7.11 Amplitude, vertical shifts and dilations of trigonometric graphs
Consider the expression \cos \theta.
Complete the table of values for different values of \theta.
| \theta | 0 | \dfrac{\pi}{3} | \dfrac{\pi}{2} | \dfrac{2\pi}{3} | \pi | \dfrac{4\pi}{3} | \dfrac{3\pi}{2} | \dfrac{5\pi}{3} | 2\pi |
|---|---|---|---|---|---|---|---|---|---|
| \cos\theta |
Sketch a graph of the function y = \cos \theta.
State the maximum value of \cos \theta.
State the minimum value of \cos \theta.
State the range of values of 4 \cos \theta.
Consider the functions y =\sin x and y =\cos x, where x is in radians.
State the domain of both functions.
State the range of both functions.
Consider the function y = 2 \cos x, where x is in radians.
State the domain of the function.
State the range of the function.
Consider the given graph of a function of the form f \left( x \right) = A \sin x:
State the amplitude of the function.
State the equation of each of the functions graphed below given that they are of the form y = a \sin x or y = a \cos x:
Consider the function f \left( x \right) = 8 \sin x, where 0 \leq x \leq \pi.
State the amplitude of the function.
Find the value of f \left( \pi \right).
Find the minimum value of the function.
The function y = k \sin x has a maximum value of 5. Find the value of k, where k > 0.
A sine function has the form y = c \sin x, a range of \left[ - 2 , 2\right] and a maximum at \dfrac{\pi}{2}. Find an expression for y.
For each of following functions:
y = 3 \sin x
y = 4 \cos x
y = 5 \cos x
Consider the function y = 3 \cos x.
State the maximum value of the function.
State the minimum value of the function.
State the amplitude of the function.
Describe the transformation required to obtain the graph of y = 3\cos x from the graph of y = \cos x.
Consider the two graphs y = \sin x and y = \sin x - 2 below:
Describe the transformation required to obtain the graph of y = \sin x -2 from y = \sin x.
Consider the two graphs y = \cos x and y = \cos x + 2 below:
Describe the transformation required to obtain the graph of y = \cos x + 2 from y = \cos x.
Describe the transformation required to obtain the graph of y = \sin x +4 from y = \sin x.
State the equation of each of the functions graphed below given that they are of the form y=\sin x + k:
State the equation of each of the functions graphed below given that they are of the form y=\cos x + k:
State the equation of the line that the function y = \sin x + 2 oscillates about.
Consider the function y = \cos x + 4.
Describe the transformation required to obtain the graph of y = \cos x+ 4 from y = \cos x.
Find the maximum value of the function.
Find the minimum value of the function.
For each of the functions below:
State the period of the function, giving your answer in radians.
State the amplitude of the function.
Find the maximum value of the function.
Find the minimum value of the function.
Sketch a graph of the function for -2 \pi \leq x \leq 2 \pi.
For each of the functions below:
Find the value of y when x = \dfrac{\pi}{2}.
State the amplitude of of the function.
Find the maximum value of the function.
Find the minimum value of the function.
Describe the transformations required to obtain the graph of the function from y = \cos x.
Sketch a graph of the function for 0 \leq x \leq 2 \pi.
The function y = \cos x + 5 is translated 4 units up.
State the equation of the new function after the translation.
Find the maximum value of the new function.
A sine function, y, has the form y = c \sin x + d and a range of \left[0, 4\right]. Find an expression for y, where c \gt 0.
A cosine function, y, has the form y = c \cos x - d and a range of \left[ - 10 , 6\right]. Find an expression for y, where c \gt 0.
Consider the given graph of f \left( x \right) = 3 \tan x:
State the domain of f \left( x \right).
State the range of f \left( x \right).
The graph of f \left( x \right) has its domain restricted to \left[0, \pi\right), state the range of the restricted graph.
State the equation of each of the functions graphed below given that they are of the form y=a\tan x:
Consider the given graph of f \left( x \right) = 2 \tan x + 1:
State the domain of f \left( x \right).
State the range of f \left( x \right).
Consider the given graph of f \left( x \right) = \tan x + 3:
State the domain of f \left( x \right).
State the range of f \left( x \right).
The graph of f \left( x \right) has its domain restricted to \left[0, \dfrac{\pi}{2}\right), state the range of the restricted graph.
For each of the functions below:
Find the y-intercept.
Find the value of y when x = \dfrac{\pi}{4}.
Find the period of the function.
Find the distance between the asymptotes of the function.
State the equation of the first asymptote of the function for x \geq 0.
State the equation of the first asymptote of the function for x \leq 0.
Sketch a graph of the function.
Consider the given graph of a tangent function:
State the transformation required to obtain the given function from y=\tan x
State the equation of the graphed function.
Sketch the following functions for -2\pi \leq x \leq 2\pi:
y = \vert 4 \sin x \vert
y = \vert 3 \cos x \vert
y = \vert 4 \cos x \vert
Sketch the following functions for 0 \leq x \leq 2\pi:
Sketch the following functions for 0 \leq x \leq 2\pi:
Explain why the graphs of the functions y=\sin x + 3 and y= \vert \sin x + 3 \vert are the same.