Quadratic
Functions

Functions like _{} are examples of
quadratic functions. Notice the presence
of a quadratic polynomial of the form _{}, *A **¹ *0 in each
of the examples of quadratic functions that are presented here.

More generally, a function of the
form _{}, where *A, B, C* are constants, *A* ¹ 0,
is called a quadratic function. The
graph of a quadratic function is called a ** parabola**. If

*y*
*y* **V** If *A* > 0, the
vertex is the *absolute minimum*

· of the function, since the
graph opens up and

no point on the graph of *f *is below the vertex.

*x* *x* If *A* < 0, the vertex is the *absolute maximum*

· of
the function, since the graph opens down

**V**
and there no point on the graph of *f *is above

the vertex.

**Figure 5 Figure 6**

The vertex is a point on the *xy*-plane, so it has the form **V = ( x, y)**,
where the

A
parabola happens to be one of a group of four curves known as ** conics**.
Conics have various applications and form the trajectory of many objects
in the Universe. The WebSite

**Examples
of Analysis of a Quadratic Function:**

1)
For the quadratic function _{},

a)
Explain why the graph of this
function is concave down.

*Answer*: Since the leading coefficient *A* = -3 is negative, the graph of *f* opens down and is
concave down.

b)
Find the components of the
vertex:

*Solution*: *x* component: _{}; *y*
component: _{} The vertex is the
point **V = (2, 3)**.

c)
Determine whether the vertex
is the absolute maximum or the absolute minimum.

*Answer*: Since the graph of *f* is concave down,
the vertex (2, 3) is the absolute maximum of the function.

d)
Write the equation of the
axis of symmetry.

*Solution: *The
equation of the axis of symmetry is *x*
= 2

f)
Sketch the graph of *f* and display all the information from parts
a)-e).

e)
Find the intercepts:
*y*

*Solution*: If *x*
= 0, then *y* = *f*(0) = -9 is the *y*-intercept. 4 (2, 3)

If *y* =
0, we must solve the equation _{} for *x*.

Factor
completely on the left side to obtain: -3(*x* - 1)(*x* - 3) = 0.
*x*

So, *x* =
1, *x* = 3 are the *x*-intercepts.
1 2 3

The complete
list of intercepts includes (0, -9), (1, 0), and (3, 0).

- 4

*
**x=2*

2)
For the quadratic function _{},

a)
Explain why the graph of this
function is concave up.

*Answer*: Since the leading coefficient *A* = 2
is positive, the graph of *f* opens up and is concave up.

b)
Find the components of the
vertex:

*Solution*: *x* component: _{}; *y*
component: _{} The vertex is point **V = (1, ****-18)**.

c)
Determine whether the vertex
is the absolute maximum or the absolute minimum.

*Answer*: Since the graph of *f* is concave up,
the vertex **(1, ****-18)** is the absolute
minimum of the function.

d)
Write the equation of the
axis of symmetry.

*Solution: *The
equation of the axis of symmetry is *x*
= 1

f)
Sketch the graph of *f* and display all the information from parts
a)-e).

e)
Find the intercepts:
*y*

*Solution*: If *x*
= 0, then *y* = *f*(0) = -16 is the *y*-intercept. *x* = 1

If *y* =
0, we must solve the equation _{} for *x*.

Factor
completely on the left side to obtain:
2(*x* -
4)(*x* + 2) = 0. *x*

So, *x* =
-2, *x* = 4 are the *x*-intercepts. -2 1 4

The complete
list of intercepts includes (0, -16), (-2,
0), and (4, 0).

*
*-16* * * *

(1, -18)

3) Suppose that
the total cost (in dollars) of manufacturing *q* units of a product is
given by _{}.

a) Find the minimizer level of production.

*Solution*: The function *C* is concave up since A
= 2 > 0. Therefore, the vertex of
the parabola is the absolute minimum and the minimizer level of production is _{} units of
production.

b) Find the minimum cost.

*Solution*: The minimum cost is _{}

4) The demand equation for a particular product
is given by _{}, where *p *is the selling price (in dollars) per unit,
0 £
*p* £
1500, and *x* is the number of units sold.

a) Find
the quantity *x* that maximizes the total revenue. That is, find the maximizer of the revenue *R.*

*Solution: *Recall that total revenue is given by *R = px*. In this case, _{}.

Thus the revenue function is _{}

Notice that in this quadratic function, A = _{} < 0. So, the function is concave down and
therefore, the vertex is the absolute maximum.

Maximizer: _{} units sold.

b) Find the maximum revenue.

*Solution: *The
maximum revenue is _{}

c) Find
the unit price *p* that should be charged in order for the total revenue
to be maximized.

