How to find the range of a parabola?

Understand the Problem

The question is asking for the method or steps to determine the range of a parabola. This involves understanding the properties of parabolas, specifically their vertex and whether they open upwards or downwards, to define the set of possible output values (y-values) for the function.

Answer

The range of the parabola is $[y_v, +\infty)$ if $a > 0$ and $(-\infty, y_v]$ if $a < 0$.

Steps to Solve

Identify the standard form of the parabola

The standard form of a parabola is given by the equation:

$$ y = ax^2 + bx + c $$

Here, $a$, $b$, and $c$ are constants. The coefficient $a$ determines the direction of the parabola (upward if $a > 0$, downward if $a < 0$).

Find the vertex of the parabola

The vertex of the parabola can be found using the formula for the x-coordinate:

$$ x_v = -\frac{b}{2a} $$

Substituting $x_v$ back into the original equation will give the y-coordinate of the vertex $y_v$.

Determine the direction of the parabola

Check the value of $a$:

If $a > 0$: The parabola opens upwards, and the vertex represents the minimum point. The range will be $[y_v, +\infty)$.
If $a < 0$: The parabola opens downwards, and the vertex represents the maximum point. The range will be $(-\infty, y_v]$.

State the range

Based on the vertex and direction determined in the previous steps, express the range of the parabola as an interval.

The range of the parabola can be expressed as:

If $a > 0$: $[y_v, +\infty)$
If $a < 0$: $(-\infty, y_v]$

More Information

The range of a parabola can help in identifying the possible outputs of a quadratic function and is essential in various applications of quadratic equations, including physics and engineering.

Tips

Forgetting to check the sign of $a$, which determines the direction of the parabola. This can lead to an incorrect range being stated.
Miscalculating the vertex coordinates, which will directly affect the range conclusion.

Thank you for voting!