91Ó°ÊÓ

A parabola is a U-shaped graph that represents a quadratic equation. When you have a quadratic equation, such as \(y = ax^2 + bx + c\), the graph of this equation is a parabola. The direction the parabola opens (up or down) depends on the coefficient \(a\):
\(\bullet\) If \(a > 0\), the parabola opens upwards.
\(\bullet\) If \(a < 0\), the parabola opens downwards.
For the equation \(y = x^2 + 3x\), we can see that \(a = 1\), which is greater than zero, so the parabola opens upwards. This means it has a bottom point, called the vertex, and the arms extend upwards to infinity. This upward-opening shape influences how the values of \(y\) behave as \(x\) changes. A parabola's symmetry means any point on one side has another point an equal distance from the vertex on the opposite side.

The vertex is an important feature of a parabola. It is either the lowest point or the highest point of the parabola, depending on the direction it opens. In a quadratic equation, the vertex can be found using the vertex formula:
\[ x = -\frac{b}{2a} \]
For our quadratic equation \(y = x^2 + 3x\), identify \(a = 1\) and \(b = 3\). By substituting these values into the vertex formula, we calculate:
\[ x = -\frac{3}{2} \]
The \(x\)-coordinate of the vertex is \(-\frac{3}{2}\).
To get the \(y\)-coordinate, substitute \(x = -\frac{3}{2}\) back into the equation:
\[ y = \left(-\frac{3}{2}\right)^2 + 3 \left(-\frac{3}{2}\right) \]
This calculation simplifies to \( -\frac{9}{4} \). Hence, the vertex of the parabola is \(\left(-\frac{3}{2}, -\frac{9}{4}\right)\). The vertex, being a minimum point here due to the parabola opening upwards, indicates the lowest point on the parabola.

The range of a function is the set of possible \(y\)-values that a function can take. With quadratic equations, the range is heavily influenced by the vertex and the direction the parabola opens. For our equation \(y = x^2 + 3x\),
\(\bullet\) The parabola opens upwards because \(a = 1\), as previously established.
\(\bullet\) This indicates that the \(y\)-values start from the lowest value at the vertex and extend to infinity.
The vertex \(y\)-value, \(-\frac{9}{4}\), is negative, which means the parabola dips below the \(x\)-axis and includes negative \(y\)-values. Therefore, the range of this quadratic function is all \(y\) such that \(y \ge -\frac{9}{4}\).
As \(x\) moves away from the vertex in either direction, \(y\) increases, making \(-\frac{9}{4}\) the smallest value in the range.