*Solution: *_{}* per unit.*

Check
that *R = px = (750)(150) =
$112,500 (this is the maximum revenue)*

5) Suppose that a farmer purchases 5000 feet of fencing and plans to fence a rectangular field with the 5000 feet of fencing. If he does not know College Algebra and chooses at random a rectangle with dimensions 1000 feet by 1500 feet, then the enclosed rectangle has area 1,500,000 squared feet. Suppose a friend advises him to use a rectangle with dimensions 1100 feet by 1400 feet because with these, the total enclosed area is 1,540,000 squared feet. Further, suppose that he later discovers that a rectangle with dimensions 1200 feet by 1300 feet still uses the entire 5000 feet of fencing and the enclosed area is bigger than the previous two (area = 1,560,000 squared feet). Clearly, by changing the dimensions of the rectangle, we obtain a rectangle with perimeter equal to 5000 feet, but the area enclosed changes, depending on the choice of the dimensions. The question is “what dimensions of the rectangle use the whole 5000 feet of fencing and produce the maximum possible enclosed area? What is the maximum possible enclosed area? We can answer these questions using quadratic functions.

*Solution:* In the figure below, *x* represents the
length of the rectangle and *y* represents the width of the
rectangle. Thus, 2*x* + 2*y*
= 5000 feet

is
the perimeter and *A* = *xy*
is the area of the rectangle.
Notice that *x* + *y* = 2500 feet can be

obtained
from the perimeter. So, *y* = -*x* + 2500, with 0 £ *x*
£
2500, so that the width does

Area
= *xy* Width = *y* not become negative.
Replace *y* into the area to get _{}. Therefore, the

area
function in terms of the length *x* is
_{}.

Length
= *x* Notice
that in this function, A = -1 < 0 and therefore the function is concave
down. So, the

maximizer of the area
*A* is _{} feet. Therefore, *y* = 1250 feet also.

The
maximum possible enclosed area is _{} squared feet. The requested dimensions are 1250 feet by
1250 feet and the maximum possible enclosed area is 1,562,500 squared feet.

6) Suppose that a farmer has 5000 feet of fencing and wants to use it for fencing a rectangular field for which one side is the wall of the farmer’s house. In this case the side along the wall does not require fencing. What dimensions yield maximum rectangular area and what is the maximum enclosed area?

*Solution:* ** Wall**
First construct a quadratic function to represent the area of the
rectangular plot.

In Figure
6, let *y* represent the length of
the side parallel to the wall of the house and let *x*

*x x*
represent the length of each of the other two sides.

Total amount of fencing:
2*x* + *y* = 5000 feet,
so *y* = 5000 - 2*x*, 0 £ *x* £ 2500.

*y * Total Area Enclosed: *A* = *xy* = *x*(5000 - 2*x*) squared feet.

**Figure 6** Quadratic Function: _{}.

Notice that this function is concave down (why?), so the vertex is the
absolute maximum. The maximizer is _{} feet. (Notice that: 0 £ 1250 £
2500).

Therefore, *y* = 5000 - 2(1250) =
2500 feet.

The maximum area enclosed is _{} squared feet.

The maximum rectangular area that the farmer can enclose with the 5000
feet of fencing is **3,125,000 squared feet.**

7) The manager of a 100-unit apartment complex knows from experience that all units will be occupied if the rent is $800 per month. A market survey suggests that, on the average, one more unit will become vacant for each $10 increase in the monthly rent. What monthly rent should the manager charge in order to maximize monthly revenue? Find the maximum monthly revenue.

SOLUTION:

Monthly rent (in
dollars): *p* = 800 + 10*x*,
where *x* = number of $10 increases).

Number of occupied
units: *q* = 100 – *x.*

Monthly revenue: _{} dollars.

Maximizer of *R*: _{}. This is 10 ten-dollar increases.

Therefore, an increase of $100 will maximize monthly revenue.

Maximum monthly
revenue: *R *(10) = -10(100)
+ 200(10) + 80000 = **$ 81,000.**

Also: Optimal monthly rent is *p* = 800 + 10(10) = **$ 900 per month.**

The optimal
number of rented units per month is *q*
= 100 – 10 = **90 units.**

The maximum
monthly revenue is *R* = *pq* = (900)(90) = **$ 81,000.**

8) An object is thrown straight up from the ground with an initial
velocity of 160 feet per second. Its
height *h* (in feet) from the ground at any time *t* (in

seconds), *t **³ *0, is given by _{}.

a) When will the object reach maximum height from the ground?

*Solution:*
Notice that since A = -16 < 0, the function is concave down and therefore,
the vertex of the parabola is the absolute maximum.

Maximizer: _{} seconds. The object will reach maximum height after 5
seconds.

b) Find the maximum height of the object.

*Solution:* The maximum height is _{} feet above the
ground.

**NOTE:** The quadratic function _{}, *A* ¹ 0,
can be transformed into _{}, where *A *¹ 0, and (*h, k*) is the vertex where _{} and _{}. We can derive
this as follows:

Completing the square: _{}.

