Slope

Scott Solomon

Slope refers to the Steepness of a Line

Slope


Basics
Slope
Slope refers to the steepness of a line.
Line 1 is steeper than Line 2.
Line 1’s slope is greater than Line 2’s slope.


Line
A line is a flat straight figure extending infinitely through two points.  (See Line L above.)
x-axis = horizontal
y-axis = vertical
x = horizontal coordinate
y = vertical coordinate
(x, y) = point representing horizontal and vertical coordinates on Line L
(o, b) = y-intercept = point where Line L crosses the y-axis
m = slope


Formulas (Mathematics Formula Sheet)
slope of a line
m = \bf\displaystyle\frac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}


Formulas (NOT on Mathematics Formula Sheet)
slope as change
m = \bf\displaystyle\frac{change\hspace{0.1cm}in\hspace{0.1cm}y}{change\hspace{0.1cm}in\hspace{0.1cm}x}


Question
What is the slope of Line 1?
Slope
Answer
m = 2

Answer Process
slope as change
m = \bf\displaystyle\frac{change\hspace{0.1cm}in\hspace{0.1cm}y}{change\hspace{0.1cm}in\hspace{0.1cm}x}

m = \bf\displaystyle\frac{10\hspace{0.1cm}boxes\hspace{0.1cm}up\hspace{0.1cm}positive\hspace{0.1cm}y\hspace{0.1cm}axis}{5\hspace{0.1cm}boxes\hspace{0.1cm}along\hspace{0.1cm}positive\hspace{0.1cm}x\hspace{0.1cm}axis} = \bf\displaystyle\frac{10}{5} = 2
Line 1


Question
What is the slope of Line 2?
Slope
Answer
m = \bf\displaystyle\frac{1}{2}

Answer Process
slope as change
m = \bf\displaystyle\frac{change\hspace{0.1cm}in\hspace{0.1cm}y}{change\hspace{0.1cm}in\hspace{0.1cm}x}

m = \bf\displaystyle\frac{5\hspace{0.1cm}boxes\hspace{0.1cm}up\hspace{0.1cm}positive\hspace{0.1cm}y\hspace{0.1cm}axis}{10\hspace{0.1cm}boxes\hspace{0.1cm}along\hspace{0.1cm}positive\hspace{0.1cm}x\hspace{0.1cm}axis} = \bf\displaystyle\frac{5}{10} = \bf\displaystyle\frac{1}{2}
Slope


Question
Using the points (3, 1) and (8, 3) in the graph below, calculate the slope of line L.
Answer
m = \bf\displaystyle\frac{2}{5}

Answer Process
(x1, y1) = (3, 1)
(x2, y2) = (8, 3)

slope of a line
m = \bf\displaystyle\frac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}

m = \bf\displaystyle\frac{3-1}{8-3} = \bf\displaystyle\frac{2}{5}

Input Display Comment
clear blinker clears screen
fraction 3 – 1 down 8 – 3  right \bf\displaystyle\frac{3-1}{8-3} m = \bf\displaystyle\frac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}
enter \bf\displaystyle\frac{2}{5} Answer
Slope


Question
Express the previous answer in decimal form.

Answer
m = 0.4

Answer Process
See Toggle.
\bf\displaystyle\frac{2}{5} ⇔ 0.4

Input Display Comment
clear blinker clears screen
fraction 3 – 1 down 8 – 3  right \bf\displaystyle\frac{3-1}{8-3} m = \bf\displaystyle\frac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}
enter \bf\displaystyle\frac{2}{5}
toggle 0.4 Answer
(Toggle)
Slope


Question
Using the points (1, 2) and (4, 9) in the graph below, calculate the slope of line L in decimal form.

Answer
m = 2.333333333

Answer Process
(x1, y1) = (1, 2)
(x2, y2) = (4, 9)

slope of a line
m = \bf\displaystyle\frac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}

m = \bf\displaystyle\frac{9-2}{4-1} = \bf\displaystyle\frac{7}{3} = 2.333333333 after toggle

Input Display Comment
clear blinker clears screen
fraction 9 – 2 down 4 – 1  right \bf\displaystyle\frac{9-2}{4-1} m = \bf\displaystyle\frac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}
enter \bf\displaystyle\frac{7}{3}
toggle 2.333333333 Answer
(Toggle)
Slope


Question
Round the previous answer to the nearest tenth.

Answer
m = 2.3

Answer Process
See Rounding.
2.333333333 → 2.3


Question
On the graph below, plot the points (2, 3) and (6, 9), draw a line through them, and calculate the slope of the line in decimal form.
Graph


Answer
m = 1.5

Answer Process
(x1, y1) = (2, 3)
(x2, y2) = (6, 9)

slope of a line
m = \bf\displaystyle\frac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}

m = \bf\displaystyle\frac{9-3}{6-2} = \bf\displaystyle\frac{3}{2} = 1.5 after toggle

Input Display Comment
clear blinker clears screen
fraction 9 – 3 down 6 – 2  right \bf\displaystyle\frac{9-3}{6-2} m = \bf\displaystyle\frac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}
enter \bf\displaystyle\frac{3}{2}
toggle 1.5 Answer
(Toggle)
Slope


