Chapter 8.5, Problem 54E

To determine

To find:The quadratic equation in the standard form.

Answer to Problem 54E

The quadratic equation in the standard form is $f\left(x\right)=-2x^2+16x$

Explanation of Solution

Given information:

The y increases as x increases when x < 4; y decreases as x increases when x> 4.

Formula used:

  $f\left(x\right)=a\left(x-p\right)\left(x-q\right)$ is the line $x=\frac{p+q}{2}$

Calculation:

The axis of symmetry of a quadratic function with equation of the form

  $f\left(x\right)=a\left(x-p\right)\left(x-q\right)$ is the line $x=\frac{p+q}{2}$

In the given function, the behaviour of the graph changes at x =4. This means that the axis of symmetry x =4.

The value of any parabola whose graph opens downward and whose axis of symmetry is x =4 will satisfy the given conditions

This means that any quadratic function whose equation is of the form $a\left(x-p\right)\left(x+q\right)$ where $\frac{p+q}{2}=4$ will meet given requirements.

Therefore one quadratic function that satisfies the given conditions is:

  $f\left(x\right)=-2\left(x\right)\left(x-8\right)$

On Converting this to standard form gives:

  $f\left(x\right)=-2x^2+16x$

The graph of the given function is:

  

Conclusion:

The quadratic equation in the standard form is $f\left(x\right)=-2x^2+16x$