Patterns B Functional Patterns
Exploration: Linear functions
Linear functions may be written in many ways:
Standard
______________________________________________________________
Slope-Intercept
______________________________________________________________
Point-Slope
______________________________________________________________
Is an algebraic expression the only way to describe a line?
x
Change in x
m
f(x)
2
4
4
9
6
14
8
19
10
24
12
29
14
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Change in y
y y2  y1


x x2  x1
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A geometric interpretation of slope:
At every point,
OK, now let’s examine lines algebraically:
For a linear function f(x), adding a constant to a given domain value results in adding a constant to the
corresponding range value.
2
y  5x  3
Try it! Use the function
Please use a different value of
x1
than your neighbor. When finished, compare answers.
x
xn
f ( x)  5 x  3
x1
x 2  x1  1
x 3  x2  1
x 3  x2  1
Please use a different value of
xn
x1
than your neighbor, then compare answers.
x
f ( x)  5 x  3
x1
x 2  x1  2
x 3  x2  2
x 3  x2  2
For a linear function f(x),
adding a constant to a given domain value results in adding a constant to the corresponding range value.
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BUT, have we proved this????
Go to the in class exercise!
4
Can you make a prediction?
Suppose our function is
2x  3 y  7  0
How will y change if 9 is added to x?
To recap,
For a linear function f(x), adding a constant to a given domain value results in adding a constant to the
corresponding range value.
What is the value of the constant added to the range function if the domain inputs change by 1 unit?
By 2 units?
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Exploration: Higher-degree polynomials
Last time, we examined data produced by linear functions and learned that the ___________difference is constant.
For example, for the data below
x
y
0
3
3
15
6
27
9
39
We have a uniform first difference in the dependent (y) variable for uniform x .
This tells us the equation describing the data is of the form Ax  By  C  0 or y  mx  b
Can you figure out the slope from the table?
If you substitute points, you can create a system of equations with unknowns A,B,C or m,b, then solve for the
coefficients.
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New functions, non linear:
Examine the new set of data below, and compute the first difference.
x
0
1
2
3
4
5
6
y
-5
0
9
22
39
60
85
First
Difference
Second
Difference
Is the function linear?
If not, compute the second difference:
If the second difference is constant, what does this tell us?
(Hint: what sort of function has a constant second derivative???)
Write the general form of the equation for the second data set:
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Now substitute points to create a system of equations. Solve this for the unknown coefficients.
Your equation:______________________________________________________________________________
Check this against your neighbor’s equation. Do your answers agree?
Check the other data points to be sure they work in your equation
(Hint: try the table in your calculator instead of repeatedly checking points by hand.)
QUESTIONS TO CONSIDER:
1) Is our equation THE one-and-only solution?
2) Can you think of a function which fits our data but differs elsewhere in its domain?
Can you extract any information about “a” from the differences table?
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Consider another set of data:
x
0
1
2
3
4
5
6
7
y
4
6
2
-14
-48
-106
-194
-318
First
Difference
Second
Difference
Third
Difference
Which difference is constant?
What degree polynomial do you and your partner believe the data represents?
What do you conjecture about the “a” in the standard equation?
Write the general form of the equation for your data:_______________________________________________________
Now select points, create a system of equations, and solve for the coefficients. Check your work against your neighbor’s.
Your solution:_______________________________________________________________________________________
Summary:
If the nth difference is constant, the data represents a(n)___________________degree polynomial.
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Exploration: Data which does NOT represent a Polynomial
In previous work, we determined:
Linear functions have a constant ________________________________ difference,
Quadratic functions have a constant ______________________________difference.
Cubic functions have a constant _________________________________difference.
In class exercise:
Choose a quadratic, make a table. Show how you found the formula on the sheet.
Not every data set represents a polynomial. Examine the data below with your neighbor:
x
0
1
2
3
4
5
6
7
y
3
6
12
24
48
96
192
384
First
Difference
Second
Difference
Third
Difference
Fourth
Difference
If you were to continue calculating a fifth, sixth, or subsequent finite differences, do you think these differences would
EVER become constant? Why?
When the independent variable is increased by a finite difference, what happens to the dependent variable?
Is a constant added to the dependent variable? Or is it multiplied? Or divided? Or is another operation taking place?
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Consider the function
f ( x)  Ab x , where A and b are real constants.
What happens when a constant is added to x? Write f ( x  c)
Now rewrite the expression in terms of f ( x ) .
What did you learn?
Does your algebraic reasoning reflect the behavior of the data in the table above?
With your partner, find constants A and b so that
f ( x)  Ab x describes the data on the previous page.
Conclusion: When adding a value to the independent variable results in multiplying the dependent variable by a
constant, the curve is _______________________________________________________________________.
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Now examine another data set:
x
3
6
12
24
48
96
192
y
0
1
2
3
4
5
6
First
Difference
What do you notice about the difference in the independent (x) variable?
What do you notice about the first difference of the dependent (y) variables?
Is this function linear?
Discuss with your neighbor.
If you believe the function is linear, please find the slope.
Compare the data above to the first data table examined in this lesson.
Pattern of first data table:
Pattern of the second data table:
What might be the relationship between the functions of the first and second data tables???
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Hypothesis:
y  A logb x
Can you find A and b so that the general expression above fits the given data?
Will the multiply-add patter always be a logarithmic function?
Let’s try an informal proof.
Examine
f ( x)  A logb x
Compute
f (kx) and rewrite in terms of f ( x ) to show that multiplication of the independent variable by a constant
results in adding a constant to the dependent variable.
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When multiplying the independent (x) variable by a constant results in the addition of a constant to
the dependent variable, the relation is LOGARITHMIC.
When adding a constant to the independent (x) variable results in multiplying the dependent variable
by a constant, the data describes an EXPONENTIAL function.
Illustrate these rules on the in class assignment page. Use a multiplier of 2 for the first one and 3 for the second one.
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