Introduction
The tourism industry thrives on being able to provide
travelers with an amazing travel experience. Specifically,
in areas known for having tropical weather, tour planners
want to maximize profit each month by identifying the
warmest and coolest months, and then plan tours
accordingly. Tour planners might use quadratic models
to determine when profits are increasing or decreasing,
when they maximized, and/or how profits change in the
earlier months versus the later months by looking at the
key features of the quadratic functions.
1
5.5.1: Interpreting Key Features of Quadratic Functions
Introduction, continued
In this lesson, you will learn about the key features of a
quadratic function and how to use graphs, tables, and
verbal descriptions to identify and apply the key
features.
2
5.5.1: Interpreting Key Features of Quadratic Functions
Key Concepts
• The key features of a quadratic function are
distinguishing characteristics used to describe, draw,
and compare quadratic functions. These key
features include the x-intercepts, y-intercept, where
the function is increasing and decreasing, where the
function is positive and negative, relative minimums
and maximums, symmetries, and end behavior of
the function.
3
5.5.1: Interpreting Key Features of Quadratic Functions
Key Concepts, continued
• The x-intercepts of
a quadratic function
occur when the
parabola intersects
the x-axis. In the
graph at right, the
x-intercepts occur
when x = 2 and
when x = –2.
4
5.5.1: Interpreting Key Features of Quadratic Functions
Key Concepts, continued
• The equation of the x-axis is y = 0;
therefore, the x-intercepts can also
be found in a table by identifying
when the y-value is 0.
• The table of values at right
corresponds to the parabola we just
saw. Notice that the same xintercepts can be found where the
table shows y is equal to 0.
x
y
–4
12
–2
0
0
–4
2
0
4
12
5
5.5.1: Interpreting Key Features of Quadratic Functions
Key Concepts, continued
• The ordered pair that corresponds to an x-intercept is
always of the form (x, 0). The x-intercepts are also the
solutions of a quadratic function.
• The y-intercepts of a quadratic function occur when
the parabola intersects the y-axis. In the next graph,
the y-intercept occurs when y = –4.
6
5.5.1: Interpreting Key Features of Quadratic Functions
Key Concepts, continued
7
5.5.1: Interpreting Key Features of Quadratic Functions
Key Concepts, continued
• The equation of the y-axis is x = 0;
therefore, the y-intercept can also be
found in a table by identifying when
the x-value is 0.
• Notice in the table of values (at right)
that corresponds to the parabola we
just saw, the same y-intercept can
be found where x is 0.
x
y
–4
12
–2
0
0
–4
2
0
4
12
8
5.5.1: Interpreting Key Features of Quadratic Functions
Key Concepts, continued
• The ordered pair that corresponds to a y-intercept is
always of the form (0, y).
• Recall that the vertex is the point on a parabola where
the graph changes direction.
• The maximum or minimum of the function occurs at
the vertex of the parabola.
9
5.5.1: Interpreting Key Features of Quadratic Functions
Key Concepts, continued
• The vertex is also the point where the parabola
changes from increasing to decreasing.
• Increasing refers to the interval of a function for
which the output values are becoming larger as the
input values are becoming larger.
• Decreasing refers to the interval of a function for
which the output values are becoming smaller as the
input values are becoming larger.
• Recall that parabolas are symmetric to a line that
extends through the vertex, called the axis of
symmetry.
5.5.1: Interpreting Key Features of Quadratic Functions
10
Key Concepts, continued
• Any point to the right or left of the parabola is
equidistant to another point on the other side of the
parabola.
• A parabola only increases or decreases as x becomes
larger or smaller.
• Read the graph from left to right to determine when
the function is increasing or decreasing.
• Trace the path of the graph with a pencil tip. If your
pencil tip goes down as you move toward increasing
values of x, then f(x) is decreasing.
11
5.5.1: Interpreting Key Features of Quadratic Functions
Key Concepts, continued
• If your pencil tip goes up as you move toward
increasing values of x, then f(x) is increasing.
• For a quadratic, if the graph has a minimum value,
then the quadratic will start by decreasing toward the
vertex, and then it will increase.
• If the graph has a maximum value, then the quadratic
will start by increasing toward the vertex, and then it
will decrease.
