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QUADRATIC FUNCTIONS
PROTOTYPE:
The leading coefficient a ≠ 0 is called the shape parameter.
SHAPE VERTEX FORMULA
One can write any quadratic function (1) as
(ShapeVertex Formula)
where and
EXAMPLE 1.
To derive the ShapeVertex Formula for f (x) we first identify the coeffi
cients:
a = 2, b = 8, c = 1.
With these identifications we have:
so the ShapeVertex 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 ShapeVertex
Formula.
We illustrate this fact with Example 1
above. In that example we started with
the function and we
found the ShapeVertex 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 ShapeVertex 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 yintercept f(0).
Step 2. Find the xintercept(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
13.)
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 yintercept is
Step 2. The xintercept(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 ShapeVertex 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 hnumber. So the
answer is
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