Math 141: Business Mathematics I
Fall 2015
§1.Q Quadratic Functions
Instructor: Yeong-Chyuan Chung
Outline
• Quadratic Functions - vertex, zeros/roots, max/min values
• Application of Quadratic Functions - nonlinear revenue functions
Quadratic Functions
A quadratic function is a function of the form
This is just one way of writing a quadratic function, and it is called the standard form.
The same quadratic function may be written in another form, called the vertex form, and
it is written in the following way:
Compare the two forms. How are the various constants related?
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The graph of a quadratic function is called a parabola. In order to sketch the graph of
a quadratic function (without using a graphing calculator), we need to know some of its
characteristic features. These include
• whether it opens upwards or downwards;
• the vertex;
• the maximum/minimum value;
• the x-/y-intercept(s).
Opening upward/downward
The positivity/negativity of a (the coefficient of x2 ) in a quadratic function determines
whether the graph opens upward or opens downward. More precisely,
• the graph opens upward if
• the graph opens downward if
The vertex
Every parabola has a distinguished point. This is the highest/lowest point on the parabola,
and it is called the vertex.
The vertex form of a quadratic function gets its name from the fact that we can read off the
vertex of its graph easily. If the equation is y = a(x − h)2 + k, then the vertex is simply
If a quadratic function is given in standard form instead, we need to do a little bit of work to
determine the vertex. In this case, if the equation is y = ax2 + bx + c, then the x-coordinate
of the vertex is given by
Once we know the x-coordinate of the vertex, we can then find the y-coordinate by plugging
in the x-value into the equation.
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Example. Identify the vertex of the graph for each function:
• y = (x − 1)2 + 2
• y = (x + 2)2 − 1
Example. Determine the x-coordinate of the vertex of the graph for each function:
• y = 2x2 + 5x + 1
• y = −x2 + 3x − 2
• y = −x2 − 3x − 2
The maximum/minimum value
This is the y-coordinate of the vertex. In other words, it is the largest/smallest y-value
attained by the quadratic function.
Whether a quadratic function has a maximum value or a minimum value depends on whether
its graph opens upward or opens downward, which in turn depends on whether the constant
a in the equation is positive or negative.
As we saw above, if the equation is given in vertex form, say y = a(x − h)2 + k, then the
y-coordinate of the vertex is simply k. If the equation is given in standard form instead,
then we can first find the x-coordinate of the vertex and then plug it into the equation to
find the y-coordinate.
Example. In the previous example, which functions have maximum values and which have
minimum values? Compute these values.
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The x-/y-intercept(s)
The y-intercept can be found in the usual way, namely plugging in x = 0 into the equation.
The x-intercepts are also called the zeros/roots of the function. If the equation is given in
standard form, then we can use the quadratic formula to find the x-intercepts.
Recall the quadratic formula:
How many x-intercepts can there be for any given quadratic function?
Example. Given the function y = 3x2 − 7x + 4,
(a) Determine whether the graph opens upward or downward. State the reason.
(b) Find the vertex of the graph.
(c) Find the maximum value.
(d) Find the minimum value.
(e) Find the x-intercept(s).
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Application
Earlier on, we considered revenue functions that were linear. Recall that these were of the
form R(x) = px where x represents the number of units sold and p is the price of each unit.
However, in real life, the price is often not constant, and may depend on the demand. In
this case, the price is a function of x, and is what we called the demand function earlier. We
will still assume that the demand function is linear.
Suppose we write p(x) for the demand function. Then the revenue function can be written
as
Since we are assuming that the demand function p(x) is linear, the revenue function will
now be quadratic. We can then use the techniques discussed in this section to analyze the
revenue function.
Example. Market research has shown that the demand for a certain computer is given by
p(x) = 4000 − 50x. How many computers must be sold to maximize the company’s revenue?
What is the maximum revenue? What will be the unit price in this case?
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Review/Self-assessment:
In the previous example, if you are also given the cost function, say C(x) = 10000 + 1000x,
• are you able to determine the profit function?
• are you able to determine the maximum profit?
• are you able to determine the break-even point?