91Ó°ÊÓ

A parabola is a beautiful mathematical curve that follows the shape of the equation \( y = ax^2 + bx + c \). Parabolas open upwards or downwards depending on the sign of the coefficient \( a \). In this exercise, we are dealing with the parabola \( y = x^2 + 2x \), which opens upwards. This happens because the coefficient of \( x^2 \) is positive.

Parabolas have a symmetric property around a vertical line known as the axis of symmetry. For this specific equation, the axis is at \( x = -1 \), found using the formula \( x = -\frac{b}{2a} \), where \( a = 1 \) and \( b = 2 \). Due to this symmetry, the parabola is identical on both sides. However, our task is to focus only on the left half, which involves all the negative values of \( x \). To achieve this, it is essential to focus on the part returning negative \( x \)-values, which requires a particular approach in parametrization.

Parametric equations are an elegant way to represent curves by introducing a third variable, often [latex] t [latex], called a parameter. These equations express the coordinates \((x, y)\) of the curve as functions of \( t \). For a parabola like \( y = x^2 + 2x \), we can use parametric equations to separately express \( x \) and \( y \) in terms of \( t \).

A common method for parameterizing is to replace one of the variables in the function with \( t \). In our case, we parameterized \( x = -t \) to ensure we capture only the left side of the parabola. This implies that as \( t \) varies, both \( x \) and \( y \) change accordingly. The parametrization becomes:

• \( x(t) = -t \), keeping \( x\) negative as desired.
• \( y(t) = t^2 - 2t \), reflecting the original equation with modified values.

In this setup, \( t \) takes on values starting from zero going towards positive infinity, effectively scanning through the negative \( x \)-values on the parabola.

The focus of this problem is on the left half of the parabola, where \( x \) values are negative. Ensuring that the parametrization covers these negative \( x \)-values effectively is key. By letting \( x = -t \) with \( t \geq 0 \), the condition for negative \( x \) values is naturally satisfied.

Here’s why this works:

• As \( t \) approaches increasing values, \( x = -t \) ensures that \( x \) moves in the negative direction, such as \( -1, -2, -3, \) and so on.
• The parameter \( t \), starting from zero, guarantees all \( x \)-values along the left parabola are covered.

Parametric representation not only helps visualize the curve but also confirms that the entire left portion is accounted for without including any right-side (positive \( x \)-values) inadvertently. This is crucial in applications where negative values play a significant role. Understanding how to constrain \( x \)-values through parametrization provides insight into controlling and analyzing sections of mathematical curves.