Question

How do you tell whether the graph opens up or down, find the vertex and the axis of symmetry of $y = 3{(x + 4)^2} - 2?$

Answer 1

Hint : As we can see that the given equation is a quadratic equation. The general form of quadratic equation is $a{x^2} + bx + c$ , but here in the given question we have the vertex form of the equation which is $y = a{(x - h)^2} + k$ and we have tell the direction of the graph and then find the vertex of the given equation.

Complete step-by-step answer :
The graph of the quadratic equation is in the shape of the parabola and the direction of it can be determined by the value of $a$ . If the value of $a$ is negative then the parabola is downward shaped i.e. $\bigcap {}$ .
But if the value of $a$ is positive then the parabola is upward shape i.e. $\bigcup {}$ .
Here the value of $a = 3$ which is positive. So the shape of the graph or parabola is upward facing.
Now the vertex of the form $y = a{(x - h)^2} + k$ is $h,k$
Since the value is negative so $x -$ vertex is $- 1 \times 4 = - 4$ and the other is $- 2$ .
Hence the vertex is $\left( { - 4, - 2} \right)$ and the axis of symmetry is $x = - 4$ .

Note : We should note that finding the nature of the curve should be the first approach. The graph of a quadratic equation is always $U -$ shaped curve which is parabola. The sign on the coefficient $a$ of the quadratic equation affects whether the graph opens up or down. If the value of $a < 0$ , then the graph opens down.