Justify Your Answer (Unit 1)
What is the concavity of a function if the AROC is increasing over consecutive intervals?
Concave Up
Double angle for sin(2x).
2sin(x)cos(x)
Find the inverse: f(x) = ln(x - 5).
f^(-1)(x) = e^x + 5
A population P triples every 5 years. Write an exponential model P(t) if the initial population is 100.
P(t) = 100(3)^(t/5)
What is the maximum value of the function f(x) = 5\sin(x) - 2?
3 (because you start at the midline of -2 and go up by the amplitude of 5.)
Explain the end behavior of -5x^4 + x.
Since the degree is even and the leading coefficient is negative, f(x) --> -infinity as x --> +-infinity
Simplify: (sin(x) + cos(x)^2 - 1
sin (2x) or 2sin(x)cos(x)
Find the inverse: f(x) = (2x - 3)/(x + 4)
f^(-1)(x) = (-4x - 3)(x - 2) (or equivalent)
If a linear regression is performed on the data (x, ln y) and the resulting line is y = 0.5x + 2, what type of function models the original (x, y) data?
An exponential function.
State the amplitude and midline of f(x) = -3\cos(2x) + 5.
Amplitude is 3; Midline is y = 5
Use AROC to explain if f(x) is increasing on [1,3] if f(1)=10, f(3)=5.
Decreasing because the AROC is negative (-2.5).
Solve 2cos^2(x) - 1 = 0 on [0, pi]
pi/4, 3pi/4
If (3, 8) is a point on f(x), what point must be on the graph of f^(-1)(x), and what does this represent graphically?
(8, 3) this represents a reflection over the line y = x
Given a table where x increases by 1 and the second differences of y are constant, what model best fits the data?
A quadratic model.
Why does tan(x) have vertical asymptotes at x = pi/2 + pi(n)?
Because tan(x) = sin(x)/cos(x) and cos(x) = 0 at those values, causing the function to be undefined.
Why does f(x) = (x-2)/(x-2) have a hole, not a V.A. at x=2
The limit as x --> 2, the factor (x-2) is in both the numerator and the denominator.
Solve for cos(2x) using only sin(x) in your answer
1 - 2sin^2(x)
If f(x) has a domain of (2, infinity) and a range of (infinity, 5), what are the domain and range of f^(-1)(x)?
Domain is (-infinity, 5) and Range is (2, infinity). (The inputs and outputs of inverse functions are swapped.
If a semi-log plot (linear (x), log (y)) shows a line with a negative slope, what does this tell you about the original exponential function?
The original function is decreasing (exponential decay).
Find the period of f(x) = sin(pi/4x).
Period = 8 (Because 2pi/(pi/4)).
Is f(x) = x^2 invertible on (-infty, +infty)? Explain.
No. The function is not one-to-one because distinct inputs result in the same output.
Simplify: tan(x))/sec(x)
sin(x)
Solve for f^(-1)(x) given f(x) = 2^(x-1) + 3.
f^(-1)(x) = log(sub2)(x - 3) + 1
In the model y = a dxb^x, if b = 1.07, interpret the rate of change in the context of a percentage.
The output increases by a constant 7% for every 1-unit increase in x.
Write a sine function with a midline of y=10, amplitude of 2, period of 4pi, and no phase shift.
f(x) = 2sin(0.5x) + 10 (or 2sin(frac(1)(2)x) + 10).