Explain how each transformations below would alter the graph of f(x).?

Question

1. Explain how each transformations below would alter the graph of f(x). (10 points)
2. f(x + 2):
3. f(x) + 2::
4. Use the f(x) given below to determine the transformed functions. (3 points each)
5. Given: f(x) = 3x + 1,
6. f(x + 2) =
7. f(x) + 2 =
8. Include a graph of each function on the same grid. Here is a link to a graphing grid. (10 points)
9. Find -f(x) for f(x) = 2x - 5. (4 points)

Answer 1

Hello Naomi!

These problems deal with manipulating functions and seeing how those changes affect the produced graph.

For the first part of this problem, we simply seek to determine what happens when we scale the input to a function versus changing the function outright. 2 describes a situation where we replace x with the equation x + 2, where as 2a describes adding 2 to the values produced by our normal function. Since we have only been given f(x), we can assume our graph is just a diagonal line, with a slope of 1 and a y-intercept of 0. Specifically, the equations produced by these operations would be...

1. f(x) = (x + 2)

1a. f(x) = (x) + 2

As we are working with a very simple equation, these will produce the same graph.

For 3, we are given a function to work with; f(x) = 3x + 1. This produces a diagonal line, with a slope of 3 and a y-intercept of 1. When we apply the given operations to the function this time, the results are different - again, in the first instance, we change our inputs by replacing x with (x + 2), whereas in the second, we add something to the output. The equations of the graphs produced would be...

4. f(x) = 3*(x + 2) + 1, which is f(x) = 3x + 6 + 1

4a. f(x) = (3(x) + 1) + 2, which is f(x) = 3x + 3

As you might already expect, these produce slightly different graphs; 4 produces a line with a slope of 3 and a y-intercept of 7, whereas 4a produces a graph with a slope of 3 and a y-intercept of 3. You should be able to graph all of these equations relatively easily, though be sure to label them with their equations.

The final question asks what happens when we multiply an equation by a negative value, which comes down to simple algebra. We can rewrite the equation like this...

-(2x - 5), or -1 * (2x - 5)

We then multiply the -1 through the equation; that is, we multiply each element by -1. This produces the final equation which, is -f(x) = -2x + 5.

Hopefully this aids in your understanding!