Introduction
Relationships can be presented through different forms:
tables, equations, graphs, and words. This section
explores how to compare linear, quadratic, and
exponential functions presented in all four forms.
1
2.4.2: Comparing Properties of Functions Given in Different Forms
Key Concepts
Linear Functions
• A linear function is a function that can be written in
the form f(x) = mx + b, in which m is the slope, b is the
y-intercept, and the graph is a straight line.
• Linear functions have a constant slope.
• As the value of x increases by 1, f(x) will increase by a
constant value.
2
2.4.2: Comparing Properties of Functions Given in Different Forms
Key Concepts, continued
• The rate of change of a linear function will always
remain constant; however, the function will either
increase, decrease, or remain constant.
• In a set of data, the change in y when x increases by
1 is called a first difference.
3
2.4.2: Comparing Properties of Functions Given in Different Forms
Key Concepts, continued
Quadratic Functions
• A quadratic function is a function that can be written in
the form f(x) = ax2 + bx + c, where a ≠ 0.
• The graph of any quadratic function is a U-shaped
curve known as a parabola.
• The value of a determines whether the quadratic has
a maximum or a minimum.
• If a is negative, the quadratic function will have a
maximum.
• If a is positive, the quadratic function will have a
minimum.
4
2.4.2: Comparing Properties of Functions Given in Different Forms
Key Concepts, continued
• There is neither a constant rate of change nor a
constant multiple in a quadratic function.
• In a quadratic model, the change in the first
differences is constant.
• The change in first differences is called a second
difference.
• Compare the y-values of the vertices of quadratic
functions to determine the greatest maximum or least
minimum values between two or more quadratic
functions.
5
2.4.2: Comparing Properties of Functions Given in Different Forms
Key Concepts, continued
Exponential Functions
• Exponential functions are written in the form
f(x) = abx.
• The variable of an exponential function is part of the
exponent.
• As the value of x increases, the value of f(x) will
increase by a multiple of b.
• Exponential functions either increase or decrease.
• There is a constant multiple in the rate of change
between y-values.
6
2.4.2: Comparing Properties of Functions Given in Different Forms
Key Concepts, continued
• An interval is the set of all real numbers between two
given numbers.
• The rate of change of an exponential function varies
depending on the interval observed.
• Graphs of exponential functions of the form f(x) = abx,
where b is greater than 1, will increase faster than
graphs of linear functions of the form f(x) = mx + b.
• A quantity that increases exponentially will always
eventually exceed a quantity that increases linearly or
quadratically.
7
2.4.2: Comparing Properties of Functions Given in Different Forms
Common Errors/Misconceptions
• misidentifying the appropriate type of function (linear,
quadratic, or exponential)
• thinking that an equation with a higher y-intercept has a
higher maximum value
• incorrectly determining the rate of change
• assuming that the rate of change of a function is linear
by only referencing one interval
8
2.4.2: Comparing Properties of Functions Given in Different Forms
Guided Practice
Example 2
Three students are shooting wads of paper with a rubber
band, aiming for a trash can in the front of the room. The
height of each student’s paper wad in feet is given as a
function of the time in seconds. Which student’s paper
wad flies the highest? The functions for each of the
three students are described on the next slide.
9
2.4.2: Comparing Properties of Functions Given in Different Forms
Guided Practice: Example 2, continued
• The path of Alejandro’s paper wad is modeled by the
equation f(x) = –x2 + 2x + 7.
• Melissa’s paper wad is estimated to reach the heights
shown in the table below.
x
0
2
3
4
y
3
6
7
6
• After 3 seconds, Connor’s paper wad achieves a
maximum height of 6.5 feet above the floor.
10
2.4.2: Comparing Properties of Functions Given in Different Forms
Guided Practice: Example 2, continued
1. Determine if each function represents a
quadratic.
Alejandro’s path is represented by a quadratic
function written in standard form.
Melissa’s table has symmetry about x = 3, which
implies a quadratic relationship.
In general, a projectile follows a parabolic path,
gaining height until reaching a maximum, then
descending. We can assume Connor’s paper wad
will follow a parabolic path.
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2.4.2: Comparing Properties of Functions Given in Different Forms
Guided Practice: Example 2, continued
2. Verify that the extremum for each function
is the maximum value of each function.