Question
On the graph below, plot the points (4, 1) and (10, 2) and calculate their slope in decimal form rounded to the nearest hundredth (without drawing a line).
Graph


Answer
m = 0.17

Answer Process
(x1, y1) = (4, 1)
(x2, y2) = (10, 2)

slope of a line
m = \bf\displaystyle\frac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}

m = \bf\displaystyle\frac{2-1}{10-4} = \bf\displaystyle\frac{1}{6} = 0.166666667 after toggle = 0.17 after rounding

Input Display Comment
clear blinker clears screen
fraction 2 – 1 down 10 – 4  right \bf\displaystyle\frac{2-1}{10-4} m = \bf\displaystyle\frac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}
enter \bf\displaystyle\frac{1}{6}
toggle 0.166666667 Answer
(before rounding)
Slope


Question
On the graph below, plot the points (3, 2.5) and (7, 6.25) and calculate their slope in decimal form to the nearest hundredth (without drawing a line).
Graph


Answer
m = 0.94

Answer Process
(x1, y1) = (3, 2.5)
(x2, y2) = (7, 6.25)

slope of a line
m = \bf\displaystyle\frac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}

m = \bf\displaystyle\frac{6.25-2.5}{7-3} = 0.9375 =0.94 after rounding

Input Display Comment
clear blinker clears screen
fraction 6.25 – 2.5 down 7 – 3  right \bf\displaystyle\frac{6.25-2.5}{7-3} m = \bf\displaystyle\frac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}
enter 0.9375 Answer
(before rounding)
Slope


Question
The following table shows the age in years and height in inches of a boy and a girl.

On the grid below the table, plot the x-y coordinates for the boy at ages 2 and 8 and calculate their slope in decimal form to the nearest tenth (without drawing a line).

Boy   Girl  
Age (yr) Height (in) Age (yr) Height (in)
2 34.00 2 33.50
3 37.50 3 37.00
4 40.00 4 39.50
5 43.00 5 42.25
6 45.25 6 45.00
7 48.00 7 48.00
8 50.25 8 50.00
9 52.50 9 52.50
10 54.50 10 54.25

Grid

gridboy
Answer
m = 2.7

Answer Process
(x1, y1) = (2, 34.00)
(x2, y2) = (8, 50.25)

slope of a line
m = \bf\displaystyle\frac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}

m = \bf\displaystyle\frac{50.25-34.00}{8-2} = 2.708333333= 2.7 after rounding

Input Display Comment
clear blinker clears screen
fraction 50.25 – 34.00 down 8 – 2  right \bf\displaystyle\frac{50.25-34.00}{8-2} m = \bf\displaystyle\frac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}
enter 2.708333333 Answer
(before rounding)
Slope


Practice – Questions
1.  What is the slope of Line 1?

2.  What is the slope of Line 2?

3.  Using the points (3.5, 1.5) and (7, 3.25) in the graph below, calculate the slope of line L.

4.  On the graph below, plot the points (4, 1) and (8, 2), draw a line through them, and calculate the slope of the line in decimal form.
Graph

5.  On the graph below, plot the points (1, 4) and (2, 8) and calculate their slope (without drawing a line).
Graph

6.  The following table shows the age in years and height in inches of a boy and a girl.

On the grid below the table, plot the x-y coordinates for the girl at ages 3 and 9 and calculate their slope in decimal form (without drawing a line).

Boy   Girl  
Age (yr) Height (in) Age (yr) Height (in)
2 34.00 2 33.50
3 37.50 3 37.00
4 40.00 4 39.50
5 43.00 5 42.25
6 45.25 6 45.00
7 48.00 7 48.00
8 50.25 8 50.00
9 52.50 9 52.50
10 54.50 10 54.25

Grid

7.  With reference to Question 6, express your answer as a fraction.

8.  The following table shows the age in years and mileage for Car A and Car B.

On the grid below the table, plot the x-y coordinates for Car A at years 3 and 6 and calculate their slope (without drawing a line).

Car A   Car B  
Age Mileage Age Mileage
1 10000 1 11000
2 22000 2 20500
3 33500 3 29725
4 39000 4 41000
5 48000 5 50500
6 62000 6 58750
7 71000 7 69900
8 83250 8 81000
9 90500 9 89900
10 99500 10 99500

Grid

9.  The following table shows the age in years and mileage for Car A and Car B.

On the grid below the table, plot the x-y coordinates for Car B at years 2 and 9 and calculate their slope (without drawing a line).

Car A   Car B  
Age Mileage Age Mileage
1 10000 1 11000
2 22000 2 20500
3 33500 3 29725
4 39000 4 41000
5 48000 5 50500
6 62000 6 58750
7 71000 7 69900
8 83250 8 81000
9 90500 9 89900
10 99500 10 99500

Grid

10.  With reference to Question 9, round your answer to the nearest tenth.


Practice – Answers
1.  m = 1

2.  m = 2

3.  m = 0.5

4.  m = 0.25

5.  m = 4

6.  m = 2.583333333
gridgirl

7.  \bf\displaystyle m =\frac{31}{12}
See Toggle.

8.  m = 9500
slopecara

9.  m = 9914.285714
slopemileageanswer9

10.  m = 9914.3

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