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Quadratic Functions
Standard Form of a Quadratic: 𝑓(𝑥) = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 where 𝑎 ≠ 0.
Vertex:
y-intercept:
Shape:
Opens: If 𝑎 > 0, _______ and if 𝑎 < 0, _______.
Axis of Symmetry:
Domain:
Range: If 𝑎 > 0, ____________and if 𝑎 < 0, ____________.
Maximum or minimum? If 𝑎 > 0, ____________________ and if 𝑎 < 0, ____________________.
Vertex Form of a Quadratic: 𝑓(𝑥) = 𝑎(𝑥 − ℎ)2 + 𝑘 where 𝑎 ≠ 0.
Vertex:
y-intercept:
Shape:
Opens: If 𝑎 > 0, _______ and if 𝑎 < 0, _______.
Axis of Symmetry:
Domain:
Range: If 𝑎 > 0, ____________and if 𝑎 < 0, ____________.
Maximum or minimum? If 𝑎 > 0, ____________________ and if 𝑎 < 0, ____________________.
1. Find all values of c for which ℎ(𝑥) = 𝑥 2 − 4𝑥 + 𝑐 has the following number of x-intercepts.
a. one x-intercept
b. two x-intercepts
c. no x-intercept
1
2. Given the quadratic function 𝑓(𝑥) = 2 (𝑥 + 1)2 − 2, find the vertex, y-intercept, any xintercepts, domain, and range. Graph the function on the given axes. Label at least 3 points on
the graph.
10
y
5
x
10
10
5
10
3. Given the quadratic function 𝑔(𝑥) = −2𝑥 2 − 8𝑥 − 11, find the vertex, y-intercept, any xintercepts, domain, and range. Graph the function on the given axes. Label at least 3 points on
the graph.
10
y
5
x
10
10
5
10
4. Find the equation of the given quadratic function in vertex form and standard form.
a.
y
8
(0, 6)
6
Vertex form: ____________________________
4
Standard form: __________________________
2
(6, 0)
(2, 0)
5
5
x
2
(4, -2)
4
6
8
b.
y
8
6
Vertex form: ____________________________
4
( )
7
,1
2
Standard form:___________________________
2
(- 4, 0)
5
5
(- 3, 0)
2
4
6
8
x
5. The product of two consecutive odd integers is 35. What are the integers?
6. A machine producing toy trucks has a production time of 10 hours before it stops each day. Its
production of toy trucks follows a model of 𝑇(𝑥) = −120𝑥 2 + 1200𝑥, where x is the production
time in hours and 𝑇(𝑥) is the number of toy trucks the machine produces. At what time each day
will the machine produce the maximum number of toy trucks if the company starts the machine
at 7:00am daily?
7. A ball thrown into the air by a child follows a parabolic path. The ball reaches a maximum
height of 20 feet when it is at a horizontal distance of 6 feet from the child. If the ball is 5 feet off
the ground when the child releases the ball.
a. Draw a picture representing the situation. Make sure to label what you know.
b. Determine the equation of the function representing the situation.
c. Determine how far from the child the ball will be when it hits the ground.
8. A farmer has 2400 m of fencing. He wants to fence a rectangular area that borders a river (no
fence is required on that side). Find a function that models the area of the fenced-in region. Find
the dimensions of the largest possible region. What is the largest possible area?
9. A rancher has 750 m of fencing. She wants to enclose a rectangular area and then divide it into
four pens with parallel fencing to one of the sides of the rectangular pen as shown below. Find a
function that models the total area of all four pens. Determine the largest possible total area.