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Chapter 10: Problem 36

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

Short Answer

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

Step by step solution

01

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.
02

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: \[(y + 2)^2 = 4p(x - 3)\]
03

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: \[9 = -12p\]
04

Solve for p

Continue from the equation \(9 = -12p\) to find \(p\). Solve for \(p\) by dividing both sides by \(-12\):\[p = -\frac{3}{4}\]
05

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:\[(y + 2)^2 = -3(x - 3)\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

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