Inverse Variation
In this type of variation, as one variable increases, the other decreases.
What is inverse variation?
These are the vertical and horizontal asymptotes for the parent function f(x)=1/x.
What are x=0 and y=0?
In the function, this is the value of the parameter h.
g(x)=3(1/{x-4})+1
What is 4?
To find this feature, you must evaluate the function at f(0).
What is the y-intercept?
This is the specific formula used to find the constant of variation (a) when given a coordinate (x, y).
What is a=xy?
These two specific coordinates are the standard "reference points" for the parent function.
What are (-1, -1) and (1, 1)?
This is the horizontal asymptote for the function
g(x)=3(1/{x-1})+2
What is y=2.
This is how you find the horizontal asymptote when the degrees of the numerator and the denominator are equal.
What is the ratio of the leading coefficients (y=a/b)?
To find the x-intercept of a rational function, the shortcut to solve is to set this specific part of the function equal to zero and solve for x.
What is the numerator?
If you are traveling 60 miles, this is the travel time if your speed is 120 mph.
What is 0.5 hours (or 30 minutes)?
This is the domain of the parent function f(x)=1/x.
what is x does not equal 0?
These are the three transformations occurring in the function compared to the parent function.
f(x)=3(1/{x-1})+2
What are a vertical stretch of 3, shift right 1 and shift up 2?
This is the horizontal asymptote for the function
y={6x-2}/{2x+1}
What is y=3?
For the function, this is the x-intercept.
f(x)={x+6}/{x^2-1}
What is (-6, 0)?
Given the coordinate (4, 3), and knowing x and y vary inversely, this is the resulting equation for y.
What is y=12/x?
What are h and k.
This is the mathematical process used to rewrite a function like what you see below in the form given.
y={2x+3}/{x+1}
y=a(1/{x-h})+k
What is long division?
To find the vertical asymptotes of a complicated rational function, you must set this part of the fraction to zero and solve.
What is the denominator?
These are the two vertical asymptotes for the function
f(x)={4x+16}/{x^2-9}
What are x=3 and x=-3?
In the context of travel where distance is the constant, this is the constant of variation if you travel at 30 mph for 2 hours.
What is 60?
This is the formula used to calculate the "positive" transformed reference point using parameters b, h, a, and k.
Using long division, the function rewrites to this specific tranformation form.
y={2x+3}/{x+1}
y=1/{x+1}+2
This is the horizontal asymptote for the function
y={x^4+3x^2+10x-4}/{x-1}
What is none?
For a rational function, these are the three general sections you must test points in to draw an accurate graph.
What are the left, middle and right sections?