To write a quadratic function that is shifted to the right by 4 units, we need to begin with a basic quadratic function and then apply the appropriate transformation.
1. Start with the basic quadratic function:
[tex]\[ f(x) = x^2 \][/tex]
2. Shift the graph of the function to the right by 4 units:
When shifting a function horizontally, we modify the input variable [tex]\( x \)[/tex]. To shift the function to the right by 4 units, we replace [tex]\( x \)[/tex] with [tex]\( x - 4 \)[/tex]. This gives us the new function:
[tex]\[ f(x) = (x - 4)^2 \][/tex]
To understand the transformation in more detail, we can expand the quadratic expression:
3. Expand the quadratic function:
Expanding [tex]\( (x - 4)^2 \)[/tex] involves using the formula for squaring a binomial [tex]\( (a - b)^2 = a^2 - 2ab + b^2 \)[/tex]:
[tex]\[ (x - 4)^2 = x^2 - 2 \cdot 4 \cdot x + 4^2 \][/tex]
Simplifying this, we get:
[tex]\[ (x - 4)^2 = x^2 - 8x + 16 \][/tex]
So, the quadratic function shifted to the right by 4 units can be represented in its factored form as:
[tex]\[ f(x) = (x - 4)^2 \][/tex]
And, in its expanded form, the function is:
[tex]\[ f(x) = x^2 - 8x + 16 \][/tex]
Thus, the quadratic function that has been shifted to the right by 4 units is:
[tex]\[ (x - 4)^2 \][/tex]
Which expands to:
[tex]\[ x^2 - 8x + 16 \][/tex]
