# QUADRATIC FUNCTIONS

**PROTOTYPE:
**

The leading coefficient a ≠ 0 is called the

**shape parameter**.

**SHAPE- VERTEX FORMULA**

One can write any quadratic function (1) as

**(Shape-Vertex Formula)**

where and

**EXAMPLE 1.**

To derive the Shape-Vertex Formula for f (x) we first identify the coeffi-

cients:

a = 2, b = -8, c = -1.

With these identifications we have:

so the Shape-Vertex Formula for f(x) is:

**FACTS ABOUT THE GRAPH:
**

A. The graph has same shape as the graph of ax

^{2}, but shifted. The shift-

ing is determined by the numbers h and k that appear in the Shape-Vertex

Formula.

We illustrate this fact with Example 1

above. In that example we started with

the function and we

found the Shape-Vertex Formula to be

By the above Fact, we then know that the

graph of f(x) is the same as the graph of

y = 2x

^{2}, but shifted 2 units to the right,

and 3 units down.

The graph of f(x) is shown in red, while

the graph of y = 2x

^{2}is shown in blue.

B. The shape of the graph of f(x) = ax^{2} + bx + c is called
a parabola.

The parabola opens upward or downward, depending on the sign of the leading

coefficient a , as shown below.

**THE VERTEX.** The "tip" of the parabola, marked by V in the
above

pictures, is called the vertex. Its coordinates are the numbers (h; k), given in

the Shape-Vertex Formula. The vertical line through the vertex is an axis of

symmetry for the parabola.

The vertex is a "turning point" (a point where the graph changes direction ).

Moreover:

• if a > 0, then the vertex is a minimum point;

• if a < 0, then the vertex is a maximum point.

The intervals of monotonicity (where the function is increasing or decreas-

ing) are (-∞, h) and (h,∞).

**GRAPHING AND ANALYZING THE FUNCTION
**

Use the following steps when dealing with a quadratic function

f(x) = ax

^{2}+ bx + c:

**Step 1.**Find the y-intercept f(0).

**Step 2.**Find the x-intercept(s), by solving the equation

f(x) = 0.

**Step 3.**Find the coordinates of the vertex:

**Step 4.** Draw the graph. (Use the information from Steps
1-3.)

**Step 5.** Analyze the graph and extract information about the function.

- specify whether the vertex is a maximum or a minimum point;

- indicate the intervals where the function is increasing or decreasing.

**EXAMPLE.** Graph and analyze

Solution: **Step 1.** The y-intercept is

**Step 2.** The x-intercept(s) are found by solving the equation:

Using the Quadratic Formula, the solutions are

so there are two x- intercepts :
and

**Step 3. **We find the numbers h, k:

so the vertex is the point (-1, 4).

**Step 4.** The graph is shown on the

right.

**Step 5**. The vertex (-1, 4) is a maximimum point.

The function f(x) is:

• increasing on (-∞,-1);

• decreasing on (-1,∞).

**FINDING THE FUNCTION, GIVEN THE VERTEX
**

When the vertex of the graph is given, we proceed as follows.

**Step 1.**Replace h, k in the Shape-Vertex Forumula

so that we get a "preliminary" form of the function:

(Here it is understood that # mean concrete numbers.

**Step 2.**Replace x and y by the coordinates of the other point given, so

that now we would get something like :

Think of the above as an equation with

**a**as the unknown, ans solve for

**a**.

**Step 3.**Replace

**a**in the "preliminary" equation.

**EXAMPLE. **Find the quadratic function whose graph has
vertex (-1, 2)

and passes through the point (1, 10).

Solution: Here the vertex gives h = -1 and k = 2.

**Step 1.** The preliminary equation is

which is the same as

**Step 2.** We replace x = 1 and y = 10, and we get

which leads to the equation

10 = 4a + 2.

We obviously get **a = 2**.

**Step 3.** The function is then given by

**APPLIED PROBLEMS.**

The meaning of the vertex, as the maximum or minimum point for the

quadratic function, is often used to solve optimization problems.

**EXAMPLE.** The daily cost C of producing lamps at the ABC COmpany

is given by

where x is the number of units produced. How many lamps should be produced

in order to yield the minimum possible cost?

Solution: What we are dealing with here is a quadratic function

whose coefficients are and c = 900. What we need to

find is the value of x , for which f(x) takes the minimum value. Since a > 0,

we know that f(x) has a minimum point at the vertex. So what we need to

find is precisely the x -coordinate of the vertex, that is the h-number. So the

answer is

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