Day 7 Homework - answers:
1. How is the sign of the leading coefficient related to the sign of the constant value of the finite
differences? They should be the same.
2. How is the value of the leading coefficient related to the constant value of the finite differences?
Value of leading coefficient = (value of constant finite difference) ÷ (degree!)
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
a) 3
b) negative
-1
10
c) -2
0
1
1
-2
2
-11
3
-38
ii)
x
y
-3
184
a) 4
-2
54
b) positive
-1
12
c) 1
0
4
1
0
2
-6
3
4
1
4. Consider the polynomial function y = x3 + 2x2 – 4x + 5.
a) State the degree of the function. 3
b) State the sign of the leading coefficient. positive
c) State the end behaviour of this function. As x→∞, y→∞, as x→∞, y→-∞
d) What can be determined about the third differences for this function? They will be constant.
e) How do you know that there are no maximum or minimum values on the graph of this
function? The end behaviour is in opposite directions so it is not possible to have a max.
or min.
5. Consider the polynomial function y = -x4 + x2 – 1.
a) State the degree of the function. 4
b) State the sign of the leading coefficient. negative
c) State the end behaviour of this function. As x→∞, y→-∞, as x→∞, y→-∞
d) What can be determined about the fourth differences for this function? They will be
constant.
e) Does the graph of this function have a maximum or minimum value? How do you know
without graphing the function? It has a maximum as it must turn around between the
two ends.
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)? quadratic
b) Determine which of the finite differences would be constant. second
c) Determine the value of the constant finite differences. -2
d) State the restrictions on the domain in this situation. {xεΖ, x≥0}
e) What do the x-intercepts represent in this situation? The number of camera sales that
result in no profit.
f) If 200 cameras are sold, what will be the profit? -38192 dollars (a loss)
g) Will there be a maximum or minimum on the graph of this function? Explain.
There will be a maximum because it is a parabola that opens down.
7. Explain why odd-degree polynomial functions must have at least one x-intercept.
In order to connect the two ends of the graph, which go in opposite directions, it is
necessary to cross the x-axis at least once.
8. Explain why even-degree polynomial functions must have either a maximum or a minimum.
In order to connect the two ends of the graph, which go in the same direction, it is
necessary for the graph to turn around. This will result in a maximum (negative leading
coefficient) or a minimum (positive leading coefficient).
2