Question

How do you find the range of the equation $ y = - {x^2} - 6x - 13 $ ?

Answer 1

Hint: We have given an equation as $ y = - {x^2} - 6x - 13 $ , which is a parabolic equation. The standard parabolic equation is always represented as $ y = a{x^2} + bx + c $ and the expression $ - \dfrac{b}{{2a}} $ gives the x-coordinate , and then substitute x-coordinate in the original equation to obtain y-coordinate , y coordinate is the leftmost range of the equation and right most range of the parabola is infinity. Therefore, you have to find the y coordinate of the vertex.

Complete step-by-step answer:
We have the following parabolic equation,
 $ y = - {x^2} - 6x - 13 $
Here y is a quadratic function. The standard parabolic equation is always represented as $ y = a{x^2} + bx + c $ and the expression $ - \dfrac{b}{{2a}} $ gives the x-coordinate , and then substitute x-coordinate in the original equation to obtain y-coordinate.
Therefore, the x coordinate of the vertex is equal to $ - \dfrac{b}{{2a}} $
Now, compare the original equation with the standard parabolic equation, we will get,
The x-coordinate of the vertex , \[ - \dfrac{b}{{2a}} = - \dfrac{{( - 6)}}{{2( - 1)}}\]
 $ = - 3 $
Now substitute $ x = - 3 $ in the original equation to get y-coordinate as,
 $ \Rightarrow y = - {x^2} - 6x - 13 $
 $ \Rightarrow y = - {( - 3)^2} - 6( - 3) - 13 $
 $ = - 9 + 18 - 13 $
\[ = - 4\]
Therefore, the required vertex of the parabola is $ \left( { - 3, - 4} \right) $ .
The smallest value to y is $ - 4 $ ,
And y has no upper bound,
Therefore, the required range is $ [ - 4,\infty ) $ .
So, the correct answer is “ $ [ - 4,\infty ) $ ”.

Note: The equation for a parabola can also be written in vertex form as $ y = a{(x - h)^2} + k $ where $ (h,k) $ is the vertex of parabola. The point where a parabola has zero gradient is known as the vertex of the parabola. We need to get ‘y’ on one side of the “equals” sign, and all the other numbers on the other side. To solve this equation for a given variable ‘y’, we have to undo the mathematical operations such as addition, subtraction, multiplication, and division that have been done to the variables. Use addition or subtraction properties of equality to gather variable terms on one side of the equation and constant on the other side of the equation. Use the multiplication or division properties of equality to form the coefficient of the variable term equivalent to one.