Purpose: Self-check your understanding after each topic. This quiz is not graded.
Use results to identify areas needing review before the unit test. For any question you get wrong,
review the corresponding lecture video and try again.
Score: 0 / 12
Topic 1.1 — Power Functions
Question 1
Which of the following is a power function?
Solution:
\( f(x) = 3x^4 \) is a power function because it has the form \( f(x) = ax^n \) where \( a = 3 \) and \( n = 4 \).
• \( g(x) = 2^x \) is exponential (base is constant, exponent is variable).
• \( h(x) = x^3 + 2x \) is a polynomial, but not a single power function.
• \( p(x) = \sqrt{x} + 1 \) is a transformed root function, not a pure power function.
Question 2
What is the end behaviour of \( f(x) = -2x^5 \)?
Solution:
Odd degree (5) with negative leading coefficient (-2):
• As \( x \to \infty \), the large positive \( x^5 \) multiplied by \( -2 \) gives \( f(x) \to -\infty \)
• As \( x \to -\infty \), the large negative \( x^5 \) multiplied by \( -2 \) gives \( f(x) \to +\infty \)
This extends from Quadrant II to Quadrant IV.
Topic 1.3 — Characteristics of Polynomial Functions
Question 3
For \( f(x) = 2x^3 - 5x^2 + x - 7 \), what is the maximum number of turning points?
Solution:
A polynomial of degree \( n \) has at most \( n - 1 \) turning points.
Degree = 3, so maximum turning points = \( 3 - 1 = 2 \).
Question 4
True or False: A polynomial of degree 4 must have exactly 3 turning points.
Solution:
False. A degree 4 polynomial has at most 3 turning points, but it can have 1 or 3.
For example, \( f(x) = x^4 \) has only 1 turning point (at the origin).
The key word is "at most" — not "exactly."
Topic 1.4 — Finite Differences
Question 5
A table of values has constant third differences of 12. What is the degree of the polynomial?
Solution:
If the \( n \)th differences are constant, the polynomial has degree \( n \).
Constant 3rd differences → degree 3.
Question 6
If the leading coefficient of a cubic function is 2, what are the constant third differences? (Assume equally spaced x-values with step 1)
Solution:
For a polynomial of degree \( n \) with leading coefficient \( a \):
$$ \text{Constant } n\text{th differences} = a \cdot n! $$
Here: \( a = 2 \), \( n = 3 \), so \( 2 \times 3! = 2 \times 6 = 12 \).
Topic 1.5 — Graphing from Factored Form
Question 7
For \( f(x) = -2(x+1)^2(x-3) \), what is the y-intercept?
Solution:
The y-intercept is \( f(0) \):
$$ f(0) = -2(0+1)^2(0-3) = -2(1)(-3) = 6 $$
Question 8
In the function \( f(x) = -2(x+1)^2(x-3) \), what happens at \( x = -1 \)?
Solution:
At \( x = -1 \), the factor is \( (x+1)^2 \) — the exponent is 2 (even multiplicity).
At zeros with even multiplicity, the graph bounces off the x-axis (tangent).
At zeros with odd multiplicity (like \( x = 3 \) with exponent 1), the graph crosses.
Topic 1.6 — Even and Odd Functions
Question 9
Is \( f(x) = x^4 - 3x^2 \) even, odd, or neither?
Solution:
Test \( f(-x) \):
$$ f(-x) = (-x)^4 - 3(-x)^2 = x^4 - 3x^2 = f(x) $$
Since \( f(-x) = f(x) \), the function is even.
Even functions have symmetry about the y-axis. Note that all exponents are even.
Question 10
Is \( g(x) = x^3 - x \) even, odd, or neither?
Solution:
Test \( g(-x) \):
$$ g(-x) = (-x)^3 - (-x) = -x^3 + x = -(x^3 - x) = -g(x) $$
Since \( g(-x) = -g(x) \), the function is odd.
Odd functions have rotational symmetry about the origin. All exponents are odd.
Topics 1.8–1.9 — Rates of Change
Question 11
Calculate the average rate of change of \( f(x) = x^2 - 4x + 1 \) on the interval \([1, 3]\).
The instantaneous rate of change of \( f(x) = x^2 - 4x + 1 \) at \( x = 2 \) is closest to:
Solution:
Using the limit of secant slopes approaching \( x = 2 \):
$$ \frac{f(2.01) - f(1.99)}{0.02} = \frac{-2.9999 - (-2.9999)}{0.02} \approx 0 $$
The vertex of this parabola is at \( x = 2 \), so the tangent line is horizontal there.
Instantaneous rate of change at the vertex = 0.
📊 Self-Reflection
Rate your confidence on each topic (1 = need help, 4 = mastered):
Topics rated 1 or 2: Review the corresponding video lectures and retake those questions.