Which statements is true? For those that are false, provide a counterexample.
A. All odd-degree polynomial functions are odd functions.
B. Even-degree polynomial functions have an even number of x-intercepts.
C. Odd-degree polynomial functions have at least one x-intercept.
D. All even-degree polynomial functions are even.
Which statement is true? For those that are false, provide a counterexample.
A. A polynomial function with constant third differences has degree four.
B. A polynomial function with a negative leading coefficient may extend from quadrant 3 to quadrant 4.
C. A power function with even degree has point symmetry.
D. A polynomial function with four x-intercepts has degree four.
Which statement is true? For those that are false, provide a counterexample.
A. A vertical compression of factor \displaystyle
\frac{1}{3}
is the same as a horizontal stretch of factor 3.
B. When applying transformations, translations are applied before stretches and compressions.
C. Stretches must be applied before compressions.
D. A negative k-value in y = a[k(x - d)]^n + c results in a reflection in the x-axis.
E. The equation of a transformed polynomial function can be written in the form y = a[k(x - d)]^n + c.
Match each graph of a polynomial function with the corresponding equation. Justify your choice.
i) f(x) = -2x^3 + 7x + 1
ii) h(x) = x^5 -7x^3 + 2x + 1
iii) p(x) =-x^6 + 5x^2 + 1
For each polynomial function below, answer the following.
- i)
f(x) = -2x^3 + 7x + 1 - ii)
h(x) = x^5 -7x^3 + 2x + 1 - iii)
p(x) =-x^6 + 5x^2 + 1
a) Which finite differences are constant?
b) Find the values of the constant finite differences.
c) Identify any symmetry. Verify your answer algebraically.
A quartic function has zeros -1, 0, and 3 {order 2).
a) Write equations for two distinct functions that satisfy this description.
b) Determine an equation for a function satisfying this description that passes through the point (2, -18).
c) Sketch the function you found in part b). Then, determine the intervals on which the function is positive and the intervals on which it is negative.
Determine an equation for the polynomial function that corresponds to this graph.
Identify the parameters a, k, d, and c in the polynomial function \displaystyle
y = \frac{1}{3}[-2(x + 3)]^4
a) Describe how each parameter transforms the base function y =x^4
b) State the domain and range, the vertex, and the equation of the axis of symmetry of the transformed function.
c) Describe two possible orders in which the transformations can be applied to the graph of y = x^4 to produce the graph of \displaystyle
y = \frac{1}{3}[-2(x + 3)]^4 -1
d) Sketch graphs of the base function and the transformed function on the same set of axes.
Transformations are applied to y = x^3 to obtain the graph shown. Determine its equation.
Describe a real-life situation that corresponds to
a) a constant, positive average rate of change
b) a constant, negative average rate of change
c) a non-constant average rate of change
d) a zero average rate of change
In 1990, 15.5% of households had a CD player, while 76.1% of households had a CD player in 2003. Determine the average rate of change of the percent of households that had a CD player over this time period.
An oil tank is being drained. The volume, V, in litres, of oil remaining in the tank after
t minutes can be modelled by the function V(t) = 0.2(25 - t)^3, where t \in [0, 25].
a) How much oil is in the tank initially?
b) Determine the average rate of change of volume during
\displaystyle
\begin{array}{llll}
&i) \text{ the first 10 min} & &ii) \text{ the last 10 min}
\end{array}
c) Compare the values you found in part b). Explain any similarities and differences.
d) Sketch a graph to represent the volume.
e) What do the values found in part b) represent on the graph?
The distance, d, in metres travelled by a boat from the moment it leaves shore can be modelled by the function
d(t) = 0.002t^3 + 0.05t^2 + 0.3t,
where t is the time, in seconds.
a) Determine the average rate of change of the distance travelled during the first 10 s after leaving the shore.
b) Estimate the instantaneous rate of change 10 s after leaving the shore.