# Question 92542

May 14, 2016

See explanation...

#### Explanation:

Note that vertex form of a vertical parabola can be written:

$y = a {\left(x - h\right)}^{2} + k$

where $a$ is a multiplier determining the 'steepness' of the parabola and $\left(h , k\right)$ is the vertex (turning point).

We are given:

$f \left(x\right) = 1 - x$

$g \left(x\right) = 2 {x}^{2} - 9$

$h \left(x\right) = \left(g \circ f\right) \left(x\right)$

$= g \left(f \left(x\right)\right)$

$= g \left(1 - x\right)$

$= 2 {\left(1 - x\right)}^{2} - 9$

$= 2 \left(1 - 2 x + {x}^{2}\right) - 9$

$= 2 - 4 x + 2 {x}^{2} - 9$

$= \textcolor{b l u e}{2 {x}^{2} - 4 x - 7}$

$= 2 \left({x}^{2} - 4 x + 4\right) - 15$

$= 2 {\left(x - 2\right)}^{2} - 15$

$= \textcolor{b l u e}{2 {\left(x + \left(- 2\right)\right)}^{2} + \left(- 15\right)}$

The minimum value of $h \left(x\right)$ occurs when ${\left(x - 2\right)}^{2} = 0$. That is when $x = 2$. hence the turning point is at color(blue)(""(2, -15))#

If $k \left(x\right)$ is a translation of $h \left(x\right)$ then it must have the same multiplier $2$. Given that its turning point is at $\left(- 1 , - 5\right)$, its equation can be written:

$\textcolor{b l u e}{k \left(x\right) = 2 {\left(x + 1\right)}^{2} - 5}$