If we graph _{}, _{}, starting with the graph of _{} and using
transformations, we will see that the extremum of the resulting parabola is (*h,
k*), where _{} and _{}. Note: The standard form of the equation of a
parabola with vertex (*h, k*) and vertical axis of symmetry *x = h*
is _{}.

**Example:**

This is the quadratic function of example 1, above: _{}.

Use the method of completing the square to find the vertex of the parabola that it represents.

*Solution:* * *_{}

Notice
that the given function has been transformed to _{}.

Clearly the vertex is
the point (2, 3), which agrees with example 1, above.

* *

For each exercise from 1) to 8), do the following:

a) Determine whether the graph is concave up or concave down and justify your answer based on the value of the leading coefficient A.

b) Find the components of the vertex and state the vertex of the parabola.

c)
Determine whether the vertex
is the absolute maximum or the absolute minimum of the function. Justify your answer.

d) Write the equation of the axis of symmetry.

e) Find the intercepts (if any).

f) Sketch the graph and state the range of the function.

1) _{}; 2) _{}; 3) _{};
4) _{};

5) _{};
6) _{}; 7) _{} 8) _{}

In exercises 9) and 10), use the
method of completing the square to find the vertex of the parabola. Find the intercepts (if any), the equation
of the axis of symmetry, and sketch the graph using transformations on the
graph of *y = x ^{2}*.

9) *f(x) = **-*3*x ^{2}* + 18

11) Suppose
that the total cost (in dollars) of manufacturing *q* units of a product
is given by _{}.

a) Find the minimizer level of production. b) Find the minimum cost.

12) The demand equation for a particular product
is given by _{}, where *p *is the selling price (in dollars) per unit,
0 £
*p* £
8000, and *x* is the number of units sold.

a) Find
the quantity *x* that maximizes the total revenue. That is, find the maximizer of the revenue *R.*

b) Find the maximum revenue.

c) Find
the unit price *p* that should be charged in order for the total revenue
to be maximized.

13) Suppose that a farmer purchases 2000 yards of fencing and plans to use it to fence a rectangular field. What dimensions of the rectangle use the whole 2000 feet of fencing and produce the maximum possible enclosed area? What is the maximum possible enclosed area?

14) Suppose that a farmer has 3000 yards of fencing and wants to use it for fencing a rectangular field for which one side is already fenced by a neighbor’s plot, so that side does not require fencing. What dimensions of the rectangle yield maximum rectangular area and what is the maximum enclosed area?

15) A distributor of computer monitors has found that when she sells 200 monitors per week when the price is $800 apiece. She has found also that she sells 5 more monitors per week for every $10 decrease in the price of each monitor. What price should she charge per monitor in order to maximize weekly revenue? Find the maximum weekly revenue.

16) An object is thrown straight up from the ground with an initial
velocity of 80 feet per second. Its
height *h* (in feet) from the ground at any time *t* (in

seconds), *t **³ *0, is given by _{}.

a) When will the object reach maximum height from the ground?

b) Find the maximum height of the object.

The
total cost, *C,* of obtaining *q* units of a particular product can
be expressed linearly by the function *C*(*q*) = *cq* + *f,*
(in dollars), where *c* represents the cost per unit and *f*
represents the fixed cost (setup, overhead, etc.). The total revenue, *R,* from the sale of *q* units of
the product can be expressed by the function *R*(*q*) = *pq*,
(in dollars), where *p* represents the unit selling price. The total profit *P* after obtaining
and selling the *q* units of the product is the difference of the total
revenue minus the total cost and is written *P*(*q*) = *R*(*q*)
– *C*(*q*). The break-even
points, *BEP*, are the values of *q* that make the total revenue
equal to the total cost. Therefore, the
*BEP* are the same values of *q* for which *P*(*q*) = 0,
(no gain and no loss).

The owner of a retail store can obtain a particular type of digital cameras from a manufacturer at a cost of $150 apiece. The retailer has noticed that if she sells the cameras at the price of $350 apiece, consumers buy 200 cameras in one month. The retailer’s associated fixed cost amounts to $125 in one month. In an attempt to increase or stimulate sales, the retailer is planning to lower the unit selling price and estimates that for each $5 reduction in the price, 16 more cameras will be sold during one month.

a)
Explain why the *BEP* are the same values of *q* for which *P*(*q*)
= 0.

b) Express the monthly total cost of the digital cameras as
a function of *q*.

c)
Find the linear demand
function *p* = *mq* + *b*.
(This is analogous to *y = mx + b*, the slope-intercept form of the
equation of a straight line)

d)
Express the monthly total
revenue as a function of *q*.

e)
Express the retailer
monthly profit as a function of *q*.

f)
Find the optimal quantity *q*
that maximizes the retailer monthly profit.

g)
Find the maximum monthly
profit.

h)
Find the optimal selling
price per camera.

i)
Find the break-even points.

j)
Sketch the revenue, cost,
and profit functions in the same rectangular system. Display the maximum profit and the *BEP* clearly.

Prepared by: Carlos I. Gil