Section 1.Q Lecture Notes
MATH 141-501
Quadratic Functions
Many situations in business are
and are better described by
.
There are two ways to write a
.
1. Standard form
• General formula: f (x) =
• Example: f (x) =
2. Vertex form
• General formula: f (x) =
• Example: f (x) =
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Graphs of Quadratic Functions
A graph of a
is called a
.
Note: If a > 0, the parabola
If a < 0, the parabola
2
.
.
Vertex and Max/Min
The
is the point where
.
The y-value of the vertex is the
•
, if the parabola
•
, if the parabola
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How to Find the Vertex
In vertex form, it’s
Easy.
The vertex of y = (x − h)2 + k is simply (h, k).
In standard form it’s
.
The x-coordinate of the vertex is
The y-coordinate of the vertex is
4
.
.
Example: Find the vertex of the parabola given by
y = 2x2 + 3x − 1.
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The x-intercepts of a Quadratic Function
The x-intercepts of a quadratic function are
•
•
If f (x) = ax2 + bx + c, the x-coordinates of the xintercepts may be found by using
x=
.
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The quadratic formula can give
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Example
Given the quadratic equation f (x) = 3x2 − 7x + 4,
1. Determine whether the graph of y = f (x) opens upward or downward.
2. Find the vertex of the parabola.
3. Where does the minimum value of f occur?
4. What is the minimum value of f ?
5. Is there a maximum value of f ?
6. What are the x-intercepts of f ?
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Example (continued)
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Applications to Business
Previously, we discussed the revenue function
We assumed that p was
which made R(x)
.
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Nonlinear Revenue
However, often the price is a function of the demand.
If the price is a linear function of the demand (p = mx +
b), then the revenue function
R(x) =
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Example
ChalkboardCo, a manufacturer of chalkboards, has determined that its price-demand function is p(x) = −.5x+
20, where x is the number of chalkboards (in thousands)
and p is the price (in dollars).
1. Find the revenue function R(x) for ChalkboardCo
2. Find the vertex of R(x).
3. Interpret the vertex. What does this tell ChalkboardCo?
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