Section 4.1: Linear functions and their properties
# 1 – 10: Find
a) slope
b) y-intercept
c) x-intercept (if any)
d) sketch a graph
e) Determine whether the function is always increasing, always decreasing, or constant
1) f(x) = 3x – 6
2) f(x) = 2x – 10
3) g(x) = -2x – 8
5) f(x) = 7
6) f(x) = - 2
7) 𝑔(𝑥) = 3 𝑥 − 4
9) 𝑓(𝑥) =
−𝑥
3
+6
10) f(x) =
−𝑥
3
2
4) g(x) = -4x – 12
8) 𝑔(𝑥) =
3
𝑥
4
−6
+ 12
11) Suppose f(x) = 3x – 6 and g(x) = - 2x + 4
a) Solve f(x) = 0
b) Solve f(x) > 0
c) Solve f(x) = g(x)
d) Solve f(x) < g(x)
e) graph f(x) and g(x) on the same graph, and label the solution to the equation f(x) = g(x)
12) Suppose f(x) = -3x – 2 and g(x) = -2x + 8
a) Solve f(x) = 0
b) Solve f(x) > 0
c) Solve f(x) = g(x)
d) Solve f(x) < g(x)
e) graph f(x) and g(x) on the same graph, and label the solution to the equation f(x) = g(x)
13) Suppose f(x) = x-3 and g(x) = 2x + 4
a) Solve f(x) = 0
b) Solve f(x) > 0
c) Solve f(x) = g(x)
d) Solve f(x) < g(x)
e) graph f(x) and g(x) on the same graph, and label the solution to the equation f(x) = g(x)
14) Suppose f(x) = 3x – 6 and g(x) = 4x + 4
a) Solve f(x) = 0
b) Solve f(x) > 0
c) Solve f(x) = g(x)
d) Solve f(x) < g(x)
e) graph f(x) and g(x) on the same graph, and label the solution to the equation f(x) = g(x)
Section 4.1: Linear functions and their properties
15) A jeweler’s salary was $30,000 in 2009 and $33,000 in 2012. The jeweler’s salary follows a linear
growth pattern.
a) Create a linear function to model the data
b) Use the function to estimate the jeweler’s salary in 2014.
16) OSU had 28,000 students in 2008 and 29,500 students in 2010. Assume that the growth is linear.
a) Create a linear function to model the data
b) Use the function to estimate the enrollment in 2013.
17) A sub shop purchases a used oven for $1,000. After 5 years the oven will need to be replaced, and
will have a value of $200. Assume the value of the oven goes down by the same amount each year.
a) Write a linear function giving the value of the equipment for the 5 years it will be in use.
b) Use the function to find the value of the equipment after 3 years.
18) A school district purchases a high volume copier for $10,000. After 10 years the equipment will be
worth $500 and it will have to be replaced. Assume the value of the equipment goes down by the same
amount each year.
a) Write a linear function giving the value of the equipment for the 10 years it will be in use.
b) Find the value of the equipment after 3 years.
19) A real estate office manages an apartment complex with 100 units. When the rent is $600 per
month, all 100 units will be occupied. However, when the rent is $700 per month only 90 units will be
occupied. Assume the relationship between the monthly rent and the number of units occupied is
linear.
a) Write the equation giving the demand y in terms of the price p
b) Use the equation to predict the number of units occupied when the price is $675. (round to the
nearest unit if needed)
c) What price would cause 80 units to be occupied?
20) A real estate office manages an apartment complex with 1000 units. When the rent is $600 per
month, all 50 units will be occupied. However, when the rent is $700 per month only 800 units will be
occupied. Assume the relationship between the monthly rent and the number of units occupied is
linear.
a) write the equation giving the demand x in terms of the price p
b) Use the equation to predict the number of units occupied when the price is $675. (round to the
nearest unit if needed)
c) What price would cause 950 units to be occupied?
