12 multiple-choice questions, progressively harder.
What is the slope of the line through (1,−4)(1, -4)(1,−4) and (5,8)(5, 8)(5,8)?
Solution
Correct answer: A
Apply the slope formula and simplify.
m=8−(−4)5−1=124=3m = \frac{8 - (-4)}{5 - 1} = \frac{12}{4} = 3m=5−18−(−4)=412=3
So the slope is 333.
What is the slope of the line shown on the grid?
Correct answer: B
Count off the slope triangle from the left point. The run is 666 units to the right and the rise is 333 units up.
m=36=12m = \frac{3}{6} = \frac{1}{2}m=63=21
So the slope is 12\tfrac{1}{2}21.
A line has slope −3-3−3 and passes through (2,5)(2, 5)(2,5). Which point is also on the line?
Read −3-3−3 as −31\tfrac{-3}{1}1−3: a run of 111 and a rise of −3-3−3. Step right 111 and down 333 from (2,5)(2, 5)(2,5).
(2,5)⟶(2+1, 5−3)=(3,2)(2, 5) \longrightarrow (2 + 1,\; 5 - 3) = (3, 2)(2,5)⟶(2+1,5−3)=(3,2)
So (3,2)(3, 2)(3,2) is on the line.
Which line is steeper, one with slope −5-5−5 or one with slope 444?
Correct answer: C
Compare the sizes of the slopes, ignoring sign.
∣−5∣=5and∣4∣=4|-5| = 5 \quad\text{and}\quad |4| = 4∣−5∣=5and∣4∣=4
Since 5>45 > 45>4, the slope −5-5−5 line is steeper.
The points (−2,5)(-2, 5)(−2,5), (0,2)(0, 2)(0,2), and (2,−1)(2, -1)(2,−1) lie on one line. What is its slope?
Correct answer: D
Use any two of the points.
m=2−50−(−2)=−32=−32m = \frac{2 - 5}{0 - (-2)} = \frac{-3}{2} = -\frac{3}{2}m=0−(−2)2−5=2−3=−23
So the slope is −32-\tfrac{3}{2}−23.
A line has slope 25\tfrac{2}{5}52. Between two of its points the run is 151515. What is the rise?
The rise is the slope times the run.
rise=25×15=6\text{rise} = \tfrac{2}{5} \times 15 = 6rise=52×15=6
So the rise is 666.
The line through (2,−3)(2, -3)(2,−3) and (6,k)(6, k)(6,k) has slope 52\tfrac{5}{2}25. What is kkk?
Set the slope formula equal to 52\tfrac{5}{2}25 and solve for kkk.
k−(−3)6−2=52,k+34=52,k+3=10\frac{k - (-3)}{6 - 2} = \frac{5}{2}, \qquad \frac{k + 3}{4} = \frac{5}{2}, \qquad k + 3 = 106−2k−(−3)=25,4k+3=25,k+3=10
So k=7k = 7k=7.
Which of these lines is the least steep?
The least steep line has the smallest slope size (absolute value).
∣−4∣=4,∣12∣=0.5,∣3∣=3,∣−2∣=2|-4| = 4, \quad \left|\tfrac{1}{2}\right| = 0.5, \quad |3| = 3, \quad |-2| = 2∣−4∣=4,21=0.5,∣3∣=3,∣−2∣=2
The smallest is 12\tfrac{1}{2}21, so that line is the least steep.
What is the slope of the line through (−3,5)(-3, 5)(−3,5) and (3,1)(3, 1)(3,1)?
m=1−53−(−3)=−46=−23m = \frac{1 - 5}{3 - (-3)} = \frac{-4}{6} = -\frac{2}{3}m=3−(−3)1−5=6−4=−32
So the slope is −23-\tfrac{2}{3}−32.
A line has slope 000. Which statement must be true?
Slope 000 means the rise is zero between any two points, so the height never changes.
m=0run=0m = \frac{0}{\text{run}} = 0m=run0=0
A line that never changes height is horizontal.
Count off the slope triangle from the left point. The run is 666 units to the right and the line drops 222 units, a rise of −2-2−2.
m=−26=−13m = \frac{-2}{6} = -\frac{1}{3}m=6−2=−31
So the slope is −13-\tfrac{1}{3}−31.
A line has slope 32\tfrac{3}{2}23 and passes through (−4,−3)(-4, -3)(−4,−3). Stepping along it, which point is on the line?
Read 32\tfrac{3}{2}23 as a run of 222 and a rise of 333. Step twice from (−4,−3)(-4, -3)(−4,−3).
(−4,−3)→(−2,0)→(0,3)(-4, -3) \to (-2, 0) \to (0, 3)(−4,−3)→(−2,0)→(0,3)
So (0,3)(0, 3)(0,3) is on the line.
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