Basic transformations
Vertical and horizontal translations
Vertical translations
Suppose f\left(x\right) is a function and C is a positive real number. Then:
- The graph of the function f\left(x\right)+C is the vertical translation obtained by shifting the graph of f\left(x\right) up by C units.
- A point \left(a,b\right) is on the graph of f\left(x\right) if and only if the point \left(a,b+C\right) is on the graph of f\left(x\right)+C.
Suppose f\left(x\right) is a function and C is a positive real number. Then:
- The graph of the function f\left(x\right)-C is the vertical translation obtained by shifting the graph of f\left(x\right) down by C units.
- A point \left(a,b\right) is on the graph of f\left(x\right) if and only if the point \left(a,b-C\right) is on the graph of f\left(x\right)-C.
Horizontal translations
Suppose f\left(x\right) is a function and C is a positive real number. Then:
- The graph of the function f\left(x-C\right) is the horizontal translation obtained by shifting the graph of f\left(x\right) to the right by C units.
- A point \left(a,b\right) is on the graph of f\left(x\right) if and only if the point \left(a+C,b\right) is on the graph of f\left(x-C\right).
Suppose f\left(x\right) is a function and C is a positive real number. Then:
- The graph of the function f\left(x+C\right) is the horizontal translation obtained by shifting the graph of f\left(x\right) to the left by C units.
- A point \left(a,b\right) is on the graph of f\left(x\right) if and only if the point \left(a-C,b\right) is on the graph of f\left(x+C\right).
Vertical and horizontal reflections
Vertical reflexions
Suppose f\left(x\right) is a function. Then:
- The graph of the function -f\left(x\right) is the vertical reflection obtained by reflecting the graph of f\left(x\right) across the x -axis.
- A point \left(a,b\right) is on the graph of f\left(x\right) if and only if the point \left(a,-b\right) is on the graph of -f\left(x\right).
Horizontal reflexions
Suppose f\left(x\right) is a function. Then:
- The graph of the function f\left(-x\right) is the horizontal reflection obtained by reflecting the graph of f\left(x\right) across the y -axis.
- A point \left(a,b\right) is on the graph of f\left(x\right) if and only if the point \left(-a,b\right) is on the graph of f\left(-x\right).
Stretches and compressions
Suppose f\left(x\right) is a function and C\gt 1. Then:
- The graph of the function Cf\left(x\right) is the vertical translation obtained by stretching the graph of f\left(x\right) vertically by a factor of C.
- A point \left(a,b\right) is on the graph of f\left(x\right) if and only if the point \left(a,Cb\right) is on the graph of Cf\left(x\right).
Suppose f\left(x\right) is a function and 0 \lt C\lt 1. Then:
- The graph of the function Cf\left(x\right) is the vertical translation obtained by compressing the graph of f\left(x\right) vertically by a factor of C.
- A point \left(a,b\right) is on the graph of f\left(x\right) if and only if the point \left(a,Cb\right) is on the graph of Cf\left(x\right).
Suppose f\left(x\right) is a function and 0 \lt C\lt 1. Then:
- The graph of the function f\left(Cx\right) is the horizontal translation obtained by stretching the graph of f\left(x\right) horizontally by a factor of \dfrac{1}{C}.
- A point \left(a,b\right) is on the graph of f\left(x\right) if and only if the point \left(\dfrac{a}{C},b\right) is on the graph of f\left(Cx\right).
Suppose f\left(x\right) is a function and C \gt 1. Then:
- The graph of the function f\left(Cx\right) is the horizontal translation obtained by compressing the graph of f\left(x\right) horizontally by a factor of \dfrac{1}{C}.
- A point \left(a,b\right) is on the graph of f\left(x\right) if and only if the point \left(\dfrac{a}{C},b\right) is on the graph of f\left(Cx\right).
Combining the various transformations of a function allows us to picture the graphs of functions translated from common functions.
Consider the following function:
f\left(x\right)=2\left(x-3\right)^2+1
The function f\left(x\right) is a combination of transformations of the following function:
g\left(x\right)=x^2
The graph of f\left(x\right) is obtained by
- First translation the graph of g\left(x\right) horizontally by 3 units.
- Then stretching the graph vertically by a factor of 2.
