Consider the graphs of the functions y = \sin x and y = \cos x and determine the following:
Amplitude
Period
Range
Midline
Consider the graphs of y = \sin x and \\y = \sin x - 2:
Describe the transformation required to obtain the graph of y = \sin x -2 from \\y = \sin x.
Consider the graphs of y = \cos x and \\y = \cos x + 2:
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.
The function y = \sin x is translated 2 units down.
State the equation of the new function after the translation.
Find the minimum value of the new function.
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.
Consider the graph of y = \cos x + 3.
State the y-intercept.
State the coordinates of the minimum points on the domain -2\pi \leq x \leq 2\pi.
State the equation of the midline.
State the equation of the midline of the graphs of the following functions:
Consider the function y = \cos x + 4.
Describe the transformation required to obtain the graph of y = \cos x+ 4 from y = \cos x.
State the maximum value of the function.
State the minimum value of the function.
Joanna wishes to sketch the graph of y = \sin x + 3.
State the equation of the midline for the graph.
State the coordinates of the y-intercept.
State the coordinates of the maximum point on the domain 0 \leq x \leq 2\pi.
State the coordinates of the minimum point on the domain 0 \leq x \leq 2\pi.
Hence sketch the graph of the function on the domain -2\pi \leq x \leq 2\pi.
The point C with coordinates \left( \pi, 2 \right) lies on the graph of f \left( x \right). State the coordinates of C after the following transformations:
Vertical translation 4 units up.
Vertical translation 3 units down.
Consider the graphs of y = \cos x and \\y = \cos \left(x + \dfrac{\pi}{2}\right) graphed on the same axes:
State the amplitude of the functions.
State the y-intercept of y = \cos x .
How can the graph of y = \cos x be transformed into the graph of \\ y = \cos \left(x + \dfrac{\pi}{2}\right)?
The functions f \left( x \right) and g \left( x \right) = f \left( x + k \right) have been graphed on the same set of axes:
Describe the transformation required to obtain the graph of g\left(x\right) from the graph of f \left( x \right) in terms of k.
Find the smallest positive value of k.
Consider the following graph of a function in the form y = \sin \left(x - c\right):
State the coordinates of the y-intercept of the base function y=\sin x.
For y=\sin (x-c), state the coordinates of the x-intercept closest to the orgin.
Hence determine the equation of the graphed function, where c is the least positive value.
Consider the following graph of a function in the form y = \cos \left(x - c\right):
State the coordinates of the y-intercept of the base function y=\cos x.
For y=\cos (x-c), state the coordinates of the maximum point closest to the orgin.
Hence determine the equation of the graphed function, where c is the least positive value.
The graph of y = \cos x is translated \dfrac{\pi}{3} units to the left. Determine the following attributes of the new function:
Equation
Amplitude
Period
y-intercept
Find the values of c in the domain - 2 \pi \leq c \leq 2 \pi that make the graph of y = \sin \left(x - c\right) the same as the graph of y = \cos x.
The point X with coordinates \left( \pi, 1 \right) lies on the graph of f \left( x \right). State the coordinates of X after the following transformations:
Horizontal translation \dfrac{\pi}{2} units left.
Horizontal translation \pi units right.
Consider the graph of the function y = \sin x for 0 \leq x < 2 \pi.
At which value of x in the given domain would y = - \sin x have a maximum value?
At which value of x in the given domain would y = - \sin x have a minimum value?
Consider the graph of the function y = \cos x for 0 \leq x < 2 \pi.
At which value of x in the given domain would y = - \cos x have a minimum value?
At which value of x in the given domain would y = - \cos x have a maximum value?
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 y-intercept.
Find the maximum value of the function.
Consider the function y = - \cos x + 2.
State the maximum value of the function.
State the minimum value of the function.
State the amplitude of the function.
State the y-intercept.
Describe the two transformations required to obtain the graph of y = -\cos x + 2, from the graph of y = \cos x.
The point A with coordinates \left( \dfrac{\pi}{3}, 4 \right) lies on the graph of f \left( x \right). State the coordinates of A after the following transformations:
Reflection about the x-axis
Compare the graphs of y = 4 \cos x and y = \cos x in terms of:
- period
- amplitude
- maximum
- minimum
State the amplitude of each of the following functions:
y = \cos 3 x
y = 2\sin \left( x - 3 \right)
y = \cos x + 3
y = -3 \sin x
State the equation of the resulting graph if y = \sin x is vertically dilated by a factor of 9 from the x-axis.
State the amplitude of the function f \left( t \right) = - \dfrac{1}{9} \sin t.
Describe the transformation required for the following:
y = \sin x to become y = \dfrac{1}{7} \sin x.
y = \sin x to become y = 5 \sin x.
y = \cos x to become y = \dfrac{1}{4} \cos x.
y = \cos x to become y = 9 \cos x.
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 value of A.
Determine the equation of the graphed function given that it is of the form:
The function y = k \sin x has a maximum value of 5. Find the value of k when:
The point B with coordinates \left( \dfrac{\pi}{2}, 3 \right) lies on the graph of f \left( x \right). State the coordinates of B after the following transformations:
Vertical dilation factor 4
Vertical compression factor \dfrac{1}{3}
Describe the transformation required for the function y = \sin x to become the following:
y = \sin \left( 5 x\right)
y = \sin \left(\dfrac{1}{2} x \right)
y = \sin 8x
y = \sin \left( \dfrac{x}{3} \right)
For each of the following graphs of the form y = \sin b x or y = \cos b x, where b is positive:
State the period.
