Chapter 14, Problem 10CCSR

Solution

To find:The correct statement/option for parabola of given characteristic.

The option H is correct.

Given data: The parabola with characteristics:

Axis of symmetry $x=-3$

Range: All real numbers $y\le 4$ .

  $\begin{array}{l}\left(F\right)y=-{\left(x-4\right)}^2-3\left(H\right)y=-{\left(x+3\right)}^2+4\\ \left(G\right)y={\left(x-4\right)}^2-3\left(I\right)y={\left(x+3\right)}^2+4\end{array}$

Method/Formula used:

The equation of parabola with vertex is given by $y=a{\left(x-h\right)}^2+k$ , the axis of symmetry of this parabola is $x=h$ .

Calculations:

If $x=-3$ is the axis of symmetry then equation of the parabola should be of the form $y=a{\left(x+3\right)}^2+k$ .

Therefore, options ( F ) and ( G ) must be excluded.

Now, check for option ( H ):

  $\begin{array}{c}y-4=-{\left(x+3\right)}^2\\ {\left(x+3\right)}^2=4-y\\ x+3=\sqrt{4-y}\\ x=-3+\sqrt{4-y}\end{array}$

The domain of function values of x for which $4-y\ge 0$ or $y\le 4$ . Thus, the range y is less or equal to 4.

For option ( I ),

  $\begin{array}{c}y-4={\left(x+3\right)}^2\\ {\left(x+3\right)}^2=y-4\\ x+3=\sqrt{y-4}\\ x=-3+\sqrt{y-4}\end{array}$

The domain of function values of x for which $y-4\ge 0$ or $y\ge 4$ . Thus, the range y is less or equal to 4.

But range must be less or equal to 4. Hence, the option H is correct.