6-2
6-2 Comparing
ComparingFunctions
Functions
Warm Up
Lesson Presentation
Lesson Quiz
HoltMcDougal
Algebra 2Algebra 2
Holt
6-2 Comparing Functions
Warm Up
For each function, determine whether
the graph opens upward or
downward.
1. f(x) = -4x2 + 6x + 1
downward
2. f(x) = 8x2 – x - 2
upward
Write each function in slope-intercept form.
3. Y + 3x =10
y = -3x + 10
4. -6y – 12x = 24
y = -2x - 4
Holt McDougal Algebra 2
6-2 Comparing Functions
Objectives
Compare properties of two functions.
Estimate and compare rates of change.
Holt McDougal Algebra 2
6-2 Comparing Functions
The graph of the exponential function
y=0.2491e0.0081x approximates the population growth
in Baltimore, Maryland.
Holt McDougal Algebra 2
6-2 Comparing Functions
The graph of the exponential function y=0.0023e0.0089x
approximates the population growth in Hagerstown,
Maryland. The trends can be used to predict what the
population will be in the future in each city.
In this lesson you will compare the graphs of linear,
quadratic and exponential functions.
Holt McDougal Algebra 2
6-2 Comparing Functions
Example 1: Comparing the Average Rates of
Change of Two Functions.
George tracked the cost of gas from two
separate gas stations. The table shows the cost
of gas for one of the stations and the graph
shows the cost of gas for the second station.
Compare the average rates and explain what
the difference in rate of change represents.
Holt McDougal Algebra 2
6-2 Comparing Functions
The rate of change for Gas Station A is about 3.0.
The rate of change for Gas Station B is about 2.9.
The rate of change is the cost per gallon for each of
the Stations. The cost is less at Gas Station B.
Holt McDougal Algebra 2
6-2 Comparing Functions
Check It Out! Example 1
John and Mike opened savings accounts on the
same day. They did not deposit any money
initially, but deposited each week as shown by
the graph and the table. Compare the average
rates of change and explain what the rates
represent in this situation.
Holt McDougal Algebra 2
6-2 Comparing Functions
Example 1 continued
Mike’s average rate of change is 26. John’s average
rate of change is ≈ 25.57. The rate of change is the
average amount of money saved per week. In this
case, Mike’s rate of change is larger than John’s, so
he saves about $0.43 more than John per week
Holt McDougal Algebra 2
6-2 Comparing Functions
Example 1 continued
Mike’s average rate
m = 124 -20 = 104
5-1
4
John’s average rate
m = 204 -25 = 179
8-1
7
Holt McDougal Algebra 2
of change is
= 26
of change is
≈ 25.57
6-2 Comparing Functions
Example 1 continued
Helpful Hint
Remember to find the average rate of change over
a data set, find the slope between the first and
last data point.
Holt McDougal Algebra 2
6-2 Comparing Functions
Example 2: Sketching Graphs of Functions Given Key
Features.
The graph for the height of a diving bird above
the water level, h(t), in feet after t seconds
passes through the points (0, 5), (3, -1), and
(5,15). Sketch a graph of the quadratic
function that models the situation. Find the
point that represents the minimum height of
the bird.
Holt McDougal Algebra 2
6-2 Comparing Functions
Example 2 continued
Step 1 Use the points to find the values of a, b, and c
in the function h(t) = at2 + bt + c.
(t, h(t))
h(t) = at2 + bt + c
(0,5)
5 = a(0)2 + b(0) + c
(3, -1)
- 1 = a(3)2 + b(3) + c
(5, 15)
15 = a(5)2 + b(5) + c
Holt McDougal Algebra 2
System in a, b, c
5=c
-1 = 9a+3b+c
15=25a+5b+c
6-2 Comparing Functions
Example 2 continued
Step 2 Solve the system found in Step 1 and write
the equation.
5=c
-1 = 9a+3b+c
15=25a+5b+c
-1 = 9a+3b+c
15=25a+5b+c
-6 = 9a+3b
10=25a+5b
30 = -45a-15b
30=75a+15b
Holt McDougal Algebra 2
Substitute c = 5 in 2nd and 3rd
equation.
Multiply the first equation by –5 and
the second equation by 3 in order
to use elimination.
6-2 Comparing Functions
Example 2 continued
60 = 30a
2=a
15 = 25(2) + 5b + 5
-8 = b
h(t) = 2t2 – 8t + 5
Holt McDougal Algebra 2
Add equations and solve.
