Reporting Category 3: Properties of Quadratic Functions
Algebra 2: RC 3
2A.6B (R)
Marco was observing the path of a hot air balloon as it rose slightly
and then descended. He recorded the balloon’s height, y, in feet,
every minute, x, for 5 minutes. His observations are recorded in the
table below.
1.
Graph the data points.
2.
Find a quadratic equation to help Marco model this situation.
3.
When would the balloon be 174 feet above the ground?
4.
What was the height of the balloon when Marco began collecting his
data?
5.
If the balloon’s path is modeled by the equation in #2, how many
minutes after Marco sees it, will it touch the ground?
6.
How many minutes had the balloon been in the air before Marco began
recording data?
Algebra 2
STAAR Page 41
Reporting Category 3: Properties of Quadratic Functions
Algebra 2: RC 3
2A.6C (S)
1.
2.
3.
4.
5.
Which of the following quadratic functions does not have zeros of -15
and 6?
A.
f(x)=
x²+3x-30
B.
f(x)= -x²-9x+90
C.
f(x)=-
x²-6x+60
D.
f(x)= -x²-9x-90
Which polynomial function has a zero 4+i?
F.
g(x)= x²+8x+17
H.
g(x)= x²-8x-17
G.
g(x)=x²-8x+17
J.
g(x)= x²+8x-17
Find the polynomial function with zeros ±2√ .
A.
h(x)=x²-8
C.
h(x)=x²+8
B.
h(x)=x²-8x+8
D.
h(x)=x²+4x-8
If a quadratic equation is graphed and has x-intercepts at -8 and 6,
what is the quadratic function that corresponds with this situation?
F.
f(x)=(x+1)²+49
H.
f(x)=(x+1)²-49
G.
f(x)=(x-1)²+49
J.
f(x)=(x-1)²-49
p(x) has roots
and - . Find p(x).
A.
p(x)=12x²+5x+2
C.
p(x)=12x²-5x+2
B.
p(x)=12x²-5x-2
D.
p(x)=12x²+5x-2
Algebra 2
STAAR Page 42
Reporting Category 3: Properties of Quadratic Functions
Algebra 2: RC 3
2A.8A (R)
1.
2.
3.
The base of a triangle is 3 inches less than twice its height. If the area
of the triangle is 126 square inches, which of the following equations
can be used to find h, the height of the triangle in inches?
A.
2h²-3h+63=0
C.
2h²-3h+252=0
B.
2h²-3h-63=0
D.
2h²-3h-252=0
The product of two consecutive even integers is 728. If the greater of
the two integers is n+6, write a quadratic equation that can be used to
find both integers.
F.
n²+14n-680=0
H.
n²+10n-704=0
G.
n²+13n-686=0
J.
n²+11n-698=0
Write a quadratic function to model the function graphed below.
A.
y=2x²+4x+300
C.
y= -x²+300
B.
y= -2x²+4x+300
D.
y=-(x-300)²+13
Algebra 2
STAAR Page 43
Reporting Category 3: Properties of Quadratic Functions
Algebra 2: RC 3
2A.8A (R)
4.
The area of a parallelogram is 143 square centimeters. The height of
the parallelogram is one more than twice a number. The base is two
less than three times the same number. Which of the following
equations can be used to find the number, x?
F.
6x²+x-145=0
H.
-6x²-x-141=0
G.
-6x²+x-141=0
J.
6x²-x-145=0
5.
The height of an isosceles trapezoid is three more than a number, x.
The shorter base is one less than three times the same number. The
other base is four more than three times the number. If the area of
the isosceles trapezoid is 148.5 square feet, write a quadratic equation
that can be solved to find x.
6.
The largest flag flown from a flagpole is a Brazilian national flag
measuring 100 meters by 70 meters in Brasilia, the capital of Brazil.
Suppose a company wants to make a flag and increases the length
and width of the flag by x feet. The area of the flag cannot exceed
10,000 square meters. Write an inequality to determine by how much
the length and width can be increased.
Algebra 2
STAAR Page 44
Reporting Category 3: Properties of Quadratic Functions
Algebra 2: RC 3
2A.8B (S)
For #1-3, find the number of distinct roots and the nature of the roots (real,
imaginary, rational, and irrational) for each quadratic equation.
1.
A car is traveling at 26 meters per second and accelerating at
-13 m/s². After traveling 26 m, the driver brings the car to a complete
stop. The equation 26=26t -
t², where t is the time it takes to stop,
can be used to represent this situation.
2.
