416 McGRAW-HILL’S SAT
Lesson 3: Transformations
Functions with similar equations tend to have similar shapes. For instance, functions of the quadratic form f(x) =
ax^2 + bx+ chave graphs that look like parabolas. You also should know how specific changes to the function equation
produce specific changes to the graph. Learn how to recognize basic transformationsof functions: shiftsand reflections.
To learn how changes in function equations produce changes in their graphs, study the graphs below
until you understand how graphs change with changes to their equations.Horizontal Shifts
The graph of y= f(x+ k) is simply the graph of y= f(x) shifted kunits to the left. Similarly, the graph of
y= f(x– k) is the graph of y= f(x) shifted kunits to the right. The graphs below show why.Vertical Shifts
The graph of y= f(x) + kis simply the graph of y= f(x) shifted kunits up. Similarly, the graph of y= f(x) – k
is the graph of y= f(x) shifted k units downward. The graphs below show why.Reflections
When the point (3, 4) is reflected over the y-axis, it be-
comes (3, 4). That is, the xcoordinate is negated.
When it is reflected over the x-axis, it becomes (3, 4).
That is, the ycoordinate is negated. (Graph it and
see.) Likewise, if the graph of y= f(x) is reflected over
the x-axis, it becomes y= f(x).