1. Slope
Lesson 4.1.3

2. Slope 4.1.3 California Standard: What it means for you: Key words:
Lesson
4.1.3
Slope
California Standard:
Algebra and Functions 3.3
Graph linear functions, noting that the vertical change (change in y-value) per unit of horizontal change (change in x-value) is always the same, and know that the ratio ("rise over run") is called the slope of a graph.
What it means for you:
You’ll learn what the slope of a graph is and how to calculate it.
Key words:
slope
steepness
ratio

3. Lesson
4.1.3
Slope
Over the past few Lessons you’ve been graphing linear equations — which have straight-line graphs.
Some straight-line graphs you’ve drawn have been steep, and others have been more shallow. y
There’s a measure for how steep a line is — slope. x
In this Lesson you’ll learn how to find the slope of a straight-line graph.

4. Slope 4.1.3 The Slope of a Line is a Ratio
Lesson
4.1.3
Slope
The Slope of a Line is a Ratio
For any straight line, the ratio is always the same
— it doesn’t matter which two points you choose to measure the changes between.
change in y
change in x y
9 units
6 units 2 4 6 8 10
change in y
change in x
= = 1.5 9 6
3 units
2 units
change in y
change in x
= = 1.5 3 2
This ratio, , is the slope of the graph.
change in y
change in x x

5. Slope 4.1.3 Slope is a Measure of Steepness of a Line
Lesson
4.1.3
Slope
Slope is a Measure of Steepness of a Line
A larger change in y for the same change in x makes the ratio bigger, so the slope is greater.
change in y
change in x y
change in y
change in x
= = 2 6 3 2 4 6 8 10
6 units
3 units
change in y
change in x
= = 0.5 2 4
2 units
4 units
This line is steeper, and it has the bigger slope.
So a slope is a measure of the steepness of a line — steeper lines have bigger slopes. x

6. Slope 4.1.3 Slopes Can Be Positive, Negative or Zero
Lesson
4.1.3
Slope
Slopes Can Be Positive, Negative or Zero x y
positive change in y
positive change in x
A positive slope is an “uphill” slope. The changes in x and y are both positive — as one increases, so does the other.
positive change in y
positive change in x
= positive slope

7. Lesson
4.1.3
Slope x y
A negative slope is a “downhill” slope. The change in y is negative for a positive change in x.
y decreases as x increases.
negative change in y
negative change in y
positive change in x
= negative slope
positive change in x

8. Lesson
4.1.3
Slope x y
A line with zero slope is horizontal. There is no change in y.
positive change in x
0 change in y
positive change in x
= zero slope x y
change in x = 0
The slope of a vertical line is undefined. There’s a change of zero on the x-axis, and you can’t divide by zero.

9. Lesson
4.1.3
Slope
Guided Practice
1. Plot the points (1, 3) and (2, 5) on a coordinate plane. Find the slope of the line connecting the two points.
2. Does the graph of y = –x have a positive or negative slope? Explain your answer.
change in y
change in x
= = 2 4 2 1 2 3 4 5 y x
4 units
2 units 5 –5 x
y = –x
Negative — it’s downhill from left to right.
Solution follows…

10. Slope 4.1.3 Compute Slopes From Coordinates of Two Points
Lesson
4.1.3
Slope
Compute Slopes From Coordinates of Two Points
Instead of counting unit squares to calculate slope, you can use the coordinates of any two points on a line.
There’s a formula for this:
For the line passing through coordinates (x1, y1) and (x2, y2):
Slope = =
change in y
change in x
y2 – y1
x2 – x1

11. Lesson
4.1.3
Slope
Example 1
The graph below is the graph of the equation y = 2x Find the slope of the line. y 1 2 3 4 –1 –3 –4 –2 –2 4
Solution
(1, 3)
(–1, –1)
Start by drawing a triangle connecting two points on the graph. x
Choose two points that are easy to read from the graph, for example: (x1, y1) = (–1, –1) and (x2, y2) = (1, 3)
Solution continues…
Solution follows…

12. Slope 4.1.3 Slope = y2 – y1 x2 – x1 = = = = 2 3 – (–1) 1 – (–1) 3 + 1
Lesson
4.1.3
Slope
Example 1
The graph below is the graph of the equation y = 2x Find the slope of the line. y 1 2 3 4 –1 –3 –4 –2 –2 4
change in x
Solution (continued)
Slope =
y2 – y1
x2 – x1
change in y
(1, 3) x
This is the change in y.
(–1, –1)
= = = = 2
3 – (–1)
1 – (–1)
3 + 1
1 + 1 4 2
This is the change in x.
So the slope of the graph is 2.

