Question

transformations of quadratic functions, write in vertex form: $g(x) = (x - 0)^2 + 0$
find $g(x)$, where $g(x)$ is the translation 7 units right of $f(x) = x^2$.
$g(x) = $

find $g(x)$, where $g(x)$ is the translation 6 units down of $f(x) = x^2$
$g(x) = $

Explanation:

First Sub - Question: Find \(g(x)\) (translation 7 units right of \(f(x)=x^{2}\))

Step 1: Recall horizontal translation rule

For a function \(y = f(x)\), a horizontal translation \(h\) units to the right is given by \(y=f(x - h)\). Here, \(f(x)=x^{2}\) and \(h = 7\).

Step 2: Apply the translation rule

Substitute \(x\) with \(x - 7\) in \(f(x)\). So \(g(x)=f(x - 7)=(x - 7)^{2}\).

Step 1: Recall vertical translation rule

For a function \(y = f(x)\), a vertical translation \(k\) units down is given by \(y=f(x)-k\). Here, \(f(x)=x^{2}\) and \(k = 6\).

Step 2: Apply the translation rule

Subtract 6 from \(f(x)\). So \(g(x)=f(x)-6=x^{2}-6\).

Answer:

\(g(x)=(x - 7)^{2}\)

Second Sub - Question: Find \(g(x)\) (translation 6 units down of \(f(x)=x^{2}\))