Mathematics

Example 2 points out the following fact.

Slope is independent of the selected points. It does not matter which two points you pick on the line to calculate its slope.

The next example demonstrates that the slope is also independent of the order of subtraction.

188 CHAPTER 3. INTRODUCTION TO GRAPHING

You Try It!

EXAMPLE 3. Compute the slope of the line passing through the pointsCompute the slope of the line passing through the points P (−3, 1) and Q(2, 4).

P (−1,−2) and Q(3, 3). Solution: First, sketch the line passing through the points P (−1,−2) and Q(3, 3) (see Figure 3.46).

−5 5

−5

5

x

y

P (−1,−2)

Q(3, 3)

Figure 3.46: Computing the slope of the line passing through the points P (−1,−2) and Q(3, 3).

To calculate the slope of the line through the points P (−1,−2) and Q(3, 3), we must calculate the change in both the independent and dependent variables. We’ll do this in two different ways.

Warning! If you are not consistent in the direction you subtract, you will not get the correct answer for the slope. For example:

3− (−2) −1− 3 = −

5

4

In this case, we subtracted the y-coordinate of point P (−1,−2) from the y-coordinate of point Q(3, 3), but then we changed horses in midstream, subtracting the x-coordinate of point Q(3, 3) from the x-coordinate of point P (−1,−2). Note that we get the negative of the correct answer.

Subtract the coordinates of point P (−1,−2) from the coordinates of point Q(3, 3).

Slope = Δy

Δx

= 3− (−2) 3− (−1)

= 5

4

Subtract the coordinates of point Q(3, 3) from the coordinates of point P (−1,−2).

Slope = Δy

Δx

= −2− 3 −1− 3

= −5 −4

= 5

4

Note that regardless of the direction of subtraction, the slope is 5/4.Answer: 3/5

3.3. RATES AND SLOPE 189

Example 3 demonstrates the following fact.

The direction of subtraction does not matter. When calculating the slope of a line through two points P and Q, it does not matter which way you subtract, provided you remain consistent in your choice of direction.

The Steepness of a Line

We need to examine whether our definition of slope matches certain expecta- tions.

Slope and steepness of a line. The slope of a line is a number that tells us how steep the line is.

If slope is a number that measures the steepness of a line, then one would expect that a steeper line would have a larger slope.

You Try It!

EXAMPLE 4. Graph two lines, the first passing through the points P (−3,−2) Compute the slope of the line passing through the points P (−2,−3) and Q(2, 5). Then compute the slope of the line passing through the points R(−2,−1) and S(5, 3), and compare the two slopes. Which line is steeper?

and Q(3, 2) and the second through the points R(−1,−3) and S(1, 3). Calcu- late the slope of each line and compare the results.

Solution: The graphs of the two lines through the given points are shown, the first in Figure 3.47 and the second in Figure 3.48. Note that the line in Figure 3.47 is less steep than the line in Figure 3.48.

−5 5

−5

5

x

y

P (−3,−2)

Q(3, 2)

Figure 3.47: This line is less steep than the line on the right.

−5 5

−5

5

x

y

R(−1,−3)

S(1, 3)

Figure 3.48: This line is steeper than the line on the left.

Remember, the slope of the line is the rate at which the dependent variable is changing with respect to the independent variable. In both Figure 3.47 and Figure 3.48, the dependent variable is y and the independent variable is x.

190 CHAPTER 3. INTRODUCTION TO GRAPHING

Subtract the coordinates of point P (−3,−2) from the coordinates of point Q(3, 2).

Slope of first line = Δy

Δx

= 2− (−2) 3− (−3)

= 4

6

= 2

3

Subtract the coordinates of the point R(−1,−3) from the point S(1, 3).

Slope of second line = Δy

Δx

= 3− (−3) 1− (−1)

= 6

2 = 3

Note that both lines go uphill and both have positive slopes. Also, note that the slope of the second line is greater than the slope of the first line. This is consistent with the fact that the second line is steeper than the first.

Answer: The first line has slope 2, and the second line has slope 4/7. The first line is steeper.

In Example 4, both lines slanted uphill and both had positive slopes, the steeper of the two lines having the larger slope. Let’s now look at two lines that slant downhill.

You Try It!

EXAMPLE 5. Graph two lines, the first passing through the points P (−3, 1)Compute the slope of the line passing through the points P (−3, 3) and Q(3,−5). Then compute the slope of the line passing through the points R(−4, 1) and S(4,−3), and compare the two slopes. Which line is steeper?

and Q(3,−1) and the second through the points R(−2, 4) and S(2,−4). Cal- culate the slope of each line and compare the results.

Solution: The graphs of the two lines through the given points are shown, the first in Figure 3.49 and the second in Figure 3.50. Note that the line in Figure 3.49 goes downhill less quickly than the line in Figure 3.50. Remember, the slope of the line is the rate at which the dependent variable is changing with respect to the independent variable. In both Figure 3.49 and Figure 3.50, the dependent variable is y and the independent variable is x.

Subtract the coordinates of point P (−3, 1) from the coordinates of point Q(3,−1).

Slope of first line = Δy

Δx

= −1− 1 3− (−3)

= −2 6

= −1 3

Subtract the coordinates of point R(−2, 4) from the coordinates of point S(2,−4).

Slope of second line = Δy

Δx

= −4− 4 2− (−2)

= −8 4

= −2

3.3. RATES AND SLOPE 191

−5 5

−5

5

x

y

P (−3, 1)

Q(3,−1)

Figure 3.49: This line goes down- hill more slowly than the line on the right.

−5 5

−5

5

x

y

R(−2, 4)

S(2,−4)

Figure 3.50: This line goes downhill more quickly than the line on the left.

Note that both lines go downhill and both have negative slopes. Also, note that the magnitude (absolute value) of the slope of the second line is greater than the magnitude of the slope of the first line. This is consistent with the fact that the second line moves downhill more quickly than the first.

Answer: The first line has slope −4/3, and the second line has slope −1/2. The first line is steeper.

What about the slopes of vertical and horizontal lines?

You Try It!

EXAMPLE 6. Calculate the slopes of the vertical and horizontal lines Calculate the slopes of the vertical and horizontal lines passing through the point (−4, 1).

passing through the point (2, 3).

Solution: First draw a sketch of the vertical and horizontal lines passing through the point (2, 3). Next, select a second point on each line as shown in Figures 3.51 and 3.52.

−5 5

−5

5

x

y

Q(2, 3)P (−2, 3)

Figure 3.51: A horizontal line through (2, 3).

−5 5

−5

5

x

y

S(2, 3)

R(2,−3)

Figure 3.52: A vertical line through (2, 3).

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