Day 7: Characteristics of Polynomial Functions – 2
Finite Differences
First Differences:
Differences between consecutive y-values in tables of values with evenly spaced
x-values.
Second Differences: Differences between consecutive first differences.
The set of all first, second, third (and so on) differences is called the finite differences.
Example 1:
a) Create a table of values for y = 2x + 1. Calculate the first differences.
x
-3
-2
-1
0
1
2
3
y
1st differences
b) Create a table of values for y = x2. Calculate the first and second differences.
x
-3
-2
-1
0
1
2
3
y
1st differences
2nd differences
c) Create a table of values for y = x3. Calculate the first, second and third differences.
x
-3
-2
-1
0
1
2
3
y
1st differences
2nd differences
3rd differences
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d) Create a table of values for y = -2x3. Calculate the first, second and third differences.
x
-3
-2
-1
0
1
2
3
y
1st differences
2nd differences
3rd differences
Example 2:
a) What is true about the first differences of a linear function?
b) What is true about the second differences of a quadratic function?
c) What is true about the third differences of a cubic function?
d) How is the leading coefficient related to the constant value of the finite differences?
Example 3:
x
-3
-2
-1
0
1
2
3
Consider the table of values shown.
y
149
56
13
2
5
4
-19
Use finite differences to determine
a) the degree of the polynomial function.
b) the leading coefficient.
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Maximum and Minimum Values
A graph has a maximum value if there is a y-value that is greater than (or equal to) all other y-values on
the graph.
A graph has a minimum value if there is a y-value that is smaller than (or equal to) all other y-values on
the graph.
Example 4:
Review the graphs from Days 4 and 5 to answer the following questions.
a) Which type of graphs have maximum values?
b) Which type of graphs have minimum values?
c) Which type of graphs have neither maximum nor minimum values?
Example 5:
Use the graphing calculator to determine the maximum/minimum value for each graph
and the x-value where it occurs.
a) y = -x4 + 7x – 3
b) y = x2 + 5x - 6
Day 7 Homework:
1. How is the sign of the leading coefficient related to the sign of the constant value of the finite
differences?
2. How is the value of the leading coefficient related to the constant value of the finite differences?
3. Use finite differences to determine
a) the degree of the polynomial function.
b) the sign of the leading coefficient.
c) the value of the leading coefficient.
i)
x
y
-3
94
-2
37
-1
10
0
1
1
-2
2
-11
3
-38
ii)
x
y
-3
184
-1
54
-2
12
0
4
1
0
2
-6
3
4
3
4. Consider the polynomial function y = x3 + 2x2 – 4x + 5.
a) State the degree of the function.
b) State the sign of the leading coefficient.
c) State the end behaviour of this function.
d) What can be determined about the third differences for this function?
e) How do you know that there are no maximum or minimum values on the graph of this
function?
5. Consider the polynomial function y = -x4 + x2 – 1.
a) State the degree of the function.
b) State the sign of the leading coefficient.
c) State the end behaviour of this function.
d) What can be determined about the fourth differences for this function?
e) Does the graph of this function have a maximum or minimum value? How do you know
without graphing the function?
6. Navin has opened a new store to sell digital cameras. He determines that the monthly profit, P,
in thousands of dollars, for the sale of digital cameras can be modeled by the function
P(x) = -x2 + 9x – 8, where x represents the number, in hundreds, of cameras sold.
a) What type of function is P(x)?
b) Determine which of the finite differences would be constant.
c) Determine the value of the constant finite differences.
d) State the restrictions on the domain in this situation.
e) What do the x-intercepts represent in this situation?
f) If 200 cameras are sold, what will be the profit?
g) Will there be a maximum or minimum on the graph of this function? Explain.
7. Explain why odd-degree polynomial functions must have at least one x-intercept.
8. Explain why even-degree polynomial functions must have either a maximum or a minimum.
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To find maximum and minimum values (extrema) and zeros using the Graphing Calculator:
Once a function is graphed, options in the CALC menu (under TRACE) can be used to
locate maximum and minimum values and real zeros (x-intercepts) of the function.
For instance, consider the quadratic function given by Y1 = -3x2 – 4x + 1 .
To find the maximum of the function, have the calculator graph the function in the
viewing window. Then hit 2nd CALC 4: maximum.
The calculator prompts you for a left bound. Enter an x-value to the left of the maximum
(or TRACE to the left of the maximum), and then hit ENTER. The calculator then asks
for a right bound. Enter an x-value to the right of the maximum (or TRACE to the right
of the maximum), and then hit ENTER.
The calculator next asks for a guess. Enter an x-value near the
maximum, and hit ENTER. The coordinates for the approximate
maximum are given.
Using –1 and 0 as left and right bounds for the given quadratic function, the maximum
value given by the calculator is 2.3333333, and it occurs when x is -0.6666662. [Here, the
calculator is off a bit; the exact maximum point, or vertex, can be proven with algebra or calculus


to be -0.6, 2.3 .]
To find the minimum (local minimum), use option 3: minimum in the CALC menu.
To find the zeros (x-intercepts) of a function, use option 2: zero in the CALC menu.
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