Math 115--Precalculus - University of Maryland

Transformations of Graphs

Translations:
Look at the table in the translations worksheet and compare the graphs of y = x²; y = |x| and y = x^1/2 in the columns for f(x-1), f(x-2), f(x+1), f(x+2), f(x)-1, f(x)-2, f(x)+1, f(x)+2. Note what is common to each of the graphs in a column when you compare the graph to the original graph. How is the change in the function (e.g., from y = x² to y = (x + 2)²) related to the change in the graph?

Conclusion: If h > 0, then the graph of

y = f(x + h) is graph of f moved right h units
y = f(x - h) moves graph of f left h units
y = f(x) + h moves graph of f up h units (same as y - h = f(x))
y = f(x) - h moves graph of f down h units

Let f(x) = x² - x + 1. Note that f(x) = (x - 1/2)² + 3/4.

Draw the graph of f.
Write a rule for a function g whose graph is obtained from the graph of f by shifting the graph horizontally 5 units to right and vertically 3 units downward

Let f(x) = |x| or f(x) = x². Use ZDecimal window; TRACE to locate an x-value with corresponding to a nonzero value of y; look at the y values for both f and g below). Compare the graphs of f and g.

Stretching and Shrinking, Reflections
What happens if we graph

g(x) = 3 f(x)?
g(x) = .5 f(x)?
g(x) = -f(x)?
g(x) = -6f(x)?

The graph of y = -f(x) is the reflection of the graph of f in the x-axis (every point (x,y) of the graph of f is replaced with the point (x, -y) for the graph of -f(x)).

If g(x) = cf(x) with c > 0, then the graph of g is the graph of f stretched vertically away from the x-axis by a factor of a if c > 1 and shrunk vertically toward the x-axis by a factor of c if c < 1. (Do an example with c = 3, c = 1/2 and f the function in the graph shown.

  |
  |  ______  (y = 1)
  | /      \
__|/________\___
    1     4 5

If g(x) = f(-x), then the graph of g is the graph of f reflected in the y-axis because every point (x,y) on the graph of f is replaced with the point (-x, y) for the graph of f. Draw the graph f(-x) if f is the function in the graph shown. Also try y = x^1/2 , y = (-x)^1/2

Questions and/or comments should be sent to Frances Gulick.
Last modified Sunday, 10-Sep-2000 21:51:18 EDT © 2009 University of Maryland