21) A truck rental company rents a truck for $50 per day plus 40 cents per mile.
a) Write a linear model that relates the Cost C, in dollars of renting the truck for one day when x miles
are driven.
b) What is the cost of renting the truck for a day when 100 miles are driven?
c) Assume the cost of renting the truck is $130. How many miles were driven?
22) A truck rental company rents a truck for $20 per day plus 60 cents per mile.
a) Write a linear model that relates the Cost C, in dollars of renting the truck for one day when x miles
are driven.
b) What is the cost of renting the truck for a day when 200 miles are driven?
c) Assume the cost of renting the truck is $80. How many miles were driven?
23) A phone company offers a monthly domestic long distance package by charging $10 plus 5 cents per
minute.
a) Write a linear function that models the cost in dollars to the number of minutes used.
b) Use the function to compute the monthly cost when 500 minutes are used.
c) How many minutes were used if the monthly cost was $40.00?
24) A phone company offers a monthly domestic long distance package by charging $10 plus 3 cents per
minute.
a) Write a linear function that models the cost in dollars to the number of minutes used.
b) Use the function to compute the monthly cost when 300 minutes are used.
c) How many minutes were used if the monthly cost was $40.00?
Section 4.2: Linear models – building linear functions from data
Data:
Is there a linear relationship between Math
SAT scores and the number of hours spent studying
for the test? A study was conducted involving 20
students as they prepared for and took the Math
section of the SAT Examination.
Task:
a.) Determine a linear regression model
equation to represent this data.
b.) Graph the new equation.
c.) Decide whether the new equation is a
"good fit" to represent this data.
d.) Interpolate data: If a student studied for
15 hours, based upon this study, what
would be the expected Math SAT
score?
Hours Spent
Studying
4
9
10
14
4
7
12
22
1
3
8
11
5
6
10
11
16
13
13
10
Math SAT
Score
390
580
650
730
410
530
600
790
350
400
590
640
450
520
690
690
770
700
730
640
e.) Interpolate data: If a student obtained a Math SAT score of 720, based
upon this study, how many hours did the student most likely spend
studying?
f.) Extrapolate data: If a student spent 100 hours studying, what would be
the expected Math SAT score? Discuss this answer.
Any answers in relation to this problem are to be rounded to the nearest
tenth.
If rounding is not indicated in a problem, leave the full calculator entries as answers.
Step 1.
Enter the data into the lists.
For basic entry of data, see Basic
Commands.
Step 2.
Create a scatter plot of the data.
Go to STATPLOT (2nd Y=) and choose
the first plot. Turn the plot ON, set the icon
to Scatter Plot (the first one), set Xlist to L1
and Ylist to L2 (assuming that is where you
stored the data), and select a Mark of your
choice.
Step 3.
Choose Linear Regression
Model.
Press STAT, arrow right to CALC, and
arrow down to 4: LinReg (ax+b). Hit
ENTER. When LinReg appears on the
home screen, type the parameters L1, L2,
Y1. The Y1 will put the equation into Y=
for you.
(Y1 comes from VARS → YVARS, #Function, Y1)
The linear regression equation is
y = 25.3x + 353.2
(answer to part a)
Step 4.
Graph the Linear Regression
Equation from Y1.
ZOOM #9 ZoomStat to see the graph.
(answer to part b)
Step 5.
Is this model a "good fit"?
The correlation coefficient, r, is
.9336055153 which places the correlation
into the "strong" category. (0.8 or greater is
a "strong" correlation)
The coefficient of determination, r 2, is
.8716192582 which means that 87% of the
total variation in y can be explained by the
relationship between x and y. The other
13% remains unexplained.
Yes, it is a "good fit".
Hit catalog on the bottom of the calculator
and scroll down to “diagnostics on” if your
calculator does not show r.
(answer to part c)
Step 6.
Interpolate: (within the data set)
If a student studied for 15 hours, based upon
this study, what would be the expected Math SAT
score?
Step 7.
