MHF 4U Final Exam Review
Chapter 2: Rational Functions
Knowledge/Understanding:
2K1. Find the vertical asymptotes, x-intercepts, y-intercepts and the horizontal asymptotes of the function .
2K2. Create an equation with a vertical asymptote at , and a horizontal asymptote
at .
2K3. Create a function with vertical asymptotes at x = -4 and x = 5, and x intercepts at x = 4 and x = 2, and a horizontal asymptote at y = -1.
2K4. A particle’s speed can be modeled by the equation . Find the average rate of change between 0.5s and 4.0s, and estimate the instantaneous rate of change at 4.0s.
Application:
2A1. Sketch the graph of the function f(x) = .
2A2. Sketch the graph of the function h(x) = .
2A3. Sketch the graph of the function g(x) = .
2A4. Sketch the graph of the function m(x) = .
2A5. A large tank contains 1000 L of pure water. Salt water that contains 20 g of salt per L pumped into the tank at 10L/min.
a) Create an equation that expresses the concentration of salt, C, in g/L as a function of time, t, in metres.
b) What happens as t gets very large?
2A6. Solve
a) b)
2A7. Solve:
a) b)
Communication:
2C1. What is a rational function? Give an example. How does the graph of a rational function differ from the graph of a polynomial function?
2C2. Explain the different types of asymptotes of a rational function.
2C3. What is the difference between the graph of f() = and the graph of
g() =
2A4. Explain why the line y = x is an asymptote for y =.
TIPS:
2T1. Find constants a and b that guarantee that graph of will have a vertical asymptote at and a horizontal asymptote at y = -2.
Chapter 2 Answers:
Knowledge/Understanding Answers:
2K1. vertical asymptotes: x = -2, x = -3
horizontal asymptotes: y = 0
x-intercept: (2, 0)
y-intercept: (0, -1/3)
2K2.
2K3. Answers may vary. Example:
2K4.
speed at 0.5s:
speed at 4.0s:
average rate of change:
=
=
instantaneous rate of change:
speed at 4.0s:
Application Answers:
2A1, 2A2, 2A3, 2A4. Verify using a graphing calculator.
2A5. a)
b) Find the horizontal asymptote. Since the degrees are the same, divide the coefficients of t. Therefore as t gets very large, the concentration of salt approaches 20g/L.
2A6. when –3 <x < -1 or 1 < x < 3
2A7. when
Communication Answers:
2C1. A rational function has the form , where f(x) and g(x) are polynomials, and g(x)0. The domain of a rational function consists of all real numbers except the zeros of the polynomial in the denominator.
The graph of a rational function has at least one asymptote. The graph of a polynomial function does not have any asymptotes.
2C2. vertical asymptote – occurs where the function is undefined, that is, at the value(s) of x that makes the denominator equal to zero
horizontal asymptote – by comparing the degrees of the polynomial numerator (n) and the denominator (d)
1) if n < d, the horizontal asymptote is y = 0
2) if n = d, the horizontal asymptote is the ratio of the leading coefficients
3) if n > d, there is no horizontal asymptote
oblique asymptote – occurs when the degree of the numerator is one more than the degree of the denominator
2C3. The function f() is not defined at the value which is represented by a hollow dot. The function g() is a linear graph.
2C4. It has an oblique asymptote because the degree in the numerator is one greater than denominator. The equation can be found through dividing the numerator by the denominator.
TIPS Answers:
2T1. x intercept: (0,0). y-intercept: (0,0). Vertical asymptote: x=2 or x=. Horizontal asymptote: y=0.
2T2.