- Then translating the graph vertically by 1 unit.
Odd, even and periodic functions
Odd functions
Odd function
A function f\left(x\right) is said to be odd if:
f\left(-x\right)=-f\left(x\right)
The following function is an odd function:
f\left(x\right)=x^3+2x
We have:
f\left(-x\right)=\left(-x\right)^3+2\left(-x\right)\\=-x^3-2x\\=-\left(x^3+2x\right)\\=-f\left(x\right)
The following function is an odd function:
f\left(x\right)=\dfrac{1}{x}
We have:
f\left(-x\right)=\dfrac{1}{-x}\\=-\dfrac{1}{x}\\=-f\left(x\right)
The graph of an odd function is symmetric about the origin. Specifically, if a point \left(a,b\right) is on the graph of an odd function f\left(x\right), then so is \left(-a,-b\right).
The following graph is a graph of an odd function f\left(x\right). It is symmetric about the origin.
Sum of odd functions
Suppose f\left(x\right) and g\left(x\right) are odd functions. Then the sum f\left(x\right)+g\left(x\right) is also an odd function.
The following functions are odd functions:
- f\left(x\right)=x^3+2x
- g\left(x\right)=\dfrac{1}{x}
Therefore the following function is also odd:
f\left(x\right)+g\left(x\right)=x^3+2x+\dfrac{1}{x}
Let f\left(x\right) and g\left(x\right) be two odd functions.
Observe the following:
\left(f+g\right)\left(-x\right)\\=f\left(-x\right)+g\left(-x\right)\\=-f\left(x\right)-g\left(x\right)\\=-\left(f+g\right)\left(x\right)
Therefore \left(f+g\right)\left(x\right) is also an odd function.
Even functions
Even function
A function f\left(x\right) is said to be even if:
f\left(-x\right)=f\left(x\right)
The following function is an even function:
f\left(x\right)=x^4-3x^2+2
We have:
f\left(-x\right)=\left(-x\right)^4-3\left(-x\right)^2+2\\=x^4-3x^2+2\\=f\left(x\right)
The following function is an even function:
f\left(x\right)=|x|
We have:
f\left(-x\right)=|-x|\\=|-1|\cdot |x|\\=1\cdot |x|\\=|x|\\=f\left(x\right)
Graphs of even functions are symmetric about the y -axis. Specifically, if a point \left(a,b\right) is on the graph of an even function f\left(x\right) then so is \left(-a,b\right).
The following graph is the graph of an even function f\left(x\right).
If f\left(x\right) is an even function then the graph of f\left(x\right) is unchanged when reflected across the y -axis.
Sum of even functions
Suppose f\left(x\right) and g\left(x\right) are even functions. Then the sum f\left(x\right)+g\left(x\right) is also an even function.
The functions f\left(x\right)=x^4-3x^2+2 and g\left(x\right)=|x| are even functions. Therefore, the following function is also an even function:
f\left(x\right)+g\left(x\right)=x^4-3x^2+2+|x|
Suppose f\left(x\right) and g\left(x\right) are both even functions. Observe the following:
\left(f+g\right)\left(-x\right)=f\left(-x\right)+g\left(-x\right)=f\left(x\right)+g\left(x\right)=\left(f+g\right)\left(x\right)
Therefore, the sum \left(f+g\right)\left(x\right) is also an even function.
Periodic functions
Periodic functions
A function f\left(x\right) is said to be periodic if there is some real number C such that:
f\left(x+C\right)=f\left(x\right)
The constant C is the period of the periodic function of f\left(x\right).
The following function is a periodic function with period 1 :
f\left(x\right)=x-\lfloor x\rfloor
We have:
f\left(x+1\right)=x+1-\lfloor x+1\rfloor\\=x+1-\left(\lfloor x\rfloor+1\right)\\=x+1-\lfloor x\rfloor-1\\=x -\lfloor x \rfloor\\=f\left(x\right)
The following graph is the graph of f\left(x\right)=x-\lfloor x \rfloor :
Many examples of periodic functions arise as trigonometric functions.
If f\left(x\right) is periodic with period C, then the graph of f\left(x\right) is unchanged when it is shifted horizontally in either direction by C units.