Mr. Stump Algebra II W/ Trig
Name:____________________
Quadratics Practice Test
Topics, skills, and concepts to know:
 Factoring quadratics
 Finding a quadratic equation from given information:
o Vertex and one other point
o x-intercepts and one other point
o Any three points
 Graphing quadratic functions in normal, vertex, and factored form
 Changing from one form to another:
o Factored to normal (standard)
o Vertex to normal
o Normal to vertex (you need to complete the square)
o Normal to factored (you need to factor)
 Find the vertex of a quadratic function (you should know lots of ways!)
 Solving a quadratic equation (i.e. find the x-intercepts)
o Calculator
o Graphing
o Factoring (monic and nonmonic)
o Completing the square
o Quadratic formula
 Solving for the exact answers (in simplified radical form) of a quadratic function
 Deriving the quadratic formula
Note: A major focus of this unit was learning multiple ways to solve problems. Part of your learning included knowing
efficient solving techniques given certain situations. These questions are not intended to be sample test questions, nor
should you assume that the material covered in these questions covers all the topics that you may be tested on.
Review Questions
These are application problems in which the vertex is an important point. Each problem specifies one or more methods for
finding the vertex, so you get to practice all the methods.
1.
The Gateway Arch is the centerpiece of the St. Louis skyline. Its shape is a parabola
Suppose the quadratic function describing this shape is
1
x 2  6 x , where x and F(x) are in feet.
F(x) =  70
a. Find the zeros of F(x).
b. Use the zeros to find the coordinates of the vertex & the line of
symmetry.
c. Sketch a graph of F(x). Label the zeros and the vertex with their
coordinates. **SKIP**
d. What is the appropriate interval of x-values for describing the Arch?
1
x 2  6 x on your calculator (you will need to set an appropriate
e. Check your work so far by graphing F(x) =  70
window), then finding the maximum on your calculator.
f.
Visitors to the Arch can take a tram ride up to the top of the monument. How high above the ground is this?
2. You are selling cookies for a school club at a bake sale. Naturally, the higher you set the selling price, the fewer
cookies you will sell. Suppose you will sell (500 – 4x) number of cookies at (x) cents per cookie. Let R(x) represent
the total amount of money made from the cookie sale. Part A-D will ask you to determine the same information
using different methods.
a. Write a function formula for R(x).
b. Find the zeros of R(x), then use the zeros to find the coordinates of the vertex.
c. Rewrite the formula for R(x) in standard form, then use the conversions to find the coordinates of the vertex.
d. Rewrite the formula for R(x) in vertex form, and use this to identify the vertex.
e. On a graphing calculator, graph R(x) and find its vertex.
f.
You should have gotten the same x and y coordinates for the vertex from the first four parts of this problem. Do
all your answers agree?
g. What do the coordinates of the vertex tell you about the cookie sale?
3. a. Solve –2x2 + 4x + 6 = 0.
b. Using conversions, put y = –2x2 + 4x + 6 into vertex form and find the vertex.
c. Compare the calculations used for parts a and b. What steps were the same? What steps were different?
4. Suppose that (x – k) is one of the factors of a quadratic function f(x). What else does this tell you about f(x) and its
graph? What can you not discern from this information? Be specific.
5. Give an example of a quadratic whose zeros fit each description, or explain why you can’t.
a. only one real zero
b.
two real zeros
6. Find a quadratic function whose zeros would be
1
2
c.
three real zeros
& –3, and a y-intercept of 6. Give your answer in standard form.
7. Solve for x by using the conversions: 0  2x 2  kx  m
8. a. If you know the x-intercepts of a parabola, can you determine the vertex? Explain.

b. If you know the x-intercepts of a parabola, can you determine the line of symmetry? Explain.
9. For a quadratic function y = ax2 + bx + c, under what circumstances would c represent both the y-intercept and the ycoordinate of the vertex?
10. Write an equation for a parabola that rises more steeply than y = x2 and has its vertex at (2, 5).
11. We studied three equation forms for quadratic functions: standard form, factored form, and vertex form. Using these
three forms, write a set of three equations that all represent the same parabola.
12. Write a quadratic equation that has 2 distinct solutions that are:
a. both integers
b. both irrational
13. Solve by factoring:

14.

2
a. 2x  7x  3  0
2
b. 5x  17x  6
2
c. 4 x  9  12x
2
d. 7x 18x  11  0
2
e. 9x  30x  25  0
f. 4x 2  6x 

9
4
0

Suppose the equation x 15x  40  0 has roots  and  . Without solving the equation, find a

.
quadratic equation with the following roots: 2 and 2
2





2
Answers
x  0 x  420
1. a.
b. V (210,630)
d. 0  x  420
e.
window 0  x  420 0  y  630
f.
630 feet
2. a. R( x)  x(500  4 x)
b-e. V(62.5,15625)
f.
62.5 cents, make $15,625
 3.
a.


4.
5.

6.
2x 2  4 x  6  0
x 2  2x  3  0
(x 1)2  4  0
b.
y  2x 2  4 x  6
y  2(x 2  2x 1 1)  6
y  2(x 1) 2 1 6

y  2(x 1) 2  8
(x 1 2)(x 1 2)  0
x  1 , x  3
Vertex (1,8)

of the graph (and a solution, zero, and root). You
 If (x  k) is a factor, then x  k is an x-intercept

cannot discern if this is the only factor or if there is one other factor. You also don’t know the leading

 coefficient, a.



Possible answers:
2

y  (x  2)
a.
y  (x  2)(x  3)
b.
c.
Not possible. A quadratic has at most two real zeros.
y  a(x  12 )(x  3)
a.
2x 2  kx  m  0
7.

x 2  k2 x  m2  0
6  a(0  12 )(0  3)

k2
k2
6   32 a
x 2  k2 x  16
 16
 m2  0
2
2
a  4

x  4k   16k  m2  0

y  4(x  12 )(x  3)


2
2
8m
8m
y  4(x 2  52 x  32 )
x  4k  k 16
x  4k  k 16
0

2

y  4 x 10x  6
2

x  k  k4  8m



8.
9.
10.
11.
12.
13.




 a.
Average the two x-intercepts to find the
 x-coordinate of the vertex.
 b.
Average the two x-intercepts to find the line of symmetry.

When x  b
2a  0, so when b = 0
Example: y  2(x  2) 2  5
Factored: y  (x  2)(x  4) Normal: y  x 2  6x  8
Vertex: y  (x  3) 2 1
y  (x  3)(x  4) ,
x  3,4
 a.
2
y  x  3,
x  3
b.

1
x


,3
x   25 ,3
a.

b.

2
3
11
x2
x  7 ,1
d.
 c.

5
x  
x  43
f.
 e.
3
14.  x 2  30x  160  0





3