Question

write an equation in standard form of the parabola that has the same shape as the graph of f(x)=5x² or g(x)= - 5x², but with the given maximum or minimum. maximum = 2 at x = - 6 h(x)=□

Explanation:

Step1: Determine the vertex - form of the parabola

The vertex - form of a parabola is \(y = a(x - h)^2+k\), where \((h,k)\) is the vertex of the parabola. Since the maximum value of the parabola is 2 at \(x=-6\), the vertex \((h,k)=(-6,2)\). Also, since the parabola has the same shape as \(y = 5x^2\) or \(y=-5x^2\), \(|a| = 5\). Because the parabola has a maximum, \(a=-5\).

Step2: Substitute the values of \(a\), \(h\), and \(k\) into the vertex - form

Substitute \(a=-5\), \(h = - 6\), and \(k = 2\) into \(y=a(x - h)^2+k\). We get \(y=-5(x+6)^2 + 2\).

Step3: Expand to standard form

\[

$$\begin{align*} y&=-5(x^{2}+12x + 36)+2\\ &=-5x^{2}-60x-180 + 2\\ &=-5x^{2}-60x-178 \end{align*}$$

\]

Answer:

\(h(x)=-5x^{2}-60x - 178\)