• The vertex is called an extremum. Extrema are the
maxima or minima of a function.
12
5.5.1: Interpreting Key Features of Quadratic Functions
Key Concepts, continued
• The concavity of a parabola is the property of being
arched upward or downward.
• A quadratic with positive concavity will increase on
either side of the vertex, meaning that the vertex is
the minimum or lowest point of the curve.
• A quadratic with negative concavity will decrease on
either side of the vertex, meaning that the vertex is
the maximum or highest point of the curve.
13
5.5.1: Interpreting Key Features of Quadratic Functions
Key Concepts, continued
• A quadratic that has a minimum value is concave up
because the graph of the function is bent upward.
• A quadratic that has a maximum value is concave
down because the graph of the function is bent
downward.
• The graphs that follow demonstrate examples of
parabolas as they decrease and then increase, and
vice versa. Trace the path of each parabola from left
to right with your pencil to see the difference.
14
5.5.1: Interpreting Key Features of Quadratic Functions
Key Concepts, continued
Decreasing then Increasing
• Vertex: (0, –4);
minimum
• x < 0 = decreasing
• x > 0 = increasing
• Direction: concave up
15
5.5.1: Interpreting Key Features of Quadratic Functions
Key Concepts, continued
Increasing then Decreasing
• Vertex: (0, 4);
maximum
• x < 0 = increasing
• x > 0 = decreasing
• Direction: concave
down
16
5.5.1: Interpreting Key Features of Quadratic Functions
Key Concepts, continued
• The inflection point of a graph is a point on a curve
at which the sign of the curvature (i.e., the concavity)
changes. In the graph on the following slide, the
curvature starts out as concave down, but then
switches to concave up at (–1, 1). The point (–1, 1)
is the point of inflection.
• The vertex of a quadratic function is also the point of
inflection.
17
5.5.1: Interpreting Key Features of Quadratic Functions
Key Concepts, continued
18
5.5.1: Interpreting Key Features of Quadratic Functions
Key Concepts, continued
• End behavior is the behavior of the graph as x
becomes larger or smaller.
• If the highest exponent of a function is even, and the
coefficient of the same term is positive, then the
function is approaching positive infinity as x
approaches both positive and negative infinity.
• If the highest exponent of a function is even, but the
coefficient of the same term is negative, then the
function is approaching negative infinity as x
approaches both positive and negative infinity.
19
5.5.1: Interpreting Key Features of Quadratic Functions
Key Concepts, continued
Even and Positive
•
•
•
•
f(x) = x2 – 4
Highest exponent: 2
Coefficient of x2: positive
As x approaches positive
infinity, f(x) approaches
positive infinity.
• As x approaches
negative infinity, f(x)
approaches positive
infinity.
20
5.5.1: Interpreting Key Features of Quadratic Functions
Key Concepts, continued
Even and Negative
•
•
•
•
f(x) = –x2 + 4
Highest exponent: 2
Coefficient of x2: negative
As x approaches positive
infinity, f(x) approaches
negative infinity.
• As x approaches
negative infinity, f(x)
approaches negative
infinity.
21
5.5.1: Interpreting Key Features of Quadratic Functions
Key Concepts, continued
• Functions can be defined as odd or even based on
the output yielded when evaluating the function for –x.
• For an odd function, f(–x) = –f(x). That is, if you
evaluate a function for –x, the resulting function is the
opposite of the original function.
• For an even function, f(–x) = f(x). That is, if you
evaluate a function for –x, the resulting function is the
same as the original function.
22
5.5.1: Interpreting Key Features of Quadratic Functions
Key Concepts, continued
• If evaluating the function for –x does not result in the
opposite of the original function or the original
function, then the function is neither odd nor even.
• Though all quadratics have an even power, not all
quadratics are even functions.
• It is important to evaluate the function for –x when the
quadratic includes both a linear and a constant term.
23
5.5.1: Interpreting Key Features of Quadratic Functions
Common Errors/Misconceptions
• incorrectly identifying when a function is increasing or
decreasing
• making sign errors when determining if a function is
odd, even, or neither
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5.5.1: Interpreting Key Features of Quadratic Functions
Guided Practice
Example 1
A local store’s monthly revenue from T-shirt sales is
modeled by the function f(x) = –5x2 + 150x – 7. Use the
equation and the graph on the next slide to answer the
following questions: At what prices is the revenue
increasing? Decreasing? What is the maximum
revenue? What prices yield no revenue? Is the function
even, odd, or neither?
