the numerator, then x2 must be the first term in the denominator. It is also a good idea to enclose substituted negative values in parentheses to guard against careless errors.


From the formula, you can see that the slope is the ratio of the change in vertical coordinates (the rise) to the change in horizontal coordinates (the run). Thus, Image Figure 16.4 illustrates the rise and run for the slope of the line through points P1(x1, y1) and P2(x2, y2).

Image

Figure 16.4 Rise and run

You will find it helpful to know that lines that slant upward from left to right have positive slopes, and lines that slant downward from left to right have negative slopes. Also, horizontal lines have zero slope, but the slope for vertical lines is undefined.

Problem Find the slope of the line through the points shown.

Image

Solution

Image Step 1. Specify (x1, y1) and (x2, y2) and identify values for x1, y1, x2, and y2.

Let Image and Image Then Image and y2 = – 6.

Step 2. Evaluate the formula for the values from step 1.

Image

Step 3. State the slope.

The slope of the line that passes through the points (7, 5) and (– 4, – 6) is 1. Note: The line slants upward from left to right—so the slope should be positive.

Problem Find the slope of the line through the two points.

a. (– 1, 4) and (5, – 3)

b. (– 6, 7) and (5, 7)

c. (5, 8) and (5, – 3)

Solution

a. (– 1, 4) and (5, – 3)

Image Step 1. Specify (x1, y1) and (x2, y2) and identify values for x1, y1, x2, and y2.

Let Image and Image Then Image and y2 = – 3.

Step 2. Evaluate the formula for the values from step 1.

Image

Step 3. State the slope.

The slope of the line through (– 1, 4) and (5, – 3) is Image. Note: If you sketch the line through these two points, you will see that it slants downward from left to right—so its slope should be negative.

b. (– 6, 7) and (5, 7)

Image Step 1. Specify (x1, y1) and (x2, y2) and identify values for x1, y1, x2, and y2.

Let Image and Image Then Image and y2 = 7.

Step 2. Evaluate the formula for the values from step 1.

Image

Step 3. State the slope.

The slope of the line that contains (– 6, 7) and (5, 7) is 0. Note: If you sketch the line through these two points, you will see that it is a horizontal line—so the slope should be 0.

c. (5, 8) and (5, – 3)

Image Step 1. Specify (x1, y1) and (x2, y2) and identify values for x1, y1, x2, and y2.

Let Image and Image Then Image and y2 = – 3.

Step 2. Evaluate the formula for the values from step 1.

Image

Step 3. State the slope.

The slope of the line that contains (5, 8) and (5, – 3) is undefined. Note: If you sketch the line through these two points, you will see that it is a vertical line—so the slope should be undefined.

Slopes of Parallel and Perpendicular Lines

It is useful to know the following:

Image If two lines are parallel, their slopes are equal; if two lines are perpendicular, their slopes are negative reciprocals of each other.

Problem Find the indicated slope.

a. Find the slope m1 of a line that is parallel to the line through (– 3, 4) and (– 1, – 2).

b. Find the slope m2 of a line that is perpendicular to the line through (–3, 4) and (– 1, – 2).

Solution

a. Find the slope m1 of a line that is parallel to the line through (– 3, 4) and (– 1, – 2).

Image Step 1. Determine a strategy.

Because two parallel lines have equal slopes, m1 will equal the slope m of the line through (– 3, 4) and (– 1, – 2); that is, m1 = m.

Step 2. Find m.

Image

Step 3. Determine m1.

m1 = m = – 3

b. Find the slope m2 of a line that is perpendicular to the line through (–3, 4) and (– 1, – 2).

Image Step 1. Determine a strategy.

Because the slopes of two perpendicular lines are negative reciprocals of each other, m2 will equal the negative reciprocal of the slope m of the line through (– 3, 4) and (– 1, – 2); that is, Image.

Step 2. Find m.

Image

Step 3. Determine m2.

Image

Image Exercise 16

For 1—6, indicate whether the statement is true or false.

1. The intersection of the coordinate axes is the origin.

2. (2, 3) = (3, 2)

3. Image

4. The point Image is in quadrant II.

5. The point Image is in quadrant III.

6. The point (5, 0) is in quadrant I.

For 7—14, fill in the blank to make a true statement.

7. The change in y-coordinates between two points on a line is the __________.

8. The change in x-coordinates between two points on a line is the __________.

9. Lines that slant downward from left to right have __________ slopes.

10. Lines that slant upward from left to right have __________ slopes.

11. Horizontal lines have __________ slope.

12. The slope of a line that is parallel to a line that has slope Image is __________.

13. The slope of a line that is perpendicular to a line that has slope Image is __________.

14. The slope of a vertical line is __________.

15. Name the ordered pair of integers corresponding to point K in the following coordinate plane.

Image

16. Find the distance between the points (1, 4) and (5, 7).

17. Find the distance between the points (– 2, 5) and (4, – 1).

18. Find the midpoint between the points (– 2, 5) and (4, – 1).

19. Find the slope of the line through (– 2, 5) and (4, – 1).

20. Find the slope of the line through the points shown.

Image