Interpolate: (within the data
set)
If a student obtained a Math SAT score of
720, based upon this study, how many hours
did the student most likely spend studying?
Use the 2nd – Calc – value feature on your
calculator to evaluate the function at x = 15, change the y in our equation to 720 and
solve for x.
or just plug 15 in for x in the equation we
720 = 25.3x + 353.2
just created
Answer x = 14.50 hours (rounded)
Y = 732.7 should score about 733
Step 8.
Extrapolate data: (beyond the data set)
If a student spent 100 hours studying, what would be the expected Math SAT score?
Discuss this answer.
Plug 100 in for x in the function that was created, either
by hand or using the 2nd calc value technique
Our equation shows that if a student studies 100 hours, he/she
should score 2885.8 on the Math section of the SAT
examination. The only problem with this answer is that the
highest score that can be obtained is 800. So why is this score
so outrageous? ANSWER: When you extrapolate data, the
further you move away from the data set, the less accurate your
information becomes. In this problem, the largest number of
hours in the data set was 22 hours, but the extrapolation tried to
jump to 100 hours.
Section 4.2: Linear models – building linear functions from data
1. As Earth’s population continues to grow, the solid waste generated by the population grows with
it. Governments must plan for disposal and recycling of ever growing amounts of solid waste.
Planners can use data from the past to predict future waste generation and plan for enough
facilities for disposing of and recycling the waste.
Given the following data on the waste generated in Florida from 1990-1994
Year
Tons of Solid Waste
Generated (in thousands)
19,358
19,484
20,293
21,499
23,561
1990
1991
1992
1993
1994
a) Make a scatterplot of the data, letting x represent the number of years since 1990.
b)
c)
d)
e)
Use a graphing calculator to fit a linear function to the data.
Decide whether the new equation is a "good fit" to represent this data.
Graph the function of best fit with the scatterplot of the data.
Predict the average tons of waste in 2005
2. The numbers of insured commercial banks y (in thousands) in the United States for the years 1987
to 1996 are shown in the table. (Source: Federal Deposit Insurance Corporation).
Year
y
1987
13.70
1988
13.12
1989
12.71
1990
12.34
1991
11.92
1992
11.46
1993
10.96
1994
10.45
1995
9.94
1996
9.53
a) Make a scatterplot of the data, letting x represent the number of years since 1987.
b)
c)
d)
e)
Use a graphing calculator to fit a linear function to the data.
Decide whether the new equation is a "good fit" to represent this data.
Graph the function of best fit with the scatterplot of the data
Predict the average number of insured commercial banks in 2000.
Section 4.2: Linear models – building linear functions from data
3. U.S. Farms. As the number of farms has decreased in the United States, the average size of the
remaining farms has grown larger, as shown in the table below.
Year
1910
1920
1930
1940
1950
1959
1969
1978
1987
1997
a)
b)
c)
d)
e)
Average Acreage Per
Farm
139
149
157
175
216
303
390
449
462
487
Make a scatterplot of the data, letting x represent the number of years since 1900.
Use a graphing calculator to fit a linear function to the data.
Decide whether the new equation is a "good fit" to represent this data.
Graph the function of best fit with the scatterplot of the data
Predict the average acreage in 2010.
4. Sports The winning times (in minutes) in the women’s 400-meter freestyle swimming event in the
Olympics from 1936 to 1996 are given by the following ordered pairs.
(1936,5.44)
(1948,5.30)
(1952,5.20)
(1956, 4.91)
(1972, 4.32)
(1976, 4.16)
(1980, 4.15)
(1984, 4.12)
(1960, 4.84) (1988, 4.06)
(1964, 4.72) (1992, 4.12)
(1968, 4.53) (1996, 4.12)
a) Make a scatterplot of the data, letting x represent the number of years since 1936.
b)
c)
d)
d)
Use a graphing calculator to fit a linear function to the data.
Decide whether the new equation is a "good fit" to represent this data.
Graph the function of best fit with the scatterplot of the data.
Predict the winning time in 2012.
