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# Quadratic
Functions

A quadratic
function has the form f(x) = ax^{2} + bx + c , where a, b,
and c are real number constants and a ¹ 0. Its
domain is all real numbers. The constants a, b, and c are called
the coefficients of the quadratic function. The graph of a quadratic
function is a parabola opening up if the leading coefficient a >
0 or opening down if a < 0.

The
simplest quadratic functions are y = x^{2} and y = -x^{2}.
The graphs are shown below.

y = x^{2}
y = -x^{2}

To see how different values of
the coefficients affect the graph, experiment with the following interactive
example.

**Example**

The
function H(t) = -16t^{2} + 45t + 200 describes the height
in feet of a ball t seconds after it is thrown upward from the top
of a 200 foot high building. The graph below shows that the ball hits
the ground shortly after 5 seconds and reaches its maximum height
of approximately 230 feet after about 1.5 seconds. We will learn to
calculate these exact values later.

**Intercepts**

The
graph of a quadratic function will always have one y-intercept and
can have as many as two x-intercepts (recall that intercepts are points
where a graph crosses an axis). The y-intercept is the point
where the graph crosses the y-axis so it will be the value of the
function at x = 0. Since f(0) = c, the y-intercept is the point
(0,c). The x-intercept(s), if there are any, are the points
where the graph crosses the x-axis. They can be found by replacing
y with 0 and solving the resulting quadratic equation _{} for x. This equation
has one, two, or no solutions. When it has one solution, there
will be one x-intercept; when there are two solutions, there will
be two x-intercepts and when the equation has no solution, the graph
will have no x-intercepts. To see examples, expolore the interactive
example below.

**Example
**

Find
the x- and y- intercepts of the quadratic function

f(x) = 1.5x^{2} - 3.2x - 9.6.

### Solution

The y-intercept is found by replacing x with 0:

f(0) = 1.5(0)^{2} - 3.2(0) - 9.6 = -9.6.

Thus the y-intercept is the point (0, -9.6).

The x-intercepts are found by replacing y or f(x) with 0 and
solving for x:

0 = 1.5x^{2} - 3.2x - 9.6.

Using the quadratic formula, we get

_{}

The
solutions (rounded to two decimal places) are x = 3.81 and x = -1.68.
The x-intercepts are approximately the points (3.81, 0) and (-1.68,
0). A graph is shown below.

TRY
THESE PROBLEMS NOW

Decide whether each parabola
opens up or down and determine the coordinates of the x- and y-intercepts.

1.
y = .02x^{2} - 63.1x + 27

2.
P(t) = -3t^{2} + 27.5t + 13.1

**Vertex**

A
parabola which opens up has a lowest point and a parabola which opens
down has a highest point. The highest or lowest point on a parabola
is called the **vertex**. The parabola is symmetric about
a vertical line through its vertex, called the **axis of symmetry**.
The figure below shows a parabola opening up with vertex (0.75, 0.875)
and axis of symmetry x = 0.75.

**Vertex
Form of a Quadratic Function**

To find the vertex of a parabola, we will write the function
in the form

_{}. As an example, consider the function _{}. We first complete the square on the right side:

f(x) = 2(x^{2} - 4x) + 7
(factor out 2 from the terms 2x^{2}
- 8x)

= 2(x^{2} - 4x + 4) + 7 -
8 (complete the square of x^{2}
- 4x)

= 2(x - 2)^{2} - 1
(factor the
perfect square and simplify.)

Notice
that _{}for all values of x.
Thus f(x) = 2(x - 2)^{2} - 1 ³ -1 for all values
of x and the minimum value of the function is -1 when x = 2.
The point (2, -1) is the lowest point on the graph so it is the vertex
of the parabola. The vertical line x = 2 is the axis of symmetry.
See the graph below.

_{}

In general, _{} is called **vertex form** of a quadratic
function. When a quadratic function is written in vertex form,
we can easily determine the vertex *(h, k). *If the coefficient
*a > 0*, then the parabola opens upward and the vertex is
the lowest point on the parabola. We say that *k* is the minimum
value of the quadratic function. On the other hand, if the coefficient
*a < 0*, then the parabola opens downward and the vertex is
the highest point on the parabola. In this case, *k* is the maximum
value of the quadratic function. Explore the role of each coefficient
in the following interactive example.

**Example**

Write
the quadratic function _{}in vertex form. Determine the vertex and the maximum
or minimum value of the function.

**Solution**

We will complete the square to
write the function in vertex form:

The vertex form is _{}, so the vertex is *(3,
-11)*. Since *a < 0, *the parabola opens downward and the
vertex is the highest point. The function has a maximum value of 11.
Its graph is shown below.

_{}

Once we know the vertex of a parabola, we can determine the
**range** of the quadratic function. Consider the function, _{}. Previously we determined that the parabola
has a minimum value of -1, occurring when x = 2. Thus the range of
the quadratic function is {y½ y ³ -1}. As another example, lets
return to the function _{}in the above example. The graph of this function
is a parabola opening downward and the maximum value of the function
is 11. Therefore, the range of the quadratic function is _{}

**Finding
the Vertex Algebraically**

The
vertex of a quadratic function _{} can also be determined algebraically. We first
assume that the quadratic function has two x-intercepts. Then the
graph is a parabola that crosses the x-axis in two distinct points.
Since the parabola is symmetric with respect to a vertical line through
its vertex (the axis of symmetry) the x-coordinate of the vertex is
always halfway between the two x-intercepts. By the quadratic formula,
the two x-intercepts are

Notice that
the same number, _{}, is being added to and subtracted from _{}. It follows that the number
_{} is halfway between _{} This means that the x-coordinate of the vertex
is _{}. We can then find the y-coordinate of the vertex
by evaluating _{} Although we assumed that the quadratic function
had two x-interecpts when we derived our vertex formula, it also holds
in the other two cases, where the parabola has one or no x-intercepts.

**Example**

Find
the vertex of the quadratic function _{}. Use the vertex to determine the maximum or minimum
value of the function and find its range.

**Solution**

The vertex
formula gives _{} To find the second coordinate of the vertex,
we evaluate _{}The vertex of the parabola is (3, 53). Since *a
< 0*, the parabola opens downward and the vertex is the highest
point. This gives a maximum value of 53 and the range of the function
is _{} A graph of the function is shown below.

_{}

Recall the
function_{}, which describes the height in
feet of a ball t seconds after it is thrown upward from the top of
a 200 foot high building. We can now determine when the ball hits
the ground and the maximum height that it reaches, as well as the
time that it reaches that maximum height. When the ball hits the ground,
its height above ground will be zero. This gives the quadratic equation
_{}. Using the quadratic formula, we find that the
solutions are

_{}and _{}

(rounded to
two decimal places). Since the time cannot be negative, we see that
the ball strikes the ground after 5.21 seconds. The maximum height
of the ball will be given by the second coordinate of the vertex and
the time will be the first coordinate. Using the vertex formula we
find that _{} (rounded to tow decimal places). Next
we evaluate _{} This means that the ball reaches its maximum
height of 231.64 feet after 1.41 seconds.