Question

How do you sketch the graph of `y=x^2-5` and describe the transformation?

Answer 1

See below

Explanation:

If you know the graph of a function `y=f(x)`, then you can have four kind of transformations: the most general expression is

`A f(wx+h)+v`

where:

• `A` multiplies the whole function, thus stretching it vertically (expansion if if `|A|>1`, contraction otherwise)
• `w` multiplies the input variable, thus stretching it horizontally (expansion if if `|w|<1`, contraction otherwise)
• `h` and `v` are, respectively, horizontal and vertical translations.

In your case, starting from `f(x)=x^2`, you have `A=1` and `w=1`. Being multiplicative factors, they have non effect.

Moreover, `h=0`. Being ad additive factor, it has non effect.

Finally, you have `v = -5`. This means that, if you start from the "standard" parabola `f(x)=x^2`, the graph of `f(x)=x^2-5` is the same, just translated `5` units down.