Question

write an equation in standard form of the parabola that has the same shape as the graph of f(x)=2x^2, but with (9,5) as the vertex. g(x)= (type your answer in standard form.)

Explanation:

Step1: Recall the vertex - form of a parabola

The vertex - form of a parabola is $y = a(x - h)^2+k$, where $(h,k)$ is the vertex of the parabola. If two parabolas have the same shape, they have the same value of $|a|$. The given parabola is $f(x)=2x^{2}$, so $a = 2$. The vertex $(h,k)$ is given as $(9,5)$.

Step2: Substitute values into the vertex - form

Substitute $a = 2$, $h = 9$, and $k = 5$ into the vertex - form $y=a(x - h)^2+k$. We get $y=2(x - 9)^2+5$.

Step3: Expand to standard form

Expand $2(x - 9)^2+5$. First, expand $(x - 9)^2=x^{2}-18x + 81$. Then $2(x - 9)^2=2(x^{2}-18x + 81)=2x^{2}-36x+162$. So $y=2x^{2}-36x + 162+5=2x^{2}-36x+167$.

Answer:

$g(x)=2x^{2}-36x + 167$