Section 4.2: Linear models – building linear functions from data
5) The average salary S (in millions of dollars) for professional baseball players from 1996 to 2002 is
shown in the table.
Year
Salary
1996
1.1
1997
1.3
1998
1.4
1999
1.6
2000
1.8
2001
2.1
2002
2.3
a) Make a scatterplot of the data, letting x represent the number of years since 1996.
b) Use a graphing calculator to fit a linear function to the data.
c) Decide whether the new equation is a "good fit" to represent this data.
d) Graph the function of best fit with the scatterplot of the data
e) Predict the average salary in 2018. (round to the nearest 10th)
6) The table show the number of Target stores worldwide from 1997 to 2002.
Year
1997
Number of 1130
Target
stores
1998
1182
1999
1243
2000
1307
2001
1381
a) Make a scatterplot of the data, letting x represent the number of years since 1997.
b) Use a graphing calculator to fit a linear function to the data.
c) Decide whether the new equation is a "good fit" to represent this data.
d) Graph the function of best fit with the scatterplot of the data
e) Predict the number of Target stores in 2020.
2002
1476
Section 4.3: Quadratic functions and their properties
#1-12: For each problem do the following
a) describe the transformation as compared to the function f(x) = x2, specifically state if the graph is shifted left,
right, up, down and if any reflection has occurred
b) make a table of values and sketch a graph
c) state the domain and range of the function
d) state the intervals where the function in increasing and decreasing
e) state if the function has a local maximum, if it does state the local maximum value
f) state if the function has a local minimum, if it does state the local minimum value
1) f(x) = (x – 3)2 + 4
2) g(x) = (x – 2)2 + 6
4) m(x)= 3(x+1)2 + 2
5) g(x) = (𝑥 + 4)2 − 6
6) 𝑓(𝑥) = (𝑥 + 3)2 − 1
7) m(x) = -2x2 + 3
8) t(x) = 3x2 + 6
9) 𝑓(𝑥) = − (𝑥 + 5)2 − 2
10) n(x) = -3(x – 3)2 – 3
11) b(x) = 2(x+3)2 + 4
12) f(x) = -2(x+1)2 + 5
1
2
3) h(x) = 2(x+3)2 – 4
1
3
1
4
#13 – 24: For each problem do the following:
a) Use completing the square to rewrite the problem in standard form
b) Sketch a graph, make sure to label the vertex. You may use your calculator, instead of making a table of
values to create your graph
13) f(x) = x2 + 6x + 5
14) g(x) = x2 + 10x – 11
15) k(x) = x2 – 4x + 2
16) m(x) = x2 – 2x + 6
17) f(x) = 2x2 + 8x – 3
18) h(x) = 4x2 + 24x – 6
19) f(x) = -x2 + 6x + 4
20) g(x) =- x2 – 8x – 2
21) k(x) = -2x2 + 12x – 7
22) g(x) = -2x2 + 8x + 3
23) f(x) = -3x2 – 12x + 1
24) n(x) = -2x2 + 20x + 3
.
Section 4.3: Quadratic functions and their properties.
#25 – 32, determine the equation of the quadratic function
25)
26)
27)
28)
Section 4.3: Quadratic functions and their properties.
29)
31)
30)
32)
Section 4.4: applications of quadratic equations
1) When a ball is thrown straight upward into the air, the equation h= -16t2 + 80t gives the height (h) in feet
that the ball is above the ground t seconds after it is thrown.
a) How long does it take for the ball to hit the ground?
b) When does the ball reach its maximum height?
c) What is the maximum height of the ball?
2) An object fired vertically into the air with an initial velocity of 96 feet per second will be at a height (h) in feet,
t seconds after launching, determined by the equation h=96t – 16t2.
a) at which times will the object have a height of 80 feet?
b) how long it will take the object to return to the ground.
c) determine the maximum height the object reaches.