6-2 Comparing Functions
Example 2 continued
Step 3 Find the minimum height of the function by
finding the vertex. Graph the function and
approximate the vertex.
minimum height: 3 ft below water level
Holt McDougal Algebra 2
6-2 Comparing Functions
Helpful Hint
Remember, in the equation f(x) = a(x - h)2 + k,
the point (h, k) represents the vertex.
Holt McDougal Algebra 2
6-2 Comparing Functions
Check It Out! Example 2
The height of a model rocket after launch is
tracked in the table. Find and graph a
quadratic function that describes the data.
Holt McDougal Algebra 2
6-2 Comparing Functions
Check It Out! Example 2 continued
Step 1 Use the points to find the values of a, b, and
c in the function h(t) = at2 + bt + c.
(t, h(t))
h(t) = at2 + bt + c
(0.5,31)
31 = a(0.5)2 + b(0.5) + c
(1.5, 59)
59 = a(1.5)2 + b(1.5) + c
(2.5, 55)
55 = a(2.5)2 + b(2.5) + c
Holt McDougal Algebra 2
System in a, b, c
31=0.25a+0.5b+c
59=2.25a+1.5b+c
55=6.25a+2.5b+c
6-2 Comparing Functions
Check It Out! Example 2 continued
Step 2 Solve the system found in Step 1 and write
the equation.
31=0.25a+0.5b+c
59=2.25a+1.5b+c
55=6.25a+2.5b+c
28 = 2a+b
24=6a+2b
Subtract the first equation from the
second and third equations.
-56 =-4a+2b
24=6a+2b
Multiply the first equation by –2.
Holt McDougal Algebra 2
6-2 Comparing Functions
Check It Out! Example 2 continued
-32 = 2a
-16 = a
Add equations and solve.
28 = 2(-16) + b
28 = -32 + b
60 = b
31 = 0.25(-16) + 0.5(60) + c
31 = -4 + 30 + c
31 = 26 + c
5=c
h(t) = -16t2 + 60t + 5
Holt McDougal Algebra 2
6-2 Comparing Functions
Check It Out! Example 2 continued
Step 3 Find the maximum height of the function by
finding the vertex. Graph the function and
approximate the vertex.
The maximum height is approximately 61 feet.
Holt McDougal Algebra 2
6-2 Comparing Functions
Example 3: Comparing Exponential and Polynomial
Functions.
Compare the end behavior of the functions
f(x) = -x2 and
g(x) = 4 logx.
Holt McDougal Algebra 2
6-2 Comparing Functions
Example 3 continued
The end behavior for the graph of
f(x) = –x2: as x approaches positive infinity, f(x)
approaches negative infinity. As x approaches
negative infinity, f(x) approaches
negative infinity.
The end behavior for the graph of
g(x) = 4log x: as x approaches positive infinity,
g(x) approaches positive infinity, as x approaches
0, g(x) → approaches negative infinity.
Holt McDougal Algebra 2
6-2 Comparing Functions
Check It Out! Example 3
Compare the end behavior of the functions
f (x) = 4x2 and g(x) = x3.
Holt McDougal Algebra 2
6-2 Comparing Functions
Check It Out! Example 3 continued
The end behavior for the graph of f(x)= 4x2: as x
approaches positive infinity, f(x) approaches
positive infinity. As x approaches negative infinity,
f(x)approaches positive infinity.
The end behavior for the graph of g(x) = x3: as x
approaches positive infinity, g(x)approaches
positive infinity, as x approaches negative infinity,
g(x) approaches negative infinity.
Holt McDougal Algebra 2
6-2 Comparing Functions
Lesson Quiz: Part I
Compare the end behavior of each pair of
functions.
1. f(x) = x and g(x) = -x4
f(x): as x approaches positive infinity, f(x) approaches
positive infinity; as x approaches
negative infinity, f(x) approaches negative infinity.
g(x): as x approaches positive infinity, g(x)
approaches negative infinity; as x approaches
negative infinity, g(x) approaches negative infinity.
Holt McDougal Algebra 2
6-2 Comparing Functions
Lesson Quiz: Part 2
2. f(x) = 4ex and g(x) = log x
f(x): as x → ∞, f(x) → ∞; as x→ –∞, f(x) → 0.
g(x): as x →∞, g(x) → 1; as x → 0, g(x)→ –∞.
3. Find the equation of a quadratic function
that describes the data in the table.
f(x) = 3x2 -4x -10.
Holt McDougal Algebra 2