A person’s blood pressure depends on her age. For women, normal
systolic blood pressure is given by the formula P=0.01A²+0.05A+107,
where P is the normal blood pressure in millimeters of mercury and A
is the age.
3.
Bryan Starr is a 17-year old high school senior from Upper Arlington,
Ohio who started his own lawn service when he was 12. His business
has grown substantially and he has been able to put away money for
college, buy a truck, and invest in new lawn equipment to keep up
with the growing demand for his services. Suppose his weekly
revenue R can be represented by the formula R= -p²+50p-125, where
p is the average price he charges for each lawn.
For #4-7, use the quadratic functions or equations in #1-3 to answer the
following questions.
4.
How long did it take the person in Question #1 to stop the car?
5.
Find the approximate age of a woman whose blood pressure is
121 mm Hg.
6.
Explain how Bryan could earn $400 each week.
Algebra 2
STAAR Page 45
Reporting Category 3: Properties of Quadratic Functions
Algebra 2: RC 3
2A.8B (S)
7.
Is there a price he could charge that would make his weekly revenue
$600?
8.
Choose the false statement regarding the value of the discriminant of
a quadratic equation.
F.
If it is a perfect square, two real rational roots exist.
G.
If it is zero, infinitely many roots exist.
H.
If it is negative, two complex roots exist.
J.
If it is not a perfect square, two real rational roots exist.
Algebra 2: RC 3
2A.8C (S)
1.
Use the table of values for f(b) to find the roots of f(b).
b
-1 0 1 2 3 4 5 6 7 8
f(b) 3 0 -2 -3 -3 -2 0 3 7 12
2.
Use the table of values for f(x) to find the zeros of f(x).
x
-9 -6 -3
0 3 6
f(x) 36 0 -18 -18 0 36
Algebra 2
STAAR Page 46
Reporting Category 3: Properties of Quadratic Functions
3.
Use the graph below to solve 2x²+2x-4=0.
4.
Although no one has found a formula that generates prime numbers,
18th century Swiss mathematician Leonhard Euler discovered that
f(x)=x²+x+17 produces prime numbers up to a certain point. Based
on the graph of f(x), what are the roots of the function?
Algebra 2: RC 3
2A.8C (S)
5.
In 1940, I.M. Chisov of the USSR bailed out of an airplane without a
parachute at 21,980 feet and survived. The function
h(t)= -16t²+21,980 describes the relationship between height h(t) in
feet and the time t in seconds. Use the graph of the function to
determine the number of seconds he fell before reaching the ground.
Algebra 2
STAAR Page 47
Reporting Category 3: Properties of Quadratic Functions
Algebra 2: RC 3
2A.8D (R)
1.
The table below shows ordered pairs that satisfy the quadratic function
f.
Based on the table, a solution to the equation f(x)=0 is found in which
interval?
2.
A.
-2<x<-1
C.
1<x<2
B.
-1<x<1
D.
3<x<5
The profit P, in dollars, for a company is modeled by the function
P(x)= -75x²+15,000x, where x is the number of items produced. For
which values of x will the company lose money?
F.
x<2
H.
10≤x<20
G.
2<x≤10
J.
x>20
Algebra 2
STAAR Page 48
Reporting Category 3: Properties of Quadratic Functions
Algebra 2: RC 3
2A.8D (R)
3.
Which graph shows the solution set for the following inequality?
x²>x+12
4.
The distance d that an object travels can be calculated when the initial
speed , elapsed time t, and the rate of constant acceleration a are
known. A formula that relates these factors is d(t)=
t+ at². If a
motorcycle has an initial speed of 30 m/s and a constant acceleration
of 6 m/s², how much time will it take to travel 200 m?
5.
A ball is thrown straight up with an initial velocity of 56 ft/s. The
height of the ball t seconds after it is thrown is given by the formula
h(t)=56t-16t². After how many seconds will the ball return to the
ground?
6.
The Apollo 11 spacecraft propelled the first men to the moon and
contained three stages of rockets. The first stage dropped off 2
minutes 40 seconds after takeoff and the second stage ignited. The
initial velocity of the second stage was 2760 m/s with a constant
acceleration of 200 m/s². How long did it take the second stage to
travel 7040 m?
Algebra 2
STAAR Page 49
Reporting Category 3: Properties of Quadratic Functions
Algebra 2: RC 3
2A.8D (R)
7.
The Empire State Building is 1250 feet tall. If an object is thrown
upward from the top of the building at an initial velocity of 35 ft/s, its
height t seconds after it is thrown is given by the function
h(t)= -16t²+35t+1250. How long will it be before the object hits the
ground?
Algebra 2
STAAR Page 50