Chapter 3, Problem 36RE

To determine

Quadratic function for given vertex.

Answer to Problem 36RE

The quadratic function that satisfies the given condition and properties I $f\left(x\right)=-2{\left(x-3\right)}^2-4$ .

Explanation of Solution

Given information:

Vertex is $\left(3,-4\right)$ contains the points $\left(4,2\right)$ .

Calculation:

The vertex form of a quadratic function is equation is $f\left(x\right)=a{\left(x-h\right)}^2+k$ where $\left(h,k\right)$ the vertex is.

The required quadratic function has its vertex at $\left(3,-4\right)$ thus using the vertex form above, its tentative equation would be,

  $f\left(x\right)=a{\left(x-3\right)}^2+2$

The function’s graph passes through the point $\left(4,2\right)$ . This means that the coordinates of this point satisfy the equation of function.

Substitute the $x$ and $y-$ values into the tentative equation above to have:

  $\begin{array}{l}2=a{\left(4-3\right)}^2-4\\ 2=a\left(1\right)-4\\ 2-4=a\\ -2=a\end{array}$

Therefore, the quadratic function that satisfies the given condition and properties is $f\left(x\right)=-2{\left(x-3\right)}^2-4$ .