12 multiple-choice questions, progressively harder.
What is the slope of the line through (−1,2)(-1, 2)(−1,2) and (3,10)(3, 10)(3,10)?
Solution
Correct answer: D
Apply the slope formula, watching the negative coordinate.
m=10−23−(−1)=84=2m = \frac{10 - 2}{3 - (-1)} = \frac{8}{4} = 2m=3−(−1)10−2=48=2
So the slope is 222.
What is the slope of the line through (−2,−1)(-2, -1)(−2,−1) and (2,1)(2, 1)(2,1)?
Correct answer: C
Apply the slope formula and simplify.
m=1−(−1)2−(−2)=24=12m = \frac{1 - (-1)}{2 - (-2)} = \frac{2}{4} = \frac{1}{2}m=2−(−2)1−(−1)=42=21
So the slope is 12\tfrac{1}{2}21.
What is the slope of the line through (4,−2)(4, -2)(4,−2) and (1,7)(1, 7)(1,7)?
Correct answer: A
Take (4,−2)(4, -2)(4,−2) as point one and (1,7)(1, 7)(1,7) as point two.
m=7−(−2)1−4=9−3=−3m = \frac{7 - (-2)}{1 - 4} = \frac{9}{-3} = -3m=1−47−(−2)=−39=−3
So the slope is −3-3−3.
What is the slope of the line through (−3,4)(-3, 4)(−3,4) and (5,4)(5, 4)(5,4)?
Both points have the same height, y=4y = 4y=4, so the rise is zero.
m=4−45−(−3)=08=0m = \frac{4 - 4}{5 - (-3)} = \frac{0}{8} = 0m=5−(−3)4−4=80=0
So the slope is 000 (a horizontal line).
What is the slope of the line through (2,−5)(2, -5)(2,−5) and (2,3)(2, 3)(2,3)?
Correct answer: B
Both points have the same xxx-coordinate, x=2x = 2x=2, so the run is zero.
m=3−(−5)2−2=80m = \frac{3 - (-5)}{2 - 2} = \frac{8}{0}m=2−23−(−5)=08
Dividing by zero has no value, so the slope is undefined (a vertical line).
A line has slope 34\tfrac{3}{4}43 and passes through (1,2)(1, 2)(1,2). Which point is also on the line?
Read 34\tfrac{3}{4}43 as a run of 444 and a rise of 333. Step right 444 and up 333 from (1,2)(1, 2)(1,2).
(1,2)⟶(1+4, 2+3)=(5,5)(1, 2) \longrightarrow (1 + 4,\; 2 + 3) = (5, 5)(1,2)⟶(1+4,2+3)=(5,5)
So (5,5)(5, 5)(5,5) is on the line.
What is the slope of the line through (−4,−2)(-4, -2)(−4,−2) and (0,0)(0, 0)(0,0)?
m=0−(−2)0−(−4)=24=12m = \frac{0 - (-2)}{0 - (-4)} = \frac{2}{4} = \frac{1}{2}m=0−(−4)0−(−2)=42=21
Which line is steeper, one with slope −3-3−3 or one with slope 222?
Compare the sizes of the slopes, ignoring sign.
∣−3∣=3and∣2∣=2|-3| = 3 \quad\text{and}\quad |2| = 2∣−3∣=3and∣2∣=2
Since 3>23 > 23>2, the slope −3-3−3 line is steeper, even though −3-3−3 is the smaller number.
What is the slope of the line through (6,1)(6, 1)(6,1) and (2,3)(2, 3)(2,3)?
Take (6,1)(6, 1)(6,1) as point one.
m=3−12−6=2−4=−12m = \frac{3 - 1}{2 - 6} = \frac{2}{-4} = -\frac{1}{2}m=2−63−1=−42=−21
So the slope is −12-\tfrac{1}{2}−21.
A line has slope −2-2−2 and passes through (0,5)(0, 5)(0,5). Which point is also on the line?
Read −2-2−2 as −21\tfrac{-2}{1}1−2: a run of 111 and a rise of −2-2−2. Step right 111 and down 222 from (0,5)(0, 5)(0,5).
(0,5)⟶(0+1, 5−2)=(1,3)(0, 5) \longrightarrow (0 + 1,\; 5 - 2) = (1, 3)(0,5)⟶(0+1,5−2)=(1,3)
So (1,3)(1, 3)(1,3) is on the line.
What is the slope of the line through (0,0)(0, 0)(0,0) and (−2,6)(-2, 6)(−2,6)?
Apply the slope formula.
m=6−0−2−0=6−2=−3m = \frac{6 - 0}{-2 - 0} = \frac{6}{-2} = -3m=−2−06−0=−26=−3
A line has a rise of 444 over a run of −1-1−1. What is its slope?
Slope is rise over run, keeping the signs.
m=4−1=−4m = \frac{4}{-1} = -4m=−14=−4
So the slope is −4-4−4.
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