3) A ball is shot into the air. It's height , h in meters after t seconds is modeled by h=-4.9t2 + 30t +1.6.
a) How long to 2 decimal places will it take the ball to reach a height of 35m?
b) how long will it take to land (round to 2 decimals)?
c) Determine the maximum height it reaches (round to 2 decimals).
4) A student tosses a water balloon of the roof of an apartment building. The height (h) in feet of the balloon,
(t) seconds after it is launched, is given by the formula h = -16t2 +48t + 112. After how many seconds will the
balloon hit the ground?
5) A baby drops his bottle at the peak of a ferris wheel. The height (h) in feet of the bottle, t seconds after the
baby drops the bottle is given by h=-16t2 + 64t + 80. After how many seconds will the bottle hit the ground?
6) A diver jumps off a cliff to water that is 240 feet below. The diver’s height (h) in feet (t) seconds after diving
is given by h = -16t2 + 240. How long does the dive last?
7) A diver jumps off a cliff to water that is 160 feet below. The diver’s height (h) in feet (t) seconds after diving
is given by h = -16t2 + 160. How long does the dive last?
Section 4.4: Applications of quadratic equations
8) A chain store manager has been told by the main office that daily profit, P, is related to the number of clerks
working that day, x, according to the equations P = -25x2 + 300x.
a) What number of clerks will maximize the profit?
b) What is the maximum possible profit?
9) The total profit ( p(x) ) in dollars for a company to manufacture and sell x items per week is given by the
function p(x) =- x2 + 50x.
a) What number of units will maximize profit?
b) What is the maximum profit?
10) A factory produces lemon-scented widgets. You know that the more you produce the lower the unit cost is,
to a point. As production levels increase so do the labor costs, storage costs, etc. An accountant has computed
the cost for producing x thousands of units per day can be approximated by the formula
C(x) = 0.04x2 – 8.504x – 25302.
a) Find the daily production level that minimizes cost.
b) What is the minimum cost?
11) A manufacturer of lighting fixtures has a daily production cost of C(x) = 0.25x2 -10x + 800. Where x is the
number of units produced.
a) how many fixtures should be produced each day to minimize cost?
b) What is the minimum cost?
Section 4.4: Applications of quadratic equations
12) In one study the efficiency of photosynthesis in an Antarctic species of grass was investigated. Table 1 below
lists results for various temperatures. The temperature x is in degrees Celsius and the efficiency y is given as a
percent. The purpose of the research was to determine the temperature at which photosynthesis is most
efficient.
x
y(%)
-1.5
0
2.5
5
7
10
12
15
17
20
22
25
27
30
33
46
55
80
87
93
95
91
89
77
72
54
46
34
a) Make a scatterplot of the data, on your calculator. You do not need to make a copy of this
on your paper.
.b) Use a graphing calculator to fit a linear function and a quadratic function to the data.
c) Decide which equation is the best to represent this data.
d) Graph the function of best fit with the scatterplot of the data, on your calculator. You do not
need to make a copy of this on your paper.
e) Use the equation to determine the optimal temperature at which photosynthesis is most
efficient.
13) A baseball player hits a ball. The following data represents the height of the ball at different times.
Time (in
seconds)
height
1
2
3
4
5
5
50
90
70
10
a) Make a scatterplot of the data, on your calculator. You do not need to make a copy of this
on your paper.
b) Use a graphing calculator to fit a linear function and a quadratic function to the data.
c) Decide which equation is the best to represent this data.
d) Graph the function of best fit with the scatterplot of the data, on your calculator. You do not
need to make a copy of this on your paper.
e) Use the equation to find the maximum height of the ball.
14) The following data represent the birth rate (per 1000 women) for women whose age is x, in 2007
age
Birth rate
12
1
17
42
22
106
27
117
32
100
37
48
42
10
a) Make a scatterplot of the data, on your calculator. You do not need to make a copy of this
on your paper.
b) Use a graphing calculator to fit a linear function and a quadratic function to the data.
c) Decide which equation is the best to represent this data.
d) Graph the function of best fit with the scatterplot of the data, on your calculator. You do not
need to make a copy of this on your paper.
e) Predict the birth rate of 35 year old women.
