Problem 91 - A Graphical Approach to College Algebra

Question

Find an equation of a parabola that satisfies the given conditions. Horizontal axis; vertex \((-2,3) ;\) passing through \((-4,0)\)

Step-by-Step Solution

Answer

The equation is \((y-3)^2 = -\frac{9}{2}(x+2)\).

Step 1: Write the Standard Form of a Horizontal Parabola

A horizontal parabola with vertex at \((h, k)\) is given by the equation \( (y-k)^2 = 4p(x-h) \). For our problem, the vertex is \((-2, 3)\), so the equation becomes \( (y-3)^2 = 4p(x+2) \).

Step 2: Substitute the Pass-Through Point into the Parabola Equation

The parabola passes through \((-4, 0)\). Substitute \(x = -4\) and \(y = 0\) into the equation. This gives \((0-3)^2 = 4p(-4+2)\), simplifying to \(9 = 4p(-2)\).

Step 3: Solve for the Parameter p

From the equation \(9 = 4p(-2)\), solve for \(p\) by getting \(9 = -8p\), leading to \(p = -\frac{9}{8}\).

Step 4: Write the Final Equation of the Parabola

Substitute \(p\) back into the equation \((y-3)^2 = 4p(x+2)\):\((y-3)^2 = 4 \cdot \left(-\frac{9}{8}\right) (x+2)\).This simplifies to \((y-3)^2 = -\frac{9}{2}(x+2)\).

Key Concepts

Vertex Form

The vertex form of a parabola is a useful way to express the equation of a parabola because it clearly shows the vertex of the parabola. This form is particularly valuable for graphing, as it provides immediate information about the parabola's orientation and vertex location. The vertex form of a vertical parabola is typically written as \[ y = a(x-h)^2 + k \] where

• \((h, k)\) represents the vertex coordinates.
• \(a\) is a parameter that affects the width and direction of the parabola.

For a horizontal parabola, which we'll focus on here, the equation is \[ (y-k)^2 = 4p(x-h) \] where:

• \( (h, k) \) is the vertex.
• \( p \) determines the distance from the vertex to the focus and affects the orientation.

In the exercise provided, the vertex was \((-2, 3)\), so we began by substituting these values into the vertex form. This initial step is crucial as it lays the foundation for determining the specific equation of the parabola.

Horizontal Axis

Understanding the axis orientation of a parabola is essential, as it dictates the structure of the equation. For a parabola with a horizontal axis, the parabola will open either to the left or the right, unlike the traditional upward or downward opening of vertical parabolas. In a horizontal orientation:

• The equation models the horizontal displacement instead of vertical.
• We use the vertex form \[ (y-k)^2 = 4p(x-h) \] to express this specific arrangement.

The axis of symmetry for horizontal parabolas runs parallel to the y-axis, differing from vertical parabolas where the axis of symmetry is parallel to the x-axis. This distinction affects how you plot the parabola and its reflectional symmetry. In this example, since the horizontal axis is specified, it directly influences the choice of using a horizontal parabola equation format. Knowing the axis helps in understanding how changes in the parabola's equation translate into graph alterations.

Equation of a Parabola

The equation of a parabola provides a mathematical description of its curve. There are different forms, including vertex form and standard form, but the correct equation form depends on the parabola's orientation and known parameters. In this exercise:

• We adopted the equation for a parabola with a horizontal axis: \[(y-k)^2 = 4p(x-h)\]
• Initially plugged in the vertex point, \((-2, 3)\), to get \[(y-3)^2 = 4p(x+2)\]

A critical aspect of finding the equation is using points that the parabola passes through. Here, we substitute the point \((-4, 0)\) into the partially completed equation, allowing us to solve for the parameter \(p\). This step helps tailor a specific equation that fits the given constraints and conditions of the parabola.

Solving for Parameter

Finding the parameter \(p\) is a crucial step in defining the specific equation of the parabola. This parameter influences the stretch and direction of the parabola. In a horizontal parabola equation \((y-k)^2 = 4p(x-h)\), \(p\) determines how wide or narrow the parabola is and dictates its direction:

• A positive \(p\) value makes it open to the right.
• A negative \(p\) value causes the parabola to open to the left.

In this example, we substituted the pass-through point \((-4, 0)\) into the equation form. This substitution resulted in \(9 = 4p(-2)\), which we then solved to get \(p = -\frac{9}{8}\). The negative value indicates the parabola opens to the left. Finally, we incorporated this information back into the vertex form equation to produce the final tailored parabola equation. Solving for \(p\) ensures that the equation accurately reflects the graph's characteristics as described.