25
5.5.1: Interpreting Key Features of Quadratic Functions
Guided Practice: Example 1, continued
26
5.5.1: Interpreting Key Features of Quadratic Functions
Guided Practice: Example 1, continued
1. Determine when the function is
increasing and decreasing.
Use your pencil to determine when the function is
increasing and decreasing.
Moving from left to right, trace your pencil along the
function.
The function increases until it reaches the vertex,
then decreases.
The revenue is increasing when the price per shirt is
less than $15 or when x < 15.
27
5.5.1: Interpreting Key Features of Quadratic Functions
Guided Practice: Example 1, continued
The vertex of this function has an x-value of 15.
The revenue is decreasing when the price per shirt is
more than $15 or when x > 15.
28
5.5.1: Interpreting Key Features of Quadratic Functions
Guided Practice: Example 1, continued
2. Determine the maximum revenue.
Use the vertex of the function to determine the
maximum revenue.
The T-shirt price that maximizes revenue is x = 15.
The maximum is the corresponding y-value.
Since it is difficult to estimate accurately from this
graph, substitute x into the function to solve.
29
5.5.1: Interpreting Key Features of Quadratic Functions
Guided Practice: Example 1, continued
f(x) = –5x2 + 150x – 7
f(15) = –5(15)2 + 150(15) – 7
f(15) = 1118
Original function
Substitute 15 for x.
Simplify.
The maximum revenue is $1,118.
30
5.5.1: Interpreting Key Features of Quadratic Functions
Guided Practice: Example 1, continued
3. Determine the prices that don’t yield
revenue.
Identify the x-intercepts.
The x-intercepts are 0 and 30, so the store has no
revenue when the shirts cost $0 and $30.
31
5.5.1: Interpreting Key Features of Quadratic Functions
Guided Practice: Example 1, continued
4. Determine if the function is even, odd, or
neither.
Evaluate the function for –x.
f(x) = –5x2 + 150x – 7
f(–x) = –5(–x)2 + 150(–x) – 7
f(x) = –5x2 – 150x – 7
Original function
Substitute –x for x.
Simplify.
Since f(–x) is neither the original function nor the
opposite of the original function, the function is not
even or odd; it is neither.
32
5.5.1: Interpreting Key Features of Quadratic Functions
Guided Practice: Example 1, continued
5. Use the graph of the function to verify that
the function is neither odd nor even.
Since the function is not symmetric over the y-axis or
the origin, the function is neither even nor odd.
✔
33
5.5.1: Interpreting Key Features of Quadratic Functions
Guided Practice: Example 1, continued
34
5.5.1: Interpreting Key Features of Quadratic Functions
Guided Practice
Example 2
A function has a minimum value of –5 and x-intercepts
of –8 and 4. What is the value of x that minimizes the
function? For what values of x is the function
increasing? Decreasing?
35
5.5.1: Interpreting Key Features of Quadratic Functions
Guided Practice: Example 2, continued
1. Determine the x-value that minimizes the
function.
Quadratics are symmetric functions about the vertex
and the axis of symmetry, the line that divides the
parabola in half and extends through the vertex.
The x-value that minimizes the function is the
midpoint of the two x-intercepts.
36
5.5.1: Interpreting Key Features of Quadratic Functions
Guided Practice: Example 2, continued
Find the midpoint of the two points by taking the
average of the two x-coordinates.
x=
-8 + 4
2
= -2
The value of x that minimizes the function is –2.
37
5.5.1: Interpreting Key Features of Quadratic Functions
Guided Practice: Example 2, continued
2. Determine when the function is increasing
and decreasing.
Use the vertex to determine when the function is
increasing and when it is decreasing.
The minimum value is –5 and the vertex of the
function is (–2, –5).
From left to right, the function decreases as it
approaches the minimum and then increases.
The function is increasing when x > –2 and
decreasing when x < –2.
✔
38
5.5.1: Interpreting Key Features of Quadratic Functions
Guided Practice: Example 2, continued
39
5.5.1: Interpreting Key Features of Quadratic Functions