Problem 17 - Beginning and Intermediate Algebra

Question

Find the vertex of each parabola. For each equation, decide whether the graph opens up, down, to the left, or to the right, and whether it is wider, narrower, or the same shape as the graph of \(y=x^{2}\). If it is a parabola with a vertical axis of symmetry, find the discriminant and use it to determine the number of \(x\) -intercepts. $$ x=\frac{1}{3} y^{2}+6 y+24 $$

Step-by-Step Solution

Answer

The vertex is (-3, -9). The parabola opens to the right and is wider than the graph of y = x^2.

Step 1 - Rewrite the equation in standard form

We start by rewriting the given equation in the standard form of a parabola that opens sideways: \[ x - h = a(y - k)^2 \] The given equation is: \[ x = \frac{1}{3} y^2 + 6y + 24 \] We will complete the square for the term involving y.

Step 2 - Complete the Square

Group the terms involving y and complete the square: \[ x = \frac{1}{3}(y^2 + 18y) + 24 \] First, factor out \(\frac{1}{3}\) from the terms involving y:\[ x = \frac{1}{3}(y^2 + 18y + 81 - 81) + 24 \] Add and subtract \(81\) because \( (\frac{18}{2})^2 = 81 \). Now simplify inside the parentheses:\[ x = \frac{1}{3}((y + 9)^2 - 81) + 24 \] Distribute \(\frac{1}{3}\):\[ x = \frac{1}{3}(y + 9)^2 - 27 + 24 \] Combine constants:\[ x = \frac{1}{3}(y + 9)^2 - 3 \]

Step 3 - Identify the Vertex

The standard form of the equation \[ x - h = a(y - k)^2 \] shows that:The equation after completing the square is: \[ x = \frac{1}{3} ( y + 9 )^2 - 3 \] So, the vertex \( (h, k) \) is \[ (-3, -9) \]

Step 4 - Determine the Direction and Shape

The given form \[ x = \frac{1}{3} ( y + 9 )^2 - 3 \] has the coefficient of \( (y + 9)^2 \) as \( \frac{1}{3} \), which is positive. Therefore, the parabola opens to the right. \( \frac{1}{3} \) is less than 1, so the parabola is wider than the graph of \( y = x^{2} \)

Step 5 - Find the Discriminant

Since step 4 established this parabola has a horizontal axis of symmetry we skip finding the discriminant and determining the number of \(x \)-intercepts.

Key Concepts

Completing the Square

When dealing with quadratic equations, especially when finding the vertex of a parabola, completing the square is a crucial algebraic technique. The objective is to transform a quadratic equation into a form that makes it easy to identify and work with.

Let's start with the given quadratic equation: \[ x = \frac{1}{3} y^2 + 6y + 24 \] To complete the square, follow these steps:

• Group the terms involving y and introduce a constant to create a perfect square trinomial.
• Factor out the coefficient of the squared term, if it is not 1.
• Add and subtract the square of half the coefficient of y (inside the parenthesis).
• Simplify and rewrite the equation in its transformed form.

For this exercise, we group and manipulate the equation: \[ x = \frac{1}{3}(y^2 + 18y) + 24 \] Factoring out \(\frac{1}{3}\), we get: \[ x = \frac{1}{3}(y^2 + 18y + 81 - 81) + 24 \] That allows us to complete the square inside the parentheses: \[ x = \frac{1}{3}((y + 9)^2 - 81) + 24 \] Distribute the \(\frac{1}{3}\) and combine constants: \[ x = \frac{1}{3}(y + 9)^2 - 27 + 24 = \frac{1}{3}(y + 9)^2 - 3 \] This gives us the quadratic equation in a form where the vertex is easily identifiable.

Parabola Direction

To determine the direction in which a parabola opens, look at the coefficient of the squared term in the equation. For a quadratic equation of the form \[ x - h = a(y - k)^2 \],

• If \(a > 0\), the parabola opens to the right.
• If \(a < 0\), the parabola opens to the left.

In our completed square form: \[ x = \frac{1}{3}(y + 9)^2 - 3 \] The coefficient of \((y + 9)^2\) is \(\frac{1}{3}\), which is positive. Therefore, the parabola opens to the right. This information helps you determine the orientation of the parabola on the graph, which is critical for graphing and understanding its behavior in the context of different problems.

Parabola Shape

The shape or width of a parabola is influenced by the coefficient of the squared term in its equation. For a parabola \[ x - h = a(y - k)^2 \], compare the value of \(a\) with 1 to deduce the shape:

• If \(|a| > 1\), the parabola is narrower than the graph of \(y = x^2\).
• If \(|a| < 1\), the parabola is wider than the graph of \(y = x^2\).
• If \(|a| = 1\), the parabola has the same shape as the graph of \(y = x^2\).

Our equation has \(a = \frac{1}{3}\), and since \(\frac{1}{3} < 1\), the parabola is wider than the graph of \(y = x^2\). Understanding the shape of the parabola helps in predicting how it will look compared to the standard parabola graph. This knowledge is useful for sketching and analyzing parabolic graphs in various mathematical applications.