the graph shows g(x), which is a translation of f(x) = x². write the function rule for g(x). write your…

the graph shows g(x), which is a translation of f(x) = x². write the function rule for g(x). write your answer in the form a(x - h)² + k, where a, h, and k are integers or simplified fractions.

the graph shows g(x), which is a translation of f(x) = x². write the function rule for g(x). write your answer in the form a(x - h)² + k, where a, h, and k are integers or simplified fractions.

Answer

Explanation:

Step1: Identify the vertex - form of a quadratic function

The vertex - form of a quadratic function is $y = a(x - h)^2+k$, where $(h,k)$ is the vertex of the parabola. For the parent function $f(x)=x^{2}$, $a = 1$, $h = 0$, and $k = 0$.

Step2: Determine the vertex of $g(x)$

From the graph, the vertex of the parabola $g(x)$ is at the point $(0,6)$. So, $h = 0$ and $k=6$.

Step3: Determine the value of $a$

Since the shape of the parabola $g(x)$ is the same as that of $f(x)=x^{2}$ (no stretching or shrinking), $a = 1$.

Step4: Write the function rule for $g(x)$

Substitute $a = 1$, $h = 0$, and $k = 6$ into the vertex - form $y=a(x - h)^2+k$. We get $g(x)=1(x - 0)^2+6$.

Answer:

$g(x)=(x - 0)^2+6$ or simply $g(x)=x^{2}+6$