Question

How do you find the domain and range of `f(x) = (x + 8)^2 - 7`?

Answer 1

Inspect using the formula `y=a(x-h)^2+k`

Explanation:

From the equation given `f(x)=(x+8)^2-7` we can see that:

h = -8
k = -7

From the original equation `y=x^2`, if h is negative the graph the will shift left or negative x and if k is negative the graph will shift down or negative y.

graph{(x+8)^2-7 [-27.05, 12.95, -8.72, 11.28]}

Since x will keep increasing to infinity regardless of any x-axis transformations the domain will be the same as `y=x^2`

Domain: All real numbers

However, since a minimum applies to the range, if the graph shifts in the y-axis the range will be different from `y=x^2`

Range: y ≥ -7