Section 4.4: Applications of quadratic equations
15)
The concentration (in milligrams per liter) of a medication in a patient’s blood as time passes is given by
the data in the following table:
Time Concentration
(Hours)
(mg/l)
0
0
0.5
78.1
1
99.8
1.5
84.4
2
50.1
2.5
15.6
a) Make a scatterplot of the data, on your calculator. You do not need to make a copy of this
on your paper.
b) Use a graphing calculator to fit a linear function and a quadratic function to the data.
c) Decide which equation is the best to represent this data.
d) Graph the function of best fit with the scatterplot of the data, on your calculator. You do not
need to make a copy of this on your paper.
e) What is the concentration of medicine after 1.75 hours?
Chapter 4 Review:
1)
a)
b)
c)
d)
Suppose f(x) = 3x – 12 and g(x) = -5x + 4
Solve g(x) = 0
Solve g(x) > 0
Solve f(x) = g(x)
Solve g(x) < f(x)
2) A school district purchases 100 new televisions, one for each classroom for $50,000. After 8 years
the televisions will be worth $2,000 and they will have to be replaced. Assume the relationship between
the value of the televisions and the number of years in use in linear.
a) Write a linear function giving the value of the equipment for the 8 years it will be in use.
b) Find the value of the televisions after 6 years.
3) A real estate office manages an apartment complex with 1000 units. When the rent is $1000 per
month, all 1000 units will be occupied. However, when the rent is $1300 per month only 900 units will
be occupied. Assume the relationship between the monthly rent and the number of units occupied is
linear.
a) Write the equation giving the demand y in terms of the price p
b) Use the equation to predict the number of units occupied when the price is $1200. (round to the
nearest unit if needed)
c) What price would cause 800 units to be occupied?
4) The numbers of insured commercial banks y (in thousands) in the United States for the years 1987 to
1996 are shown in the table. (Source: Federal Deposit Insurance Corporation). (this is the same problem
that is in section 4.2)
Year
y
1987
13.70
1988
13.12
1989
12.71
1990
12.34
1991
11.92
1992
11.46
1993
10.96
1994
10.45
1995
9.94
1996
9.53
a) Make a scatterplot of the data, letting x represent the number of years since 1987, on your
calculator. You do not need to make a copy of this on your paper.
b) Use a graphing calculator to fit a linear function to the data.
c) Decide whether the new equation is a "good fit" to represent this data.
d) Graph the function of best fit with the scatterplot of the data, on your calculator. You do not
need to make a copy of this on your paper.
e) Predict the average number of insured commercial banks in 2000.
Chapter 4 review continued
#5-7: For each problem do the following
a) describe the transformation as compared to the function f(x) = x2, specifically state if the graph is shifted left,
right, up, down and if any reflection has occurred
b) make a table of values and sketch a graph
c) state the domain and range of the function
d) state the intervals where the function in increasing and decreasing
e) state if the function has a local maximum, if it does state the local maximum value
f) state if the function has a local minimum, if it does state the local minimum value
5) h(x) = 2(x+1)2 +3
1
6) g(x) = 2 (𝑥 + 2)2 − 4
7) m(x) = -3x2 + 8
#8-9: For each problem do the following:
a) Use completing the square to rewrite the problem in standard form
b) Sketch a graph, make sure to label the vertex. You may use your calculator, instead of making a table of
values to create your graph
8) m(x) = x2 – 4x + 5
9) h(x) = -4x2 + 16x – 6
10) When a ball is thrown straight upward into the air, the equation h= -16t2 + 80t + 6 gives the height (h) in
feet that the ball is above the ground t seconds after it is thrown.
a) How long does it take for the ball to hit the ground?
b) When does the ball reach its maximum height?
c) What is the maximum height of the ball?
