compared to the graph of f(x) = x2 , the graph of g(x) = (x-2)2 + 3 is the result of translating f(x)

Answer 1

Hi John!

I assume this question is asking how the graph is translated.

If the original function is f(x) = x² then it's what's called the "parent function" which just means it's a function with no transformations. The parent function actually takes the form of f(x) = a(x-h)² + k.

Therefore, the other way we can see this parent function is f(x) = 1(x-0)² + 0 where h and k are both 0 and a is 1.

"a" determines the width of the parabola and whether it is facing up or down.

"h" determines if the vertex (the bottom or top point) of the parabola is moving left or right and by how much

"k" determines if the vertex of the parabola is moving up or down

Always think "opposite for h, regular for k". What that means is that if you have (x-5)² that means that the vertex is moving to the right 5 units and if you have (x+5)², the vertex is moving to the left 5 units. This is because the parent function is already written as (x - h)² meaning that if you plug in a positive 5, the function will be subtraction and if you plug in a negative 5 for h, it'll be plus a positive. K on the other hand does exactly what it sounds like. -5 will mean going down five and +5 will mean going up 5 units.

Therefore g(x) = (x-2)² +3 means that you have moved the vertex of the parabola 2 units to the right and 3 units up.

Hope that helps!