### 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.

### Vertical Shifts

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.

1. Describe the difference between the two graphs.

2. What do you think the difference between the graphs of f(x) = sqrt(x) and f'(x) = sqrt(x) + 2 will be?

3. Erase the graphs on your screen and graph y = sqrt(x) and y = sqrt(x) + 2. Was your prediction correct?

4. 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.

5. 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.

### Horizontal Shifts

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 that function.

Graph the equation y = sqrt(x).

1. 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)?

2. How do you think the graph of y = sqrt(x+1) will differ from the graph of y = sqrt(x)?

3. Check your answer by graphing y = sqrt(x+1). What change do you observe?

4. 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.

5. In general, what is the effect of adding a constant to the input of a function like sqrt(x) or x2?

6. What do you think the effect of subtracting a constant from the input of a function will be?

7. If g(x) = x2 + x, what is g(x-2)?  (Your answer should be a polynomial in standard form.)

8. 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 calculation.

9. 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.

### Reflections

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).

1. In general, what do you think happens to the graph when you multiply the output of a function by -1? Why?

2. Graph y = sqrt(x) and y = sqrt(-x). What is the effect of multiplying the input of a function by -1?

3. The graph of y = -1(1/(-x)) should be upside-down and backwards. However, it looks exactly like the graph of y = 1/x. Why?

### 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.

### Further Exploration

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.)