How is the graph of the parent quadratic function transformed to produce the graph of y = negative (2 x + 6) squared + 3?

Mathematics

2 answers:

bearhunter [10]3 years ago

5 0

We start with the parent function

$f(x)=x^2$

The first child function would be

$g(x)=(2x)^2$

We have multiplied the input of the function by a constant: we have

$g(x)=f(2x)$

This kind of transformation result in a horizontal stretch/compression. If the multiplier is greater than 1, we have a compression. So, this first child causes a horizontal compression with compression rate 2.

The second child function would be

$h(x)=(2x+6)^2$

We added 6 to the input of the function: we have

$h(x)=g(x+6)$

This kind of transformation result in a horizontal translation. If the constant added is positive, we translate to the left. So, this second child causes a translation 6 units to the left.

The third child function would be

$l(x)=-(2x+6)^2$

We changed the sign of the previous function (i.e. we multiplied it by -1): we have

$l(x)=-h(x)$

This kind of transformation result in a vertical stretch/compression. If the multiplier is greater than 1 we have a stretch, if it's between 0 and 1 we have compression. If it's negative, we reflect across the x axis, and then apply the stretch/compression. In this case, the multiplier is -1, so we only reflect across the x axis.

The fourth child function would be

$m(x)=-(2x+6)^2+3$

We added 3 to previous function: we have

$m(x)=l(x)+3$

This kind of transformation result in a vertical translation. If the constant added is positive, we translate upwards. So, this last child causes a translation 3 units up.

Recap

Starting from the parent function $y=x^2$ , we have to: