I need help with transformations of graphs of functions...?

Question

so basically i lost my pre-calc notes, and i need to know a few things about transformations, like how different changes to the equation effects the graph f(x). so far i have:

f(x+1) shifts the graph 1 unit to the left

f(x)+1 shifts the graph 1 unit up

f(-x) reflects the graph on the y-axis

-f(x) reflects the graph on the x-axis

but i don't know how these changes to f(x) affect the graph:

f(-3x)

-3f(x)

|f(x)|

f(|x|)

f(3-x)-4

i really really appreciate any help, partial or full answers! thank you :)

Answer 1

First of all, if you have younger brothers or sisters in K-12, you should advise them to take a full year of trigonometry are receive good algebra skill.  This is far more important than precalculus.  If the high school insists on teaching precalculus, complain to your parents to help your younger relatives.

Second, the four transformations that you do understand apply to three of the five that you question.

What does f(-3x) do to f(x)?

You correctly stated that f(-x) reflects the graph of f(x) on the y-axis.

By transforming from f(-x) to f(-3x), x would only have to go up one third as fast for the same effect to be seen in the y values.  If f(x) = mx + b and f(-x) = -mx + b, then f(-3x) = -3mx + b and the slope of f(-x) is tripled (going up and down faster).  If f(x) is a quadratic equation, f(-3x) will sort of go up and down nine times faster (with 3^2 = 9).

-3f(x) is similar to f(-3x).

f(3-x) -4 can be reached by going from 1) f(x) to f(x - 3) to

2) f(-(x - 3)) = f(3 - x) to 3) f(3 - x) - 4.

Step 1), as you know, shifts the graph 3 units to the right.

Step 2), as you know, reflects the graph of f(x - 3) on the vertical line

where x - 3 = 0.

Step 3), as you know, shifts the graph of f(3 - x) four units down.

If(x)I reflects the portion of the graph of f(x) that is below the x-axis on the

x-axis and leaves the rest untouched.

For f(IxI), copy f(x) for x >= 0 and make a mirror image of this portion of f(x) in quadrants II and III.  Note: f(I-xI) = f(x).