ACT Math : How to find transformation for an analytic geometry equation
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Example Questions
Example Question #1 : How To Find Transformation For An Analytic Geometry Equation
If this is a sine graph, what is the phase displacement?
π
0
2π
4π
(1/2)π
0
The phase displacement is the shift from the center of the graph. Since this is a sine graph and the sin(0) = 0, this is in phase.
Example Question #1 : Transformations
If this is a cosine graph, what is the phase displacement?
π
(1/2)π
0
4π
2π
π
The phase displacement is the shift of the graph. Since cos(0) = 1, the phase shift is π because the graph is at its high point then.
Example Question #2 : Transformation
A regular pentagon is graphed in the standard (x,y) coordinate plane. Which of the following are the coordinates for the vertex P?
Regular pentagons have lines of symmetry through each vertex and the center of the opposite side, meaning the y-axis forms a line of symmetry in this instance. Therefore, point P is negative b units in the x-direction, and c units in the y-direction. It is a reflection of point (b,c) across the y-axis.
Example Question #3 : Transformation
If g(x) is a transformation of f(x) that moves the graph of f(x) four units up and three units left, what is g(x) in relation to f(x)?
To solve this question, you must have an understanding of standard transformations. To move a function along the x-axis in the positive direction, you must subtract the value from the operative x-value. For example, to move a function, f(x), five units to the left would be f(x+5).
To shift a function along the y-axis in the positive direction, you must add the value to the overall function. For example, to move a function, f(x), three units up would be f(x)+3.
The question asks us to move the function, f(x), left three units and up four units. f(x+3) will move the function three units to the left and f(x)+4 will move it four units up.
Together, this gives our final answer of f(x+3)+4.
Example Question #4 : Transformation
What is the period of the function?
2π
3π
π
1
4π
4π
The period is the time it takes for the graph to complete one cycle.
In this particular case we have a sine curve that starts at 0 and completes one cycle when it reaches 
Therefore, the period is
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