# What is the equation of a parabola that is a vertical translation of y=-5x^2+4x-3 of -12 and a horizontal translation of -9?

May 16, 2018

$y = - 5 {\left(x + 9\right)}^{2} + 4 \left(x + 9\right) - 15$
y=−5x^2−86x−384

#### Explanation:

To ma(x+e this easier, let's call our function $f \left(x\right)$

To vertically translate the function by $a$ we just add $a$, $f \left(x\right) + a$.

To horizontally translate a function by $b$, we do $x - b$, $f \left(x - b\right)$

The function needs to be translated 12 units down and 9 units to the left, so we will do:
$f \left(x + 9\right) - 12$

This gives us:
$y = - 5 {\left(x + 9\right)}^{2} + 4 \left(x + 9\right) - 3 - 12$

$y = - 5 {\left(x + 9\right)}^{2} + 4 \left(x + 9\right) - 15$

After expanding all the brackets, multiplying by factors and simplifying, we get:
y=−5x^2−86x−384