2.1.4 What Information determines a line?
Name:________________________ Date:_____
Objective: In previous lessons, you found the slope and starting values of linear relations.
You connected growth and start to their representations in patterns, tables, equations, and
graphs. Today you will complete your focus on finding slope as well as using slope and the
y-intercept to find the equation of a line. During this lesson, keep the following questions in
mind:
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How can you find the growth? How can you find the starting value?
Is there enough information to graph the line?
How can you find the slope of a line without graphing it?
What effect does m have on a graph of the line? What effect does b have?
The Slope of a Line, m
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The slope of a line is the ratio of the vertical distance to the horizontal distance of a slope triangle
formed by two points on a line. The vertical part of the triangle is called y (Δy ) (read “change in y”),
while the horizontal part of the triangle is called x (Δx ) (read “change in x”). It indicates both how
steep the line is and its direction, upward or downward, left to right.
Note that “Δ ” is the Greek letter “delta” that is often used to represent a difference or a change.
Note that lines pointing upward from left to right have positive slope, while lines pointing downward
from left to right have negative slope. A horizontal line has zero slope, while a vertical line has
undefined slope.
To calculate the slope of a line, pick two points on the line, draw a slope triangle (as shown in the
example above), determine Δy and Δx, and then write the slope ratio. You can verify that your slope
correctly resulted in a negative or positive value based on its direction. In the example above,
Δy = 2 and Δx = 5, so the slope is
.
2-36. Equations for linear patterns can all be written in the form y = mx + b.
a) x and y represent variables. When you wrote equations relating the figure
number to the number of tiles, what did x represent? What did y represent?
b) m and b are parameters – they do not change within the situation of a given linear pattern. m is also
called a coefficient since it multiplies the variable x. What do m and b represent in a linear pattern like
the tile patterns?
2-37. THE LINE FACTORY
If the description below describes one and only one line, then write the equation and graph the line. However, if
you do not have enough information to draw one specific line, draw at least two lines .
Standardizes graphs by scaling the axes from –10 to 10 in both directions.
a) Line A goes through the point (2, 5).
b) Line B has a slope of –3 and goes through the origin.
b)
c) Line C goes through points (–3, –2) and (3, 10).
d)Line D has the following table.
x
y
2
1
3
3.5
4
6
5
8.5
f)Line F goes through the point (8, –1) and has a slope of –
e)Line E grows by 4.
g) Customer G sent the following table.
x
y
2-38.
−2
−1
0
1
2
3
1
3
9
27
FINDING THE SLOPE OF A LINE WITHOUT GRAPHING
While finding the slope of a line that goes through the points (6, 5) and (3, 7),
Gloria figured that Δy = −2 and Δx = 3 without graphing.
a. Explain how Gloria could find the horizontal and
vertical distance of the slope triangle without
graphing.
b. Draw a sketch of the line and validate her method.
c. What is the slope of the line?
.
d. Use Gloria’s method (without graphing) to find the slope of the line that goes through the points
(4, 15) and (2, 11).
e. Use Gloria’s method to find the slope of the line that goes through the points (28, 86) and (34,
83).
f. Another student found the slope from part (d) to be 2. What error or errors did that student
make?
2-39. SLOPE CHALLENGE
What is the steepest line possible? What is its slope? Be ready to justify your statements.
HW 12 : 2-41 – 2-45