The value of a in the equation for Alejandro’s paper
wad is a negative value. Therefore, the function has a
maximum value.
The table for Melissa’s paper wad increases then
decreases, so its path will have a maximum value.
The maximum height of the path of Connor’s paper
wad is given.
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2.4.2: Comparing Properties of Functions Given in Different Forms
Guided Practice: Example 2, continued
3. Determine the vertex of each function.
Find the vertex of Alejandro’s function using
æ -b æ -b ö ö
ç 2a , f ç 2a ÷ ÷ .
è øø
è
x=
x=
-b
2a
-(2)
2(-1)
x=1
Formula to find the xcoordinate of the vertex
Substitute –1 for a and
2 for b.
Simplify.
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2.4.2: Comparing Properties of Functions Given in Different Forms
Guided Practice: Example 2, continued
æ -b ö
Use the given function to find f ç ÷ = f (1).
è 2a ø
f(x) = –x2 + 2x + 7
Original equation
f(1) = –(1)2 + 2(1) + 7
Substitute 1 for x.
f(1) = –1 + 2 + 7
Simplify.
f(1) = 8
The vertex is at (1, 8).
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2.4.2: Comparing Properties of Functions Given in Different Forms
Guided Practice: Example 2, continued
Find the vertex for Melissa’s paper wad using
symmetry. Her paper wad’s function is symmetric
about the vertex at (3, 7).
The vertex for Connor’s paper wad is given as
(3, 6.5).
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2.4.2: Comparing Properties of Functions Given in Different Forms
Guided Practice: Example 2, continued
4. Use the vertices to determine whose
paper wad goes the highest.
Compare the y-values of the vertices.
8 > 7 > 6.5
Alejandro’s paper wad flies the highest, at 8 feet.
✔
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2.4.2: Comparing Properties of Functions Given in Different Forms
Guided Practice: Example 2, continued
17
2.4.2: Comparing Properties of Functions Given in Different Forms
Guided Practice
Example 3
Which of the following quadratic functions has a vertex
with a larger y-value: f(x) = 2x2 – 12x + 25, or g(x) as
presented in the table?
x
g(x)
–4
7
–3
8
–2
7
0
–1
2
–17
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2.4.2: Comparing Properties of Functions Given in Different Forms
Guided Practice: Example 3, continued
1. Verify for each function whether the
vertex is a minimum or maximum.
For f(x), a is positive, which means that the vertex is
a minimum.
For g(x), as x increases, the y-values first increase
and then decrease. This indicates the vertex is a
maximum.
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2.4.2: Comparing Properties of Functions Given in Different Forms
Guided Practice: Example 3, continued
2. Find the vertex for each function.
æ -b
For f(x), the vertex is of the form ç ,
è 2a
æ -b ö ö
f ç ÷÷.
è 2a ø ø
Use the original function, f(x) = 2x2 – 12x + 25, to
find the values of a and b in order to find the
x-value of the vertex.
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2.4.2: Comparing Properties of Functions Given in Different Forms
Guided Practice: Example 3, continued
x=
x=
-b
2a
-(-12)
x=3
2(2)
Formula to find the x-coordinate of
the vertex of a quadratic
Substitute 2 for a and –12 for b.
Simplify.
The x-coordinate of the vertex is 3.
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2.4.2: Comparing Properties of Functions Given in Different Forms
Guided Practice: Example 3, continued
Substitute 3 into the original function to find the
y-coordinate.
f(x) = 2x2 – 12x + 25
Original function
f(3) = 2(3)2 – 12(3) + 25
f(3) = 7
Substitute 3 for x.
Simplify.
The y-coordinate of the vertex is 7.
The vertex of f(x) is (3, 7).
For g(x), the points are symmetric about x = –3.
Therefore, the line of symmetry contains (–3, 8),
which must be the vertex.
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2.4.2: Comparing Properties of Functions Given in Different Forms
Guided Practice: Example 3, continued
3. Determine which vertex has a larger
y-value.
Compare the y-values of each function.
7<8
The function g(x) has a vertex with a larger y-value.
✔
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2.4.2: Comparing Properties of Functions Given in Different Forms
Guided Practice: Example 3, continued
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2.4.2: Comparing Properties of Functions Given in Different Forms