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Math 103 – Graphs – Functions – Average Rate of Change
1) Complete the table using the graphs shown below
Graph 1
Graph 2
Graph 3
Graph 4
Yes, it passes the
vertical line test
(every x has only
one image)
Yes, it passes the
vertical line test
(every x has only
one image)
NO
An x has infinitely
many images
Yes, it passes the
vertical line test
(every x has only
one image)
a) Is it a function?
EXPLAIN
b) Give the domain.
All real numbers
All real numbers
X=3
All real numbers
c) Give the range.
All real numbers
All real numbers
All real numbers
Y=4
d) What is the slope
of this line?
M=3
M = -3/2
M is undefined
M=0
P(0,0)
b=0
P(0,5)
b=5
none
P(0,4)
b=4
f) Write the
equation for the
graph
Y = 3x
Y = (-3/2)x + 5
X=3
Y=4
g) What is the xintercept? Use the
graph.
X=0
X ~ 3.5
X=3
none
e) Give the yintercept
Graph 3
Graph 1
Graph 4
Graph 2
1
2) Problem (1) continued: Use the results obtained in problem 1-f and write the equations for the lines of graphs
1 and 2 here.
Equation of graph 1 in the slope-intercept form
Equation of graph 2 in the slope-intercept form
Y = 3x
Y = (-3/2)x + 5
3) Problem (2) continued: For each of the two lines shown in problem (2), use algebra to find the x- and yintercepts.
Work for graph 1
Work for graph 2
y-intercept
If x = 0,
y = 3*0 = 0
y-intercept
If x = 0,
y = (-3/2)*0 + 5 = 5
x-intercept
If y = 0,
0 = 3x
X=0
x-intercept
If y = 0,
0 = (-3/2)x + 5
Multiply by 2 both sides
0 = -3x + 10
-10 = -3x
-10/-3 = x
X= 10/3
4) Use function notation to rewrite the two equations from part (2). Use f(x) for graph 1 and g(x) for graph 2.
Graph 1: f(x) = 3x
Graph 2: g(x) = (-3/2)x + 5
5) Use the functions from part (4), USE ALGEBRA to find each of the following. After getting the answers by
using algebra, check on the graphs from the previous page to see if answers are correct!
a) f(1) = 3(1) = 3
b) g(-2) = (-3/2)*(-2) + 5 = 3 + 5 = 8
c) Solve f(x) = 5
5 = 3x
X = 5/3
d) Find x when g(x) = -1
- 1 = (-3/2) x + 5
multiply by 2 both sides
-2 = -3 x + 10
-12 = - 3x
X=4
2
6) Complete the table using the graphs shown below
Graph 1
Graph 2
Yes, it passes the
vertical line test (every x
has only one image)
Yes, it passes the
vertical line test (every x
has only one image)
[-4, 4]
(-∞, 4]
[0, 4]
[0, ∞)
d) Give the y-intercept
Y=4
Y=2
e) Give the x-intercept(s)
X = -4, x = 4
X=4
a) Is it a function?
EXPLAIN
b) What is the domain?
(Write as an inequality,
and with interval
notation)
c) What is the range?
(Write as an inequality,
and with interval
notation)
f(x) - Graph 1
g(x) - Graph 2
7) Use the graphs from problem (6) to estimate each of the following:
(a) f(2) = ~3.3
(c) Solve f(x) = -4
(b) Solve f(x) = 2 when x ~ -3.3
no solution
(d) f(-4) = 0
(e) g(3) = 1
(f) Solve g(x) = 3 when x = -5
(h) Solve g(x) = 2, when x = 0
(i) g(2) ~ 1.4
3
8) The graph shows the population in millions of bacteria t minutes after a bactericide is introduced into a
culture.
a) Over what time interval is the population increasing?
(0, 3) (during the first 3 minutes)
b) Over what time interval is the population decreasing?
(3, ∞)
c) Find the average rate of change of population with respect to time for the given intervals. (To find the
average rate of change we find the slope of the line connecting the two points)
Show work:
(i) Find ARC over [1, 2]
Find the slope of the line through the points (1,3) and (2,5)
m = (5 – 3) / (2 – 1) = 2 / 1 = 2 million bacteria per minute
Interpret in words within context:
From one to two minutes after the bactericide was introduced, the number of bacteria is
increasing at a rate of 2 million bacteria per minute.
(ii) Find ARC over [4, 6]
Find the slope of the line through the points (4, 4) and (6, 2)
m = (2 - 4) / (6 - 4) = - 2 / 2 = - 1 million bacteria per minute
Interpret in words within context:
From two to six minutes after the bactericide was introduced, the number of bacteria is decreasing
at a rate of 1 million bacteria per minute.
d) Give an interval over which the number of bacteria is decreasing at a rate of two million bacteria per
minute
From t = 3 to t = 4 the slope is m = (4 – 6) / (4 – 3) = -2 million bacteria per minute
Over the time interval (3, 4) the number of bacteria is decreasing at a rate of two million bacteria
per minute
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