Determine the equation of the function.
The functions f\left(x \right) and g\left(x \right) = f \left(kx \right) have been graphed on the same set of axes below:
State the period of f \left( x \right).
State the period of g \left( x \right).
Describe the transformation required to obtain the graph of g\left(x\right) from the graph of f \left( x \right).
Hence find the value of k.
The point Y with coordinates \left( \dfrac{\pi}{2}, 4 \right) lies on the graph of f \left( x \right). State the coordinates of Y after the following transformations:
Horizontal dilation factor 2
Horizontal compression factor \dfrac{1}{2}
The graph of y = \sin x undergoes the following series of transformations:
Reflected about the x-axis.
Horizontally translated to the left by \dfrac{\pi}{4} radians.
Vertically translated downwards by 2 units.
Determine the equation of the transformed graph.
The graph of y = \cos x undergoes the following series of transformations:
Reflected about the x-axis.
Horizontally translated to the left by \dfrac{\pi}{6} radians.
Vertically translated upwards by 5 units.
Determine the equation of the transformed graph.
Determine the equation of the new function after performing the following transformations:
The curve y = \sin x is reflected about the x-axis and translated 4 units down.
The curve y = \cos x is translated \dfrac{\pi}{3} units to the right, and then reflected about the x-axis.
For each of the following functions:
Describe the transformations that could be applied to y = \cos x to form the function.
The point \left(0, 1 \right) lies on the graph of y = \cos x. Determine the coordinates of the point, after the transformations have been applied.
Consider the functions f\left(x \right) = \cos x and g\left(x \right) as displayed on the following graph:
Determine whether the following statements are true or false in transforming f\left(x \right) to g\left(x \right):
f\left(x \right) is compressed vertically.
f\left(x \right) is expanded vertically.
f\left(x \right) is compressed horizontally.
f\left(x \right) is expanded horizontally.
f\left(x \right) is shifted up.
f\left(x \right) is shifted down.
f\left(x \right) is reflected about the y-axis.
f\left(x \right) is reflected about the x-axis.
Describe the transformations required to turn the graph of f\left(x \right) into g\left(x \right).
Hence write the equation for g\left(x \right).
Consider the function y = 3 \sin \left(\dfrac{x}{2}\right).
Determine the period of the function in radians.
Find the x-intercepts of the function within the domain 0 \leq x \leq 4 \pi.
Determine the coordinates of the maximum value of the function within the domain \\0 \leq x \leq 4 \pi.
Describe the transformations required for y = \sin x to become y = 3 \sin \left(\dfrac{x}{2}\right).
Consider the graphs of y = \sin x and y = 5 \sin \left(x + \dfrac{\pi}{4}\right).
Describe the series of transformations required to obtain the graph of y = 5 \sin \left(x + \dfrac{\pi}{4}\right) from the graph of y=\sin x.
Consider the graphs of y = \cos x and y = 3 \cos \left(x - \dfrac{\pi}{4}\right).
Describe the series of transformations required to obtain the graph of y = 3 \cos \left(x - \dfrac{\pi}{4} \right) from the graph of y=\cos x.
Describe the series of transformations required to obtain the graph of y = \sin \left(x - \dfrac{\pi}{4}\right) + 2 from the graph of y = \sin \left(x\right).
Consider the functions f \left( x \right) and g \left( x \right) = jf \left( kx \right), graphed on the same set of axes:
Determine the equation for f \left( x \right).
Describe the transformations required to obtain the graph of g \left( x \right) from the graph of f \left( x \right).
Determine the value of j.
Determine the value of k.
Hence state the equation for g \left( x \right).
Consider the functions f \left( x \right) and g \left( x \right) = f \left( x - k \right) - j, graphed on the same set of axes:
Determine the equation for f \left( x \right).
Describe the series of transformations required to obtain the graph of g\left(x\right) from the graph of f\left(x\right).
Determine the value of j.
Determine the smallest positive value of k.
Hence state the equation for g \left( x \right).
Consider the graph of y = \sin x.
The first maximum point for x \geq 0 is indicated at \left(\dfrac{\pi}{2}, 1\right).
By considering the transformation that has taken place, find the coordinates of the first maximum point of each of the given functions for x \geq 0:
y = 5 \sin x
y = - 5 \sin x
y = \sin x + 2
y = \sin 3 x
y = \sin \left(x - \dfrac{\pi}{4}\right)
y = 5 \sin x + 2
Consider the graph of y = \cos x.
Its first maximum point for x \geq 0 is indicated at \left(0, 1\right).
By considering the transformation that has taken place, state the coordinates of the first maximum point of each of the given functions for x \geq 0:
y = \cos \left(x + \dfrac{\pi}{3}\right)
y = 5 \cos \left(x - \dfrac{\pi}{3}\right)
y = 2 - 5 \cos x
y = \cos \left(\dfrac{x}{4}\right)
y = 5 \cos 4 x - 2
y = \cos \left(x - \dfrac{\pi}{3}\right) + 2
Consider the graphs of y = \sin x and y = g \left( x \right) graphed on the same set of axes:
Describe the series of transformations required to obtain the graph of y = g \left( x \right) from the graph of y=\sin x.
Hence state the equation of g \left( x \right) .
Consider the graphs of y = \cos x and y = g \left( x \right) graphed on the same set of axes:
Describe the series of transformations required to obtain the graph of y = g \left( x \right) from the graph of y=\cos x.
Hence state the equation of g \left( x \right) .