Therefore, the transformed function a(x) is given by the following formula.
a(x) = x^2-5
Vertical translations occur when we add or subtract a value from the output of a function. We can observe it more clearly in the table below.
Vertical Translations
Translation up k units, k>0
y=f(x)+ k
Translation down k units, k>0
y=f(x)- k
We can describe our transformation as vertical translation 2 units down. It means that the graph of the given function was shifted 2 units down.
b Using the function f(x), we want to find the expression b(x) = - 2 * f(x). To do this, we will multiply the given function by - 2 and simplify.
Therefore, the transformed function b(x) is given by the following formula.
b(x) = -2x^2+6
Let's name our transformation using the table below.
Reflections
In the x-axis
y=- f(x)
In the y-axis
y=f(- x)
Vertical Stretch or Shrink
Vertical stretch, a>1
y=af(x)
Vertical shrink, 0
We can describe our transformations as a reflection in the x-axis and a vertical stretch by a factor of 2.
c Using the function f(x), we want to evaluate for the given value, c(x) = f( x-2). To do this, we need to substitute x-2 for x in each instance of the x-variable.
Therefore, the transformed function c(x) is given by the following formula.
c(x) = (x-2)^2-3
Horizontal translations occur when we add or subtract a value from the input of a function. We can observe it more clearly in the table below.
Horizontal Translations
Translation right h units, h>0
y=f(x- h)
Translation left h units, h>0
y=f(x+ h)
We can describe this transformation as horizontal translation 2 units right. It means that the graph of the given function was shifted 2 units right.
d Using the function f(x), we want to evaluate for the given value, d(x) = f( 2x). To do this, we need to substitute 2x for x in each instance of the x-variable and simplify.
