Advanced Geometry : How to find transformation for an analytic geometry equation
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Example Questions
Example Question #3 : Transformation
Let f(x) = x3 – 2x2 + x +4. If g(x) is obtained by reflecting f(x) across the y-axis, then which of the following is equal to g(x)?
x3 + 2x2 + x + 4
–x3 + 2x2 – x + 4
–x3 – 2x2 – x + 4
–x3 – 2x2 – x – 4
x3 – 2x2 – x + 4
–x3 – 2x2 – x + 4
In order to reflect a function across the y-axis, all of the x-coordinates of every point on that function must be multiplied by negative one. However, the y-values of each point on the function will not change. Thus, we can represent the reflection of f(x) across the y-axis as f(-x). The figure below shows a generic function (not f(x) given in the problem) that has been reflected across the y-axis, in order to offer a better visual understanding.
Therefore, g(x) = f(–x).
f(x) = x3 – 2x2 + x – 4
g(x) = f(–x) = (–x)3 – 2(–x)2 + (–x) + 4
g(x) = (–1)3x3 –2(–1)2x2 – x + 4
g(x) = –x3 –2x2 –x + 4.
The answer is –x3 –2x2 –x + 4.
Example Question #21 : Transformations
What is the period of the function?
1
π
4π
3π
2π
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
Example Question #21 : How To Find Transformation For An Analytic Geometry Equation
Explain how the below function translates:
5 units up, 7 units left
5 units right, 7 units down
5 units left, 7 units down
5 units down, 7 units right
5 units left, 7 units down
When estimating the translations for a quadratic function we must remember what vertex form for a parabola looks like:
In order to end up with:
We must have the below to end up with a positive 5:
This is what gives us the translation left 5 spaces and down 7 spaces.
Example Question #22 : How To Find Transformation For An Analytic Geometry Equation
Assume we have a triangle, 
If 
When we reflect a point across the line, 
The correct answer is the following:
The other answer choices are incorrect because we only use the negatives of the coordinate points if we are flipping across either the x- or y-axis.
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