Problem 18 - Intermediate Algebra

Question

The graph of each equation is a parabola. Find the vertex of the parabola and then graph it. $$x=y^{2}-6 y+6$$

Step-by-Step Solution

Answer

The vertex of the parabola is \((-3, 3)\). The parabola opens to the right.

Step 1: Reorganize the Equation

The given equation is \( x = y^2 - 6y + 6 \). We want to express this in a form that makes it easy to identify the vertex. Our aim is to complete the square for the \( y \) terms.

Step 2: Complete the Square

To complete the square, we consider the \( y^2 - 6y \) portion. Take half of the \( y \) coefficient, which is -6, divide it by 2 to get -3, then square it to get 9. Add and subtract 9 inside the equation: \( x = (y^2 - 6y + 9) - 9 + 6 \).

Step 3: Rewrite the Equation

Now, rewrite the equation with the perfect square: \( x = (y - 3)^2 - 3 \). This shows the equation of the parabola in vertex form.

Step 4: Identify the Vertex

The vertex form of a parabola \((y - k)^2 = 4p(x - h)\) gives the vertex at \((h, k)\). Here, \((y - 3)^2 = x + 3\), and the vertex is thus \((-3, 3)\).

Step 5: Graph the Parabola

Plot the vertex \((-3, 3)\) on a coordinate plane. Since the coefficient of the squared term \((y - 3)^2\) is positive, the parabola opens to the right. The equation can be rewritten as \(x + 3 = (y - 3)^2\), indicating the shift by -3 along the x-axis.

Key Concepts

Completing the Square

Completing the square is a fundamental technique that helps transform quadratic expressions into a more manageable form. Let's dive into why it's useful and how it works. For a quadratic expression like \( y^2 - 6y \), completing the square involves converting this into a perfect square trinomial. Here's how:

• Take the coefficient of the linear term, which in this case is -6.
• Divide this coefficient by 2 to get -3.
• Square the result, obtaining \((-3)^2 = 9\).

By doing this, you effectively "complete the square," allowing you to rewrite \( y^2 - 6y \) as \((y - 3)^2 - 9\). This transformation is useful because it simplifies the process of graphing parabolas, by making it easier to identify key features like the vertex. It's a handy tool that brings clarity to what might otherwise be complex algebraic problems.

Vertex Form of a Parabola

The vertex form of a parabola is particularly advantageous when you need to quickly identify important properties of the graph, such as the vertex. A parabola expressed in vertex form looks like \((y - k)^2 = 4p(x - h)\) for horizontal parabolas, where \((h, k)\) represents the vertex.
In the exercise example, after completing the square, the equation becomes \(x = (y - 3)^2 - 3\). This reveals the parabola's vertex, \((-3, 3)\).

• The vertex form directly tells you the point which is essentially the 'tip' or 'turning point' of the parabola.
• Understanding the structure of this form helps in predicting the direction the parabola opens – to the right or left for horizontal parabolas.

In vertex form, graphing becomes straightforward, as it visually and algebraically centers the parabola around its vertex, simplifying further manipulations or interpretations of the parabola's characteristics.

Coordinate Plane Graphing

Graphing on the coordinate plane involves plotting points and understanding how algebraic expressions translate into visual representations. With simpler forms like the vertex form, we can efficiently plot parabolas by finding key features such as the vertex and direction of opening.
Here's how you address graphing using the example equation \(x = (y - 3)^2 - 3\):

• First, take the vertex \((-3, 3)\) and plot it on the graph.
• Since the equation shows a positive coefficient for the squared term, \((y - 3)^2\), the parabola opens to the right.
• Consider additional points by choosing values for \(y\) and solving for \(x\) to complete the parabola's shape.

This method enables a precise and systematic approach to sketching parabolas. The vertex provides a foundation point, from which the symmetry and direction of the parabola can be accurately drawn, offering a comprehensive view of its layout and behavior.