Chapter 4.1, Problem 66PPSE

Solution

To find: The equation of second parabola shown in the figure shown below if one parabola has the equation $y={\left(x-4\right)}^2+2$ .

  

The equation $y=-{\left(x-4\right)}^2+2$ represents the second parabola in right and option $\left(A\right)$ is correct.

Given:

The equation of one parabola shown below is $y={\left(x-4\right)}^2+2$ .

  

Concept used:

The vertex form of parabola is $y=a{\left(x-h\right)}^2+k$ where $a\ne 0$ and the point $\left(h,k\right)$ represents the vertex of the parabola which either open upwards or downwards. And the parabola opens upwards if $a>0$ and downwards if $a<0$ .

Calculation:

From the graph of function, it can be interpreted that the graph of parabolic function shown above in blue colour represents the one parabola $y={\left(x-4\right)}^2+2$ because $a>0$ and it is opened upwards.

And, the graph shown in red colour curve is just the reflection of curve $y={\left(x-4\right)}^2+2$ and it is opened downwards. So, the coefficient of red colour curve will be negative.

Hence, the equation of red colour curve will be $y=-{\left(x-4\right)}^2+2$ .So, option $\left(A\right)$ is correct.