11) An object fired vertically into the air with an initial velocity of 96 feet per second will be at a height (h) in
feet, t seconds after launching, determined by the equation h= – 16t2 + 96t + 5
a) at which times will the object have a height of 85 feet?
b) how long it will take the object to return to the ground.
c) determine the maximum height the object reaches.
Chapter 4 review continued
12) A baseball player hits a ball. The following data represents the height of the ball at different times.
Time (in
seconds)
height
1
2
3
4
5
5
20
40
25
5
a) Make a scatterplot of the data on your calculator. You do not need to make a copy of this
graph on your paper.
b) Use a graphing calculator to fit a linear function and a quadratic function to the data.
c) Decide which equation is the best to represent this data.
d) Graph the function of best fit with the scatterplot of the data, on your calculator. You do not
need to make a copy of this on your paper.
e) Use the equation to find the maximum height of the ball.
Ch 4 practice test.
1) Suppose f(x) = -2x - 6 and g(x) = 5x + 20
a) Solve g(x) = 0 b) Solve f(x) > 0
c) Solve f(x) = g(x)
d) Solve g(x) < f(x)
2) A school district purchases new televisions, one for each classroom for $25,000. After 8 years the televisions will be worth $1,000 and
they will have to be replaced. Assume the relationship between the value of the televisions and the number of years in use in linear.
a) Write a linear function giving the value of the equipment for the 8 years it will be in use.
b) Find the value of the televisions after 4 years.
3) A real estate office manages an apartment complex with 1000 units. When the rent is $900 per month, all 1000 units will be occupied.
However, when the rent is $1100 per month only 900 units will be occupied. Assume the relationship between the monthly rent and the
number of units occupied is linear.
a) Write the equation giving the demand y in terms of the price p
Hint: Create points in the form (rent, number units rented)
b) Use the equation to predict the number of units occupied when the price is $1400. (round to the nearest unit if needed)
c) What price would cause 700 units to be occupied?
4) h(x) = -(x+3)2 + 2
a) describe the transformation as compared to the function f(x) = x2, specifically state if the graph is shifted left, right, up, down and if
any reflection has occurred
b) make a table of values and sketch a graph
c) state the domain and range of the function
d) state the intervals where the function in increasing and decreasing
5) m(x) = x2 – 6x + 5
a) Use completing the square to rewrite the problem in standard form
b) Sketch a graph, make sure to label the vertex. You may use your calculator, instead of making a table of values to create your graph
6) When a ball is thrown straight upward into the air, the equation h= -8t2 + 20t gives the height (h) in feet that the ball is above the
ground t seconds after it is thrown. (window x min 0 x max 10 y min 0 y max 20)
a) How long does it take for the ball to hit the ground?
b) When does the ball reach its maximum height? (expect a fraction answer)
c) What is the maximum height of the ball?
7) A baseball player hits a ball. The following data represents the height of the ball at different times.
Time (in
seconds)
height
1
2
4
5
12
19
27
28
a) Use a graphing calculator quadratic function to the data. (round each number to 2 decimals)
(window x min 0 x max 10 y min 0 y max 50)
b) Use the equation to find the maximum height of the ball. (round to 2 decimals)
Answers: 1a) x = -4 1b) x < -3 1c) x = -26/7 1d) x < -26/7 2a) v(t) = -3000t + 25000 or y = -3000x + 25000 2b) $13,000
3a) y = (-1/2)p + 1450 or y = -.5p + 740 3b) 750 units 3c) $1,500 4a) reflect x-axis, left 3 up 2 4b) no room for this show me
4c) domain (−∞, ∞) range (−∞, 2] 4d) increasing (−∞, −3) decreasing (−3, ∞) 5a) (x-3)2 - 4 5b) again no room, show me your graph
6a) 2.5 seconds 6b) 12.5 seconds 6c) 12.5 feet 7a) y = -7x2 +42.7x – 33.6 or h(t) = -t2 +10t+3 7b) 3.05 seconds 7c) 28 feet