By adding some number to every function value,
g(x)=f(x)+k,
its graph is vertically. To instead translate it horizontally, a number is subtracted from the input of the function rule.
g(x)=f(x−h)
The number
h is subtracted and not added, so that a positive
h translates the graph to the right.
Notice that if the quadratic function
f(x)=ax2 is translated both vertically and horizontally, the resulting function is
g(x)=a(x−h)2+k.
This is exactly the of a quadratic function. The of
f(x)=ax2 is located at
(0,0). When the graph is then translated
h units horizontally and
k units vertically, the vertex moves to
(h,k).
A function is in the
x-axis by changing the sign of all function values:
g(x)=-f(x).
Graphically, all points on the graph move to the opposite side of the
x-axis, while maintaining their distance to the
x-axis.
A graph is instead reflected in the
y-axis, moving all points on the graph to the opposite side of the
y-axis, by changing the sign of the input of the function.
g(x)=f(-x)
Note that the is preserved.
A function graph is by multiplying the function rule by some constant
a>0:
g(x)=a⋅f(x).
All vertical distances from the graph to the
x-axis are changed by the factor
a. Thus, preserving any .
By instead multiplying the input of a function rule by some constant
a>0,
g(x)=f(a⋅x),
its graph will be by the factor
a1. Since the
x-value of is
0, they are not affected by this transformation.