Problem 9 - College Algebra with Corequisite Support

Question

For the following exercises, determine whether the given equation is a parabola. If so, rewrite the equation in standard form. \((y-3)^{2}=8(x-2)\)

Step-by-Step Solution

Answer

The equation is a parabola and is already in standard form.

Step 1: Identify the Equation Type

The given equation \((y-3)^2 = 8(x-2)\) suggests a parabola because it resembles the form \((y-k)^2 = 4p(x-h)\), which represents a parabola that opens either to the left or right.

Step 2: Recognize the Standard Form

The standard form of a parabola that opens horizontally is \((y-k)^2 = 4p(x-h)\). Compare this to the given equation \((y-3)^2 = 8(x-2)\) to identify the values of \(h\), \(k\), and \(4p\).

Step 3: Determine Constants

By comparing, we see that \(h = 2\), \(k = 3\), and \(4p = 8\).

Step 4: Convert to Standard Form

Since the equation \((y-3)^2 = 8(x-2)\) already matches the form \((y-k)^2 = 4p(x-h)\), it is already in standard form for a horizontall-opening parabola.

Key Concepts

Standard form of a parabola

A parabola is a specific type of curve on a graph, and it is essential to understand how to express its equation in the standard form. For parabolas that open horizontally, the standard form of the equation is given by \[(y-k)^2 = 4p(x-h)\] Here,

• \(h\) and \(k\) represent the coordinates of the vertex of the parabola, which is the peak point around which the curve turns.
• The term \(4p\) indicates the distance between the vertex and the focus of the parabola multiplied by four.
• An important aspect of this form is that the \((y-k)^2\) term signifies that the parabola opens either to the left or to the right.

Being able to identify this structure in an equation allows us to confirm that it represents a parabola and understand its orientation with respect to the coordinate axes.

Horizontal parabola

A horizontal parabola is a curve that opens to either the left or the right instead of the usual upward or downward direction you might have seen in vertical parabolas. In the equation\[(y-k)^2 = 4p(x-h)\],

• The squared term \((y-k)^2\) in the equation signifies that the parabola is horizontal rather than vertical.
• The vertex, represented by the coordinates \((h, k)\), is used as the center point from which the parabola expands.
• The value of \(p\) determines the direction in which the parabola opens. If \(p > 0\), the parabola opens to the right. Conversely, if \(p < 0\), it opens to the left.

Grasping this concept helps in predicting where the curve will open and aligning it on the graph correctly.

Equation transformation

Transforming equations to identify if they form parabolas and rewriting them in standard form is a crucial skill. Consider the given equation:\((y-3)^2 = 8(x-2)\).Here’s how we determine its form and attributes:

• By comparing it with \((y-k)^2 = 4p(x-h)\), we find that \(h = 2\), \(k = 3\), and \(4p = 8\).
• The equation is already arranged to reflect a specific structure, making it easy to recognize it as a horizontal parabola.
• Recognizing these comparisons ensures that the equation does not need any additional mathematical manipulation.

Understanding how to transform and compare equations effectively helps in accurately identifying the form of a parabola and assures that you place the correct emphasis on the terms involved.