- Identify the standard form of a horizontal parabola with vertex at \((0,0)\) as \(y^2 = 4ax\).
- Recognize that a focus on the negative \(x\)-axis requires the coefficient of \(x\) to be negative.
- Compare the given equations to find that \(y^2 = -2x\) is the only option with a negative coefficient.
- Conclude that the equation is \(\boxed{y^2 = -2x}\).
Explanation
- Analyze the properties of the parabola equation.
A parabola with a vertex at the origin \((0,0)\) and its focus on the \(x\)-axis has a standard equation of the form \(y^2 = 4ax\). The constant \(a\) represents the directed distance from the vertex to the focus. If \(a > 0\), the focus is on the positive \(x\)-axis and the parabola opens to the right. If \(a < 0\), the focus is on the negative \(x\)-axis and the parabola opens to the left. Since the focus lies on the negative part of the \(x\)-axis, we must have \(a < 0\). Thus, the equation must be of the form \(y^2 = 4ax\) where \(a\) is a negative constant.
- Evaluate the given options to find the correct equation.
We evaluate the given options:
- \(y^2 = x\): This corresponds to \(4a = 1\), so \(a = 0.25\). Since \(a > 0\), the parabola opens to the right.
- \(y^2 = -2x\): This corresponds to \(4a = -2\), so \(a = -0.5\). Since \(a < 0\), the parabola opens to the left and the focus lies on the negative \(x\)-axis.
- \(x^2 = 4y\): This is a vertical parabola opening upward.
- \(x^2 = -6y\): This is a vertical parabola opening downward.
Comparing these with our requirement, only the second option satisfies the condition.
- State the final answer.
Based on the analysis, the only equation representing a parabola with a vertex at \((0,0)\) and a focus on the negative \(x\)-axis is \(y^2 = -2x\). The final answer is \(\boxed{y^2 = -2x}\).
Examples
Architects and engineers use the reflective properties of parabolas to design structures like satellite dishes or solar concentrators. By knowing the equation of a parabola, they can determine the exact location of its focus—the point where all incoming signals or light rays will converge. For example, in a design where a sensor must be placed on the negative x-axis relative to a dish's vertex, the equation \(y^2 = -2x\) helps calculate the precise coordinate for the optimal signal reception.