Question

How can I tell whether a parabola is upward or downward?

Answer 1

This problem deals with determining the shape of the parabola, whether it is upwards or downwards. The general equation of a parabola is ${x^2} = 4ay$ where, its vertex is the origin and doesn’t have any intercepts, or $y = a{x^2} + bx + c$ where its vertex may not be the origin, and it has intercepts intersecting the coordinate axes.

Complete step-by-step answer:
We know that the general equation of the parabola is given by: $y = a{x^2} + bx + c$.
The given parabola is upwards, when $a > 0$ in $y = a{x^2} + bx + c$. In this case the vertex is the minimum, or lowest point of the parabola. A large positive value of a makes a narrow parabola; a positive value of a which is close to zero makes the parabola wide.
The given parabola is downwards, when $a < 0$ in $y = a{x^2} + bx + c$. In this case the parabola opens upwards.
The function of the coefficient $a$ in the general equation is to make the parabola wider or skinnier, or to turn it upside down, when $a < 0$, and when the coefficient of ${x^2}$ is positive, the parabola opens up, otherwise it opens down.

**The parabola is upwards when $a > 0$ in $y = a{x^2} + bx + c$. The parabola is downwards when $a < 0$ in $y = a{x^2} + bx + c$. **

Note:
Please note that the graph of a quadratic function is a U-shaped curve which is a parabola. The sign on the coefficient $a$ of the quadratic function affects whether the graph opens up or down. If $a < 0$, the graph makes a frown (opens down) and if $a > 0$ then the graph makes a smile (opens up).