Question

The graph of f(x) = sqrt (16-x^2) is shown below. How do you sketch the graph of the function y = 3f(x)-4 based on that equation (sqrt (16-x^2)?

Answer 1

We start with the graph of `y = f(x)`:
graph{sqrt(16-x^2) [-32.6, 32.34, -11.8, 20.7]}
We then will do two different transformations to this graph—a dilation, and a translation.

The 3 next to `f(x)` is a multiplier. It tells you to stretch `f(x)` vertically by a factor of 3. That is, every point on `y = f(x)` gets moved to a point that's 3 times higher. This is called a dilation.

Here's a graph of `y = 3f(x)`:
graph{3sqrt(16-x^2) [-32.6, 32.34, -11.8, 20.7]}

Second: the `-4` tells us to take the graph of `y=3f(x)` and move every point down by 4 units. This is called a translation.

Here is a graph of `y = 3f(x) - 4`:
graph{3sqrt(16-x^2)-4 [-32.6, 32.34, -11.8, 20.7]}

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Quick method:

Fill in the following table for a few values of `x`:

`x"   |   "f(x)"   |   "3f(x)-4`
`"———————————"`
`"     |              |"`
`"     |              |"`
`"     |              |"`
`"     |              |"`

Then, plot `x` vs. `3f(x)-4` by plotting their pairs and connecting the dots.