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Analyzing Functions withSuccessive Differences
Lesson 9-6
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Over Lesson 9–5
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Over Lesson 9–5
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Understand how to identify linear, quadratic, and exponential
functions from given data and write equations that model data.
LEARNING GOAL
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Choose a Model Using Graphs
A. Graph the ordered pairs. Determine whether the ordered pairs represent a linear, quadratic, or exponential function.
(1, 2), (2, 5), (3, 6), (4, 5), (5, 2)
Answer: The ordered pairs appear to represent a quadratic equation.
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Choose a Model Using Graphs
Answer: The ordered pairs appear to represent an exponential function.
B. Graph the ordered pairs. Determine whether the ordered pairs represent a linear, quadratic, or exponential function.
(–1, 6), (0, 2),
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A. linear
B. quadratic
C. exponential
A. Graph the set of ordered pairs. Determine whether the ordered pairs represent a linear, quadratic, or exponential function.(–2, –6), (0, –3), (2, 0), (4, 3)
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A. linear
B. quadratic
C. exponential
B. Graph the set of ordered pairs. Determine whether the ordered pairs represent a linear, quadratic, or exponential function.(–2, 0), (–1, –3), (0, –4), (1, –3), (2, 0)
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Choose a Model Using Differences or Ratios
A. Look for a pattern in the table of values to determine which kind of model best describes the data.
–1 1 3 5 7
2 2 2 First differences:
Answer: Since the first differences are all equal, the table of values represents a linear function.
2
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Choose a Model Using Differences or Ratios
B. Look for a pattern in the table of values to determine which kind of model best describes the data.
–24 –8 First differences:
The first differences are not all equal. So, the table of values does not represent a linear function. Find the second differences and compare.
36 12 4__43
__49
–2 __23
__89–
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Choose a Model Using Differences or Ratios
16
First differences:
The second differences are not all equal. So, the table of values does not represent a quadratic function. Find the ratios of the y-values and compare.
1 __79
5 __13Second differences:
36 4 __49
12 __43
__13
__13
Ratios: __13
__13
–24 –8 –2 __23
__89–
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Choose a Model Using Differences or Ratios
The ratios of successive y-values are equal.
Answer: The table of values can be modeled by an exponential function.
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A. linear
B. quadratic
C. exponential
D. none of the above
A. Look for a pattern in the table of values to determine which kind of model best describes the data.
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A. linear
B. quadratic
C. exponential
D. none of the above
B. Look for a pattern in the table of values to determine which kind of model best describes the data.
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Write an Equation
Determine which kind of model best describes the data. Then write an equation for the function that models the data.
Step 1 Determine which model fits the data.
–1 –8 –64 –512 –4096
–7 –56 –448 –3584 First differences:
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Write an Equation
–7 –56 –448 –3584 First differences:
–49 –392 –3136Second differences:
× 8 × 8
The table of values can be modeled by an exponential function.
–1 –8 –64Ratios: –512 –4096
× 8 × 8
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Write an Equation
Step 2 Write an equation for the function that models the data.
The equation has the form y = abx. Find the value of a by choosing one of the ordered pairs from the table of values. Let’s use (1, –8).
y = abx Equation for exponential function–8 = a(8)1x = 1, y = –8, b = 8–8 = a(8) Simplify.–1 = a An equation that models the data
is y = –(8)x.
Answer: y = –(8)x
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A. quadratic; y = 3x2
B. linear; y = 6x
C. exponential; y = 3x
D. linear; y = 3x
Determine which model best describes the data. Then write an equation for the function that models the data.
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Write an Equation for a Real-World Situation
KARATE The table shows the number of children enrolled in a beginner’s karate class for four consecutive years. Determine which model best represents the data. Then write a function that models that data.
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Write an Equation for a Real-World Situation
Understand We need to find a model for the data, and then write a function.
Plan Find a pattern using successive differences or ratios. Then use the general form of the equation to write a function.
Solve The first differences are all 3. A linear function of the form y = mx + b models the data.
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Write an Equation for a Real-World Situation
y = mx + b Equation for linear function8 = 3(0) + b x = 0, y = 8, and m = 3b = 8 Simplify.
Answer: The equation that models the data is y = 3x + 8.
Check You used (0, 8) to write the function. Verify that every other ordered pair
satisfies the function.
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A. linear; y = 4x + 4
B. quadratic; y = 8x2
C. exponential; y = 2 ● 4x
D. exponential; y = 4 ● 2x
WILDLIFE The table shows the growth of prairie dogs in a colony over the years. Determine which model best represents the data. Then write a function that models the data.
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Homework
Page 593 #15-27 odd#41-59 odd