Problem 41 - Algebra for College Students

Question

For Problems \(31-50\), find an equation of the parabola that satisfies the given conditions. Vertex \((0,0)\), symmetric with respect to the \(x\) axis, and contains the point \((-3,5) \quad 3 y^{2}=-25 x, y^{2}=-\frac{25}{3} x\)

Step-by-Step Solution

Answer

The equation of the parabola is \(y^2 = -\frac{25}{3}x\).

Step 1: Identify the form of a parabola opening sideways

Since the parabola is symmetric with respect to the \(x\) axis, it opens either to the left or right. The standard form of such a parabola is \(y^2 = 4px\), where \(p\) is the distance from the vertex to the focus. Since the vertex is at \((0,0)\), we start with \(y^2 = 4px\).

Step 2: Plug in the given point into the equation

We know the parabola passes through the point \((-3,5)\), so we substitute these coordinates into the equation \(y^2 = 4px\). This gives \(5^2 = 4p(-3)\).

Step 3: Solve for \(p\)

Simplify the equation from Step 2: \(25 = -12p\). Solving for \(p\), we have \(p = -\frac{25}{12}\).

Step 4: Write the final equation of the parabola

Substitute \(p = -\frac{25}{12}\) back into the equation \(y^2 = 4px\) to get \(y^2 = -\frac{25}{3}x\). This equation represents the parabola with vertex at \((0,0)\), symmetric with respect to the \(x\) axis, and passing through the point \((-3, 5)\).

Key Concepts

Vertex Form of a Parabola

The vertex form of a parabola provides a neat and intuitive way to understand a parabola's characteristics. It highlights where the vertex of the parabola is located. For a parabola that opens sideways, which is the case if it is symmetric to the x-axis, the vertex form is often written as \( y^2 = 4px \). Here, \( (h, k) \) would be the vertex, but since our vertex is at \((0,0)\), the equation simplifies this way.

This form allows you to see the vertex directly from the equation without any need to transform or rearrange. It's especially useful in problems requiring quick identification of the vertex point, which is critical for graphing or deriving other attributes.

Symmetric Parabola

A symmetric parabola means the parabola is mirrored along a specific axis. In this problem, the symmetry is about the x-axis. This symmetry implies that for every point \((x, y)\), there is a corresponding point \((x, -y)\) on the opposite side.

Understanding symmetry helps significantly in graphing and analyzing parabolas because it ensures that the shape is balanced and predictable. With symmetry about the x-axis, parabolas can open either to the right or left. When dealing with an equation like \( y^2 = 4px \), if \( p \) is positive, the parabola opens to the right, whereas if \( p \) is negative, it opens to the left.

Solving for p in Parabola

The value \( p \) in the parabola's equation \( y^2 = 4px \) is crucial as it indicates the distance from the vertex to the focus. To solve for \( p \), substitute a known point on the parabola into the equation and solve.

In our exercise, the given point \((-3, 5)\) allows us to substitute these values into the parabola's equation. Plugging in gives \(25 = 4p(-3)\), which simplifies to \( p = -\frac{25}{12} \). This negative \( p \) is indicative of the parabola opening to the left.

Standard Form of a Parabola

The standard form of a parabola often comes in the general forms \( ax^2 + bx + c = 0 \) for vertical orientation, and \( y^2 = 4px \) for horizontal orientation. This form is direct and helps in quickly getting equations or transforming them from vertex form based on problem needs.

For horizontally oriented parabolas, the focus and direction of opening are centered in \( y^2 = 4px \). By comparing it to vertex form, this equation wraps in both the symmetry and the value of \( p \) needed to understand the complete parabola geometry. Here, \( y^2 = -\frac{25}{3}x \) is the specific standard form that fulfills all conditions of our problem, such as symmetry, vertex position, and passing through the given point.