Problem 36 - Precalculus: Functions and Graphs

Question

Find an equation of the parabola that satisfies the given conditions. Vertex \(V(3,-2),\) axis parallel to the \(x\) -axis, and \(y\) -intercept 1

Step-by-Step Solution

Answer

The equation of the parabola is \((y + 2)^2 = -3(x - 3)\).

Step 1: Understand the Parabola Structure

Since the parabola has a vertex at \(V(3, -2)\) and its axis is parallel to the \(x\)-axis, it is horizontally oriented. The general equation for a horizontally oriented parabola with vertex \((h, k)\) is \((y - k)^2 = 4p(x - h)\). For our parabola, \(h = 3\) and \(k = -2\). Substitute these into the equation.

Step 2: Substitute Known Values into the General Equation

Plug the vertex values \(h = 3\) and \(k = -2\) into the equation \((y - k)^2 = 4p(x - h)\). This gives us: \[(y + 2)^2 = 4p(x - 3)\]

Step 3: Use the Given y-Intercept to Find p

We know that the parabola crosses the \(y\)-axis at \(y = 1\), when \(x = 0\). Substitute \((x, y) = (0, 1)\) into the equation \((y + 2)^2 = 4p(x - 3)\). \[(1 + 2)^2 = 4p(0 - 3)\]Simplifying, this gives us: \[9 = -12p\]

Step 4: Solve for p

Continue from the equation \(9 = -12p\) to find \(p\). Solve for \(p\) by dividing both sides by \(-12\):\[p = -\frac{3}{4}\]

Step 5: Write the Final Equation of the Parabola

Now that we have \(p = -\frac{3}{4}\), substitute it back into the equation \((y + 2)^2 = 4p(x - 3)\):\[(y + 2)^2 = 4\left(-\frac{3}{4}\right)(x - 3)\]Simplify to get the final equation:\[(y + 2)^2 = -3(x - 3)\]

Key Concepts

Vertex Form

The vertex form of a parabola equation is essential for designing and manipulating parabolas easily. It generally looks like this for a horizontally oriented parabola:

• \((y - k)^2 = 4p(x - h)\)

This form is particularly useful because it immediately gives you the vertex of the parabola as

• \((h, k)\),

making it simpler to identify where the "turning point" of the parabola is. In the exercise, the given vertex is \((3, -2)\), meaning the parabola turns at this coordinate point.
The parameters \(h\) and \(k\) essentially "shift" the parabola left/right and up/down, respectively. This makes the vertex form a powerful tool in graphing parabolas.

Horizontally Oriented Parabola

A horizontally oriented parabola differs from the more familiar vertically oriented parabola in that it opens to the left or right instead of up or down.

• The axis of symmetry for this type is parallel to the \(x\)-axis.

The equation reflects this orientation, and it typically appears as:

• \((y - k)^2 = 4p(x - h)\)

This is unlike a vertical parabola, where the \(x\) and \(y\) roles are reversed. In the given problem, because the axis is parallel to the \(x\)-axis, it's understood that we must use the "horizontal" form of the parabola equation, allowing the parabola to "stretch" horizontally based on the value of \(p\).

Y-Intercept

The \(y\)-intercept of a graph is the point where it crosses the \(y\)-axis. This is expressed as \((0, c)\). It is an important property because it represents a specific relationship between the variables in the equation.
When given the \(y\)-intercept, you can substitute \(y\) and \(x\) with these values in the equation to solve for unknown constants like \(p\). In this exercise, our intercept \((0, 1)\) means when \(x = 0\), \(y = 1\).
This helps in confirming portions of your equation are handled correctly. In the derived equation

• \((y + 2)^2 = 4p(x - 3)\),

substituting the \(y\)-intercept gives \(9 = -12p\), which helps us find \(p\). This ensures we ultimately have a correctly framed equation.

Solving for P

In a horizontally oriented parabola, the variable \(p\) represents a parameter determining the "width" and "direction" of the parabola. To solve for \(p\), we rely on given conditions, such as intercepts or specific graph points.

• With a \(y\)-intercept, you can substitute known values into your equation to solve for \(p\).

In our scenario, we substitute \((x, y) = (0, 1)\) into

• \((y + 2)^2 = 4p(x - 3)\),

resulting in

• \(9 = -12p\).

Upon simplifying this, we divide both sides by \(-12\) to isolate \(p\).
Therefore, \(p = -\frac{3}{4}\), determining the degree and direction of the parabola's "openness." This is crucial for completing the equation

• \((y + 2)^2 = -3(x - 3)\),

which describes the horizontally oriented parabola.