Question

PLEASE HELP

In two or more complete sentences, compare the number of x-intercepts in the graph of f(x) = x2 to the number of x-intercepts in the graph of g(x) = (x-2)2 -3. Be sure to include the transformations that occurred between the parent function f(x) and its image g(x).

Answer 1

Ok, for f(x) = x^2 we have only one x-intercept (actually, two equal x-intercepts) at x = 0.

Now, for g(x) = (x - 2)^2 - 3

First, let's analyze the transformations.

When we have g(x) = f(x - a) this means that we moved "a" units to the right (if a is positive)

When we have g(x) = f(x) + a, this means that (if a > 0) we move the graph "a" units up.

In this case we have both those transformations:

g(x) = f(x - 2) - 3

this means that we move 2 units to the right, and 3 units down (because the number is negative)

now we can find the roots of g(x) as:

g(x) = (x - 2)^2 - 3 = x^2 - 4x + 4 - 3 = x^2 - 4x + 1 = 0

using the Bhaskara's equation:

`x = (4 +-√(4^2 - 4*1*1) )/(2*1) = (4 +- 3.5)/(2)`

then the roots are:

x = (4 + 3.5)/2 = 3.75

x = (4 - 3.5)/2 = 0.25

Here we have two different x-intercepts