Name Algebra 1B notes and homework
September 25, 2008 “Slope of a linear function” page 6
Slope of a linear function
A linear function is a function whose graph is a straight line. Every linear function has a number called its slope. The slope m tells you the direction and the steepness of the line.
· If m is positive, the graph slopes upward. If m is negative, the graph slopes downward.
· Larger m values have “steeper” graphs; smaller m values are “less steep.”
· When you move 1 unit to the right on the graph, you move up or down by m units.
If m is positive, you move up. If m is negative, you move down.
Five ways to find slopes
A. If you have a line graph, choose any two points on the line.
(It’s easiest if you choose points located at grid intersections.)
Count squares to find the rise and the run.
(If the rise goes down, use a negative number for the rise.)
Then .
B. Using any two points (x1, y1) and (x2, y2) taken from a table or a graph, you can also find the rise and the run by subtracting:
.
Example: Find the slope of the line through the points (3, –5) and (10, 4).
Solution: x1 = 3 y1 = –5 x2 = 10 y2 = 4
C. If you have a function equation in the “slope-intercept” form f(x) = mx + b or y = mx + b,
the number m is the slope.
Examples: y = –4x + 5 has a slope of –4. f(x) = x has a slope of .
D. If you have a table of values where the x-values go up by 1’s, the y-values goes up or down bythe slope.
Examples: / x / y–3 / –5
–2 / –3
–1 / –1
0 / 1
1 / 3
2 / 5
3 / 7
/ x / f(x)
–3 / 5.0
–2 / 4.5
–1 / 4.0
0 / 3.5
1 / 3.0
2 / 2.5
3 / 2.0
This table has a slope of 2. / This table has a slope of –0.5.
For tables where the x-values have a different spacing or an uneven spacing, just choose anytwopoints from the table and use method B instead.
E. Horizontal lines always have slope = 0. Vertical lines have no slope.
Problems
1. Find the slope of each line.
a./ b.
/ c.
d.
/ e.
/ f.
g.
/ h.
/ i.
2. Find the slope for the line passing through each pair of points specified.
a. (1, –2) and (4, –3) c. (–3, 1) and (5, 1)
b. (–3, 1) and (–3, 4) d. (20, 100) and (80, 1000)
3. Find the slope for each of these tables. Hint: Parts a and b can be answered just by looking at the changes in the output column; for parts c and d use the same method as in problem 2.
a. / b. / c. / d.x / y
–3 / 8
–2 / 5
–1 / 2
0 / –1
1 / –4
2 / –7
3 / –10
/ x / f(x)
–3 / –2
–2 / –0.5
–1 / 1
0 / 2.5
1 / 4
2 / 5.5
3 / 7
/ x / y
–6 / 21
–4 / 17
–2 / 13
0 / 9
2 / 5
4 / 1
6 / –3
/ x / f(x)
–8 / –15
–5 / –9
–2 / –3
0 / 1
2 / 5
5 / 11
7 / 15
4. Identify the slope for each of these equations.
a. f(x) = – 7
b. y = –4x
c. f(x) = 2 – 6x Hint: Need to rearrange into mx + b form before answering.
d. y = 5
e. y + 1 = 3x Hint: Need to solve the equation for y before answering.
f. y = x
Interpreting the slope number
· The slope of a line tells you which direction the line is going.
□ A positive slope means that a line is increasing (goes up when moving from left to right).
□ A negative slope means that a line is decreasing (goes down when moving from left to right).
□ A zero slope means that the line is constant (in other words: flat, horizontal).
□ Note that vertical lines do not have a slope.
· The slope of a line gives a measurement of how steep the line is.
□ For positive slopes, a larger slope number indicates a line that is steeper (rises more quickly).
□ For negative slopes, slope numbers further from zero indicate steeper slopes
(for example, slope = –3 would be a steeper line than slope = –2).
Problems
5. a. Make up an equation for a line that is steeper than the line y = 4x.
b. Make up an equation for a line the slopes downward.
c. Make up a graph of a line that has a slope of –3. / f. Make up a graph of a line that has a slope of .e. Make up an input-output table for a line that slopes downward. / f. Make up an input-output table for a line that has a slope of .
x / f(x)
/ x / f(x)
6. Here are equations for several lines. Rewrite them in order according to their direction and steepness. (Begin with the steepest increasing line, end with the steepest decreasing line.)
y = 2x + 1
y = –4x + 2
y = x + 6
y = –x + 5
y =
y = –x + 4
y = 8