Quadratic Forms
y  ax 2  bx  c
Names
Standard
form
Vertex
form
y  ax  h   k
2
y  ax  r1 x  r2 
Factor
form
If a  0 then concave up 
If a  0 then concave down 
Used best for
y-intercept (0, c)
doesn’t help with graphing by hand
vertex ( h, k ) line of symmetry x  h
used to find x-intercepts
y intercept ah 2  k
finding max and min values: k
helps for graphing by hand
x-intercepts r1 ,0 r2 ,0
y-intercept a(r1 )(r2 )  c
because of symmetry
r r
h  x coordinate of vertex = 1 2
2
Helps for graphing by hand (don’t know min or
max value)
Vertex is a min and k is the minimum value
Vertex is a max and k is the maximum value
1. For each of the following
State the vertex, State the y-intercept,
Indicate if there is a maximum or minimum value and give the value
Sketch the function without your calculator; include vertex and y-intercept
From the sketch how many x-intercepts are there?
a. f ( x)  ( x  3) 2  5
b. g  x    x  2   4
2
2
 1  13
c. ht   7 t   
4
 2
2
d. s t   4t  6 
2. Write a quadratic equation for the following scenarios
a) A ball is tossed in the air from a height of 5 feet and the following data is
recorded
Time (seconds)
Height (feet)
0
5
0.25
8
0.5
9
0.75
8
b) The vertex is (-5,100) and the y-intercept is 0
c) The x-intercepts are -2 and 7 and the y-intercept is 6
1.0
5
1.25
0
3. Solve each quadratic equation algebraically:
You should be using different techniques to solve these problems to be efficient.
a. 2x  5  6  0
1
2
b. t  1  3  0
3
1
c.  m  2 m  1  5  0
2
2
d.  p  2  16 p
2
e. 40  0.025 2  0
f. 3 y 2  21y  18
g. Solve for x : a x  h   k  0
2
4. In winter with heavy snow falls, the animals in Yellowstone cannot freely roam to find
food, so hay is dropped for them from helicopters. A helicopter 180 feet above the
ground drops a bale of hay while the helicopter itself is raising 32 feet per second.
The height, s (t ) (in feet) of the bale as a function of time is given by
s(t )  16t 2  32t  180 where t is the time since drop off (in seconds).
a)
b)
c)
d)
What is the height of the bale 2 seconds after it is released?
How many seconds will it take the bale to reach the ground?
When does the bale reach a maximum height? What is the maximum height?
Sketch a graph of the height of the bale as a function of time. Explain the shape
of the graph.
5. A motorcycle stunt rider jumped across Snake River. The path of his motorcycle was
given approximately by H  0.005 x 2  2.39 x  600 where H was measured in feet
above the river and x was the distance from his launch ramp.
a) What was the rider’s maximum height above the river?
b) How far (horizontal distance) was the rider from the ramp when he reached his
maximum height?
c) How high above the river was the launch ramp?
6. The revenue (in dollars) earned by producing n items can be modeled by the function
100

n
0  n  350

7
R ( n)  
2 2

n  240n  54,500 350  n  895
 10
a) Find the revenue earned when 200 items are produced
b) Find the revenue when 400 items are produced
c) What is the largest possible revenue? How many items must be produced to
maximize the revenue?