A kosarad üres
Systematisation of function transformations
We got familiar with function transformations through specific examples.
In the next pages – just as before – we are dealing with functions which map from a subset of the set of real numbers to the set of real numbers. We do not specify their domain, we always assume that the functions have a meaning at the places in question.
The function transformations already seen with the specific functions can basically be divided into two groups.
Value transformations
We know the function \latex{ f } with its graph.
- Let the following be fulfilled for function \latex{ f_{1}:f_{1}(x)=f(x)+c }.
The domain of the function \latex{ f_{1} } is the same as the domain of function \latex{ f }. The graph of \latex{ f_{1} } is derived from the graph of \latex{ f } by shifting/translating it along the \latex{ y- }axis by \latex{ c }, in the case of positive \latex{ c } upwards, i.e. into the positive direction, in the case of negative \latex{ c } downwards, i.e. into the negative direction. (Figure 89)
(Figure 89)
\latex{y=f\left(x\right)}
\latex{f_{1} \left(x\right) =f\left(x\right) +c}
\latex{y=f_{1}\left(x\right) }
\latex{y=f\left(x\right) }
\latex{ y }
\latex{ y }
\latex{ x }
\latex{ x }
\latex{ c }
\latex{ c }
\latex{ c }
\latex{ c }
- Let \latex{ f_{2}(x)=-f(x) } .
The graph of the function \latex{ f_{2} } is derived from the graph of \latex{ f } by reflecting it about the \latex{ x- }axis. (Figure 90)
(Figure 90)
\latex{y=f\left(x\right) }
\latex{f_{2} \left(x\right) =-f\left(x\right)}
\latex{y=f\left(x\right) }
\latex{y=f_{2}\left(x\right) }
\latex{ x }
\latex{ y }
\latex{ y }
\latex{ x }
- Let c be a positive number, and \latex{ f_{3}(x)=c\times f(x) } .
The graph of \latex{ f_{3} } is derived from the graph of \latex{ f } by changing the \latex{ y- }coordinate of every point of the curve \latex{ c } times greater while leaving the \latex{x-}coordinate unchanged. (Figure 91)
(Figure 91)
\latex{y =f\left(x\right)}
\latex{y =f_{3}\left(x\right)}
\latex{y =f\left(x\right)}
\latex{f_{3}=c\times f\left(x\right) }
\latex{ y }
\latex{ x }
\latex{ x }
\latex{ y }
- Let \latex{ f_{4}(x)=|f(x)| }.
The graph of \latex{ f_{4} } is derived from the graph of \latex{ f } by reflecting the part of the curve which is below the \latex{ x }-axis, i.e. where \latex{ f } takes negative values, about the \latex{ x }-axis. We leave unchanged the curve part where \latex{ f } takes non-negative values. (Figure 92)
(Figure 92)
\latex{y=f\left(x\right) }
\latex{f_{4}=\left|f\left(x\right) \right| }
\latex{y=f_{4}\left(x\right) }
\latex{y=f\left(x\right) }
\latex{ x }
\latex{ y }
\latex{ y }
\latex{ x }
We can observe for these four function transformations that the original graph was changed “in the direction of the \latex{ y }-axis”, i.e. we transformed the values of the original function. This is why these are called value transformations.
Variable transformations
- Let \latex{ f_{5}(x)=f(x-c) }.
The graph of \latex{ f_{5} } is derived by shifting the graph of \latex{ f } along the \latex{ x }-axis by \latex{c}, for positive \latex{ c } values into the positive direction, for negative \latex{ c } values into the negative direction. (Figure 93)
(Figure 93)
\latex{y=f\left(x\right) }
\latex{f_{5} \left(x\right) =f\left(x-c\right) }
\latex{y=f_{5}\left(x\right) }
\latex{y=f\left(x\right) }
\latex{ c }
\latex{ c }
\latex{ c }
\latex{ c }
\latex{ y }
\latex{ y }
\latex{ x }
\latex{ x }
- Let \latex{ f_{6}(x)=f(-x) }.
The graph of the function \latex{ f_{6} } can be derived from the graph of\latex{ f } by reflecting it about the \latex{ y }-axis. (Figure 94)
(Figure 94)
\latex{y=f\left(x\right) }
\latex{f_{6}\left(x\right)=f\left(-x\right) }
\latex{y=f\left(x\right) }
\latex{y=f_{6}\left(x\right) }
\latex{ y }
\latex{ x }
\latex{ x }
\latex{ y }
- Let \latex{ c } be a positive number, and \latex{ f_{7}(x)=f(c\times x) }
The graph of \latex{ f_{7} } can be derived from the graph of \latex{ f } by shrinking it perpendicular to the \latex{ y }-axis (i.e. in the direction of the \latex{ x }-axis) to times. It means that the \latex{ x }-coordinate of every curve point changes to \latex{\frac{1}{c} } times, its \latex{ y } -coordinate stays unchanged. (Figure 95)
(Figure 95)
\latex{y=f\left(x\right)}
\latex{y=f_{7}\left(x\right)}
\latex{y=f\left(x\right)}
\latex{f_{7} \left(x\right) =f\left(c\times x\right) }
\latex{ x }
\latex{ x }
\latex{ y }
\latex{ y }
- Now we assume that the domain of f contains positive numbers too. Let \latex{ f_{8}(x)=f(|x|) }.
The graph of the function \latex{ f_{8} } can be derived from the graph of \latex{ f } by reflecting the curve part belonging to the places \latex{x\geq 0} about the \latex{ y }-axis, and we omit the part belonging to the negative \latex{ x } places. (Figure 96)
(Figure 96)
\latex{y=f\left(x\right)}
\latex{y=f_{8}\left(x\right)}
\latex{y=f\left(x\right)}
\latex{f_{8}\left(x\right) =f\left(\left|x\right| \right) }
\latex{ y }
\latex{ x }
\latex{ x }
\latex{ y }
We can observe for the latter four function transformations that the original graph was changed “in the direction of the \latex{ x }-axis”, i.e. we transformed the elements of the domain of the original function. This is why these are called variable transformations.