13. Lesson
4.1.3
Slope
Example 2
Find the slope of the line connecting the points C (–2, 5) and D (1, –4).
Solution
You don’t need to draw the line to calculate the slope — you are given the coordinates of two points on the line. (x1, y1) = (–2, 5) and (x2, y2) = (1, –4).
Substitute the coordinates into the formula for slope:
Slope =
y2 – y1
x2 – x1
= = =
–4 – 5
1 – (–2)
1 + 2 –9 3
Slope = –3
Solution continues…
Solution follows…

14. Lesson
4.1.3
Slope
Example 2
Find the slope of the line connecting the points C (–2, 5) and D (1, –4).
Solution (continued) 1 2 3 4 –1 –3 –4 –2 y x
C (–2, 5)
D (1, 4)
If you plot these points and draw a line through them, you can see that the slope is negative (it’s a “downhill” line).

15. Lesson
4.1.3
Slope
Guided Practice
3. Plot the points (–2, 3) and (2, 5) on a coordinate plane. Find the slope of the line connecting the two points.
y2 – y1 = 4
x2 – x1
= 0.5
5 – 3
2 – (–2) y 1 2 3 4 –1 –3 –4 –2
4. Plot the graph of the equation y = 4x – 2 and find its slope. x x y –2 1 2
Find two points on the line:
y2 – y1 = 4 1
x2 – x1
= 4
2 – (–2)
1 – 0
Solution follows…

16. Slope 4.1.3 Independent Practice
Lesson
4.1.3
Slope
Independent Practice
1. Identify whether the slope of each of the lines below is positive, negative, or zero. y y y –4 –2 2 4 –4 –2 2 4 –4 –2 2 4 x x x
zero
positive
negative
Solution follows…

17. Slope 4.1.3 Independent Practice
Lesson
4.1.3
Slope
Independent Practice
On a coordinate plane, draw lines with the slopes given in Exercises 2–5.
2. 3
3. 6
4. –1
5. –4 1 2 3 4 –1 –3 –4 –2 y 5 4 3 2 x
NB. Any parallel line will have the same slope
Solution follows…

18. Slope 4.1.3 Independent Practice
Lesson
4.1.3
Slope
Independent Practice
In Exercises 6–9, find the slope of the line passing through the two points.
6. W (3, 6) and R (–2, 9) 7. Q (–5, –7) and E (–11, 0)
8. A (–12, 18) and J (–10, 6) 9. F (2, 3) and H (–4, 6) –3 5 –7 6 –1 2 –6
Solution follows…

19. Slope 4.1.3 Independent Practice
Lesson
4.1.3
Slope
Independent Practice
10. The move required to get from point C to D is up six and left eight units. What is the slope of the line connecting C and D?
11. Point G with coordinates (7, 12) lies on a line with a slope of Write the coordinates of another point that lies on the same line. 3 4 –3 4
Answers will vary. Example: (11, 15)
Solution follows…

20. Slope 4.1.3 Independent Practice
Lesson
4.1.3
Slope
Independent Practice
12. On the coordinate plane, draw a line through the points E (–2, 5) and S (4, 1). Find the slope of this line.
On the same plane, draw a line through the points P (–2, –2) and N (4, –6). Find the slope of this line.
What can you say about the two lines you have drawn and their slopes? 2 4 6 8 –2 –6 –8 –4 –2 3
Slope of the line through E and S = E S P N –2 3
Slope of the line through P and N =
They’re parallel and their slopes are the same.
Solution follows…

21. Slope 4.1.3 Independent Practice
Lesson
4.1.3
Slope
Independent Practice
13. Consider the statement: “The slope of a line becomes less steep if the distance you have to move along the line for a given change in y increases.” Determine whether this statement is true or not.
14. Is it possible to calculate the slope of a vertical line? Explain your answer.
True
No — it’s undefined, as the change in x is zero, and you cannot divide by zero.
Solution follows…

22. Lesson
4.1.3
Slope
Round Up
The slope of a line is the ratio of the change in the y-direction to the change in the x-direction when you move between two points on the line — it’s basically a measure of how steep the line is.
Positive slopes go “uphill” as you go from left to right across the page, and negative slopes go “downhill.”
Slope is actually a rate — and you’ll be looking at rates over the next few Lessons.