MATH 100 Transformation Activity
Names of people you worked with:
Work in groups of two or three to fill in answers to the questions
below. Each group member should contribute equally to answering the
questions. This will count as one quiz grade.
The purpose of this activity is to observe how changes in the equation
of a function affect the graph of the function. By the end of the
activity, all students should understand how to change the equation of
a function to shift or reflect its graph; advanced students may also
explore nonrigid transformations.
In this exercise, we explore how adding a constant value to a
function's output affects the graph of that function.
Graph y = x^2.
Without erasing the graph of y = x^2, graph y = x^2 + 1.
- Describe the difference between the two graphs.
- What do you think the difference between the graphs of f(x) =
sqrt(x) and f'(x) = sqrt(x) + 2 will be?
- Erase the graphs on your screen and graph y = sqrt(x)
and y = sqrt(x) + 2. Was your prediction correct?
- In general, how do you think adding a constant to the output of a function affects the
graph of the function? Write your conjecture below. Then
erase the graphs on your screen and test your conjecture on the
graphs of g(x) = 1/x and g'(x) = 1/x + 1.
- How do you think changing a function by subtracting a constant
will affect the graph of the function? Test your conjecture by
graphing the functions h(x) = x3 and h'(x) =
x3 - 2.
In the last exercise we learned how changing a function's
output by adding or subtracting a constant value affected the
graph of that function. In this exercise we find out how adding a
constant value to a function's input affects the graph of
Graph the equation y = sqrt(x).
- What are the allowable inputs to (the domain of) the function f(x) = sqrt(x)?
What is the domain of h(x) = sqrt(x+1)?
- How do you think the graph of y = sqrt(x+1) will
differ from the graph of y = sqrt(x)?
- Check your answer by graphing y = sqrt(x+1). What
change do you observe?
- How do you think the graph of g(x) = (x+3)2
will differ from the graph of f(x) = x2? Check your answer by graphing.
- In general, what is the effect of adding a constant to the
input of a function like sqrt(x) or x2?
- What do you think the effect of subtracting a constant from the
input of a function will be?
- If g(x) = x2 + x, what is g(x-2)? (Your answer
should be a polynomial in standard form.)
- Check your answers to the previous two problems by graphing
g(x) and g(x-2). Are the graphs related as you
predicted? If not, find the error in your prediction or
- Graph the equation y = x3 - 4x. What changes
must you make to the equation to shift the graph to the left by 2
units? Write your "shifted" equation below and check your work by
graphing. (Here you need not simplify your equation.) If you're still not sure of your
answer, ask other students what they got.
You've explored changes in a function's graph caused by adding to
or subtracting from the function's input and output. What happens if
you multipy the input by -1? What if you multiply the output
by -1? Your answers to these questions will be related to what you
have learned about even and odd functions.
Use your computer to graph y = x^2
and y = -1(x^2).
- In general, what do you think happens to the graph when you
multiply the output of a function by -1? Why?
- Graph y = sqrt(x) and y = sqrt(-x). What is the
effect of multiplying the input of a function by -1?
- The graph of y = -1(1/(-x)) should be upside-down and
backwards. However, it looks exactly like the graph of y =
Putting it into Practice
You have just learned how changes to the equation of a function affect
the graph of the function. Next you will practice predicting the
equation of a function by looking at the graph of the function. Use
the list of common functions and their graphs from your text to guess
the equations that generated each of the graphs shown below. Check
your work by graphing.
Once you understand how the graph of a function changes as you add
or subtract values to the input or output, see if you can describe the
changes that occur when you multiply the input or output by a
constant. What if you divide by a constant? (Answering
these questions will earn you up to 5 bonus points.)