To determine the transformation of a given quadratic function when it is shifted up by 2 units and to the right by 3 units, let's start with the standard quadratic function:
[tex]\[ f(x) = x^2 \][/tex]
### Step-by-Step Transformation:
1. Shift Up by 2 Units:
When a function is shifted up by a certain number of units, you add that number to the original function. In this case, we add 2 to the quadratic function [tex]\(f(x)\)[/tex].
[tex]\[ f(x) = x^2 \rightarrow f(x) = x^2 + 2 \][/tex]
2. Shift to the Right by 3 Units:
When a function is shifted to the right by a certain number of units, you replace [tex]\(x\)[/tex] with [tex]\( (x - \text{number of units}) \)[/tex] in the function. Here, we replace [tex]\(x\)[/tex] with [tex]\((x - 3)\)[/tex].
[tex]\[ f(x) = x^2 \rightarrow f(x) = (x - 3)^2 \][/tex]
3. Combining Both Transformations:
To apply both transformations (shifting up by 2 units and to the right by 3 units), combine the effects of both steps. Take the function from step 2 and add 2 to it.
[tex]\[ f(x) = (x - 3)^2 + 2 \][/tex]
So, the quadratic function shifted up by 2 units and to the right by 3 units is:
[tex]\[ f(x) = (x - 3)^2 + 2 \][/tex]
