Transcript Slope Lesson 4.1.3 Lesson 4.1.3 Slope California Standard: What it means for you: Algebra and Functions 3.3 Graph linear functions, noting that the vertical change (change in y-value) per unit.

Slope
Lesson 4.1.3
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Lesson
4.1.3
Slope
California Standard:
What it means for you:
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.
You’ll learn what the slope of a
graph is and how to calculate it.
Key words:
• slope
• steepness
• ratio
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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.
There’s a measure for how
steep a line is — slope.
y
x
In this Lesson you’ll learn how to find
the slope of a straight-line graph.
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Lesson
Slope
4.1.3
The Slope of a Line is a Ratio
For any straight line, the ratio
change in y
is always the same
change in x
— it doesn’t matter which two points you choose to measure
y
10
change in y
8
change in y
change in x
This ratio,
=
6
3
2
= 1.5
2 units
6
= 1.5
change in y
,
change in x
is the slope of the graph.
9 units
change in x
=
9
6 units
4
3 units
the changes between.
2
x
0
0
2
4
6
8
10
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Lesson
Slope
4.1.3
Slope is a Measure of Steepness of a Line
A larger change in y for the same change in x makes the ratio
change in x
bigger, so the slope is greater.
change in y
change in x
change in y
change in x
=
=
6
3
2
4
10
y
3 units
=2
8
6 units
2 units
change in y
= 0.5 6
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.
4
2
4 units
x
0
0
2
4
6
8
10
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Lesson
4.1.3
Slope
Slopes Can Be Positive, Negative or Zero
y
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 change in y
positive change in x
= positive slope
x
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Lesson
4.1.3
Slope
y
negative change in 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
positive change in x
positive change in x
= negative slope
x
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Lesson
4.1.3
Slope
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
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.
change in x = 0
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x
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.
Negative — it’s downhill
from left to right.
y
5
5
y = –x
4
4 units
change in y
4
= =2
change in x
2
3
0
2–5
1
0
x
5
2 units
0
0
1
–5
2
x
3
4
5
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Solution follows…
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):
change in y y2 – y1
Slope =
=
change in x x2 – x1
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Lesson
4.1.3
Example
Slope
1
The graph below is the graph of the equation y = 2x + 1.
Find the slope of the line.
y
Solution
Start by drawing a triangle
connecting two points on
the graph.
Choose two points that are easy to
read from the graph, for example:
(x1, y1) = (–1, –1) and (x2, y2) = (1, 3)
4
3
(1, 3)
2
1
x
0
–4 –3 –2 –1 0 1 2 3 4
–1
(–1, –1)
–2
–3
–4
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Solution
Solution
continues…
follows…
Lesson
Slope
4.1.3
Example
1
The graph below is the graph of the equation y = 2x + 1.
Find the slope of the line.
y
y2 – y1
Slope =
x2 – x1
This is the change in y.
3 – (–1) 3 + 1 4
=
=
= =2
1 – (–1) 1 + 1 2
This is the change in x.
change
4 in x
3
(1, 3)
2
1
x
0
–4 –3 –2 –1 0 1 2 3 4
–1
(–1, –1)
–2
–3
–4
change in y
Solution (continued)
So the slope of the graph is 2.
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Lesson
4.1.3
Example
Slope
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:
y2 – y1 –4 – 5 –4 – 5 –9
Slope =
=
=
=
x2 – x1 1 – (–2) 1 + 2
3
Slope = –3
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Solution
Solution
continues…
follows…
Lesson
4.1.3
Example
Slope
2
Find the slope of the line connecting the points
C (–2, 5) and D (1, –4).
Solution (continued)
If you plot these points and
draw a line through them, you
can see that the slope is
negative (it’s a “downhill” line).
C (–2, 5)
y
4
3
2
1
x
0
–4 –3 –2 –1 0 1 2 3 4
–1
–2
–3
D (1, 4)
–4
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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
5–3
=
=
= 0.5
x2 – x1 2 – (–2)
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4. Plot the graph of the equation
y = 4x – 2 and find its slope.
Find two points
on the line:
x
0
1
y
–2
2
y2 – y1
4
2 – (–2)
x2 – x1 = 1 – 0 = 1 = 4
y
4
3
2
1
x
0
–4 –3 –2 –1 0 1 2 3 4
–1
–2
–3
–4
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Solution follows…
Lesson
Slope
4.1.3
Independent Practice
1. Identify whether the slope of each of the lines
below is positive, negative, or zero.
y
4
2
–4
–2
0
y
4
2
x
0
2
4
–4
–2
0
y
4
2
x
0
2
4
–4
–2
0
x
0
–2
–2
–2
–4
–4
–4
zero
positive
2
4
negative
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Solution follows…
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
5
4
3
2
y
4
3
2
1
x
0
–4 –3 –2 –1 0 1 2 3 4
–1
–2
–3
–4
NB. Any parallel line will have the same slope
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Solution follows…
Lesson
Slope
4.1.3
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)
–3
5
8. A (–12, 18) and J (–10, 6)
–6
7. Q (–5, –7) and E (–11, 0)
–7
6
9. F (2, 3) and H (–4, 6)
–1
2
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Solution follows…
Lesson
Slope
4.1.3
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? –3
4
11. Point G with coordinates (7, 12) lies on a line with
a slope of 3 . Write the coordinates of another point
4
that lies on the same line.
Answers will vary. Example: (11, 15)
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Solution follows…
Lesson
Slope
4.1.3
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
Slope of the line through E and S =
3
8
E6
4
S
2
0
–8 –6 –4 –2 0
–2
2
4
6
8
–2
Slope of the line through P and N =
3
P–4
–6
–8
N
They’re parallel and their slopes are the same.
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Solution follows…
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.
True
14. Is it possible to calculate the slope of a vertical line?
Explain your answer.
No — it’s undefined, as the change in x is zero,
and you cannot divide by zero.
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Solution follows…
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.
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