# A parabola has a maxima at (-4,7) and passes through (1,1). Which is the other point with same ordinate that it passes through?

Nov 2, 2016

Parabola also passes through $\left(- 9 , 1\right)$

#### Explanation:

We can use here vertex form of equation for parabola. As it has a maximum at $\left(- 4 , 7\right)$, it will be of the form

$y = - a {\left(x + 4\right)}^{2} + 7$

Note that at $\left(- 4 , 7\right)$, $y$ has a maximum value of $- 7$ as $a {\left(x + 4\right)}^{2} = 0$ and at other places as it is negative, value of $y$ is far less.

Now as $y = - a {\left(x + 4\right)}^{2} + 7$ passes through $\left(1 , 1\right)$, we have

$1 = - a {\left(1 + 4\right)}^{2} + 7$ or $1 = - 25 a + 7$ i.e. $25 a = 6$ and

$a = \frac{6}{25}$ and equation of parabola is $y = - \frac{6}{25} {\left(x + 4\right)}^{2} + 7$

Note that this form of equation is symmetric w.r.t. $x + 4 = 0$ and hence The point symmetric to $\left(1 , 1\right)$ will have abscissa given by $\frac{x + 1}{2} = - 4$ i.e. $x = - 9$ and then

$y = - \frac{6}{25} {\left(- 9 + 4\right)}^{2} + = - \frac{6}{25} \times 25 + 7 = 1$ i.e.

Parabola also passes through $\left(- 9 , 1\right)$
graph{-6/25(x+4)^2+7 [-14.17, 5.83, -2.2, 7.8]}