Problem 18 - 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. $$ x=\frac{1}{2} y^{2}+10 y-5 $$

Step-by-Step Solution

Answer

The parabola with equation \( x = \frac{1}{2} y^2 + 10 y - 5 \) opens to the right. It is wider than \(y = x^2\). The vertex is at \((-55, -10)\).

Step 1: Identify the type and orientation of the parabola

Given equation is in the form of a horizontal parabola since it is solved for x. The general form of a horizontal parabola is \(x = a y^2 + b y + c\).

Step 2: Complete the square

Rewrite the equation \(x = \frac{1}{2} y^2 + 10 y - 5\). To complete the square, factor out \(\frac{1}{2}\) from the quadratic and linear term: \(x = \frac{1}{2} (y^2 + 20y) - 5\). Add and subtract \(100\) inside the parenthesis: \(x = \frac{1}{2} (y^2 + 20y + 100 - 100) - 5 = \frac{1}{2} (y+10)^2 - 50 - 5 = \frac{1}{2} (y+10)^2 - 55\).

Step 3: Determine the vertex

The vertex form of the parabola is \(x = \frac{1}{2} (y + 10)^2 - 55\), so the vertex is at \((-55, -10)\)

Step 4: Determine the direction and shape

The coefficient of \y^2 \ is positive \(\frac{1}{2} > 0\), indicating the parabola opens to the right. The coefficient \(\frac{1}{2}\) means the parabola is wider than the graph of \ y = x^2 \.

Step 5: No need for Discriminant

Since the parabola opens horizontally (to the right or left), computing the discriminant is unnecessary for determining x-intercepts.

Key Concepts

Completing the Square

Completing the square is a technique used to simplify quadratic equations and convert them into a form that makes it easy to identify the vertex of a parabola. It involves creating a perfect square trinomial from a quadratic equation.
Here’s how you can complete the square for a quadratic equation:

• Factor out the coefficient of the quadratic term if it is not 1.
• Add and subtract the square of half the coefficient of the linear term inside the parentheses.
• Simplify the equation to get a squared binomial and constant.

In our example, we have the equation:ewlineewline \(x = \frac{1}{2} y^2 + 10y -5\)ewlineewlineewline

• First, we factor out \frac{1}{2} from the quadratic and linear terms inside.ewline\(x = \frac{1}{2}(y^{2} + 20y) - 5 \)ewlineewlineewline
• Next, we add and subtract 100 inside the parenthesis.ewlineewline\(x = \frac{1}{2} (y^{2} + 20y + 100 - 100) - 5\)ewline = \frac{1}{2} ((y+10)^2 - 100) - 5 = \frac{1}{2} (y+10)^2 - 50 - 5 = \frac{1}{2} (y+10)^2 - 55 ewline

This equation is now in vertex form, making it simple to identify the vertex of the parabola.

The vertex form will help us to easily spot important attributes of the parabola.

Horizontal Parabola

A horizontal parabola opens to the right or left, unlike the more familiar vertical parabolas which open upwards or downwards.
When a quadratic equation is given in the form of \(x = a y^2 + b y + c\) or is converted to a similar structure after completing the square, it represents a horizontal parabola.
In our example, after completing the square, we get ewline\(x = \frac{1}{2} (y + 10)^2 - 55\)
This is in the form ewline\(x = a(y-h)^2 + k\)
which confirms that it is a horizontal parabola. The value of \(a\) dictates the opening direction and shape.

Parabola Orientation

The orientation of a parabola depends on the coefficient of its quadratic term. Specifically for a quadratic equation in the form of \(x = a y^2 + b y + c\):

• If \(a\) is positive, the parabola opens to the right.
• If \(a\) is negative, the parabola opens to the left.

For our given equation, after completing the square, ewlineewline\(x = \frac{1}{2} (y + 10)^2 - 55\)
We observe that \(\frac{1}{2}\) (which is \(a\)) is positive. Therefore, the parabola opens to the right. The direction tells us how the graph of the parabola behaves as we move along the y-axis.

Vertex Form

The vertex form is a simplified way of writing the equation of a parabola, which makes it easy to identify the vertex.
For a vertical parabola, the vertex form is \(y = a (x - h)^2 + k\), while for a horizontal parabola, it is \(x = a (y - k)^2 + h\). The vertex \((h, k)\) gives the highest or lowest point of the parabola.
In our example, the equation was converted to ewline\(x = \frac{1}{2} (y + 10)^2 - 55\)
The vertex here is \((-55, -10)\)
The vertex gives us critical information about the position and the direction of the parabola.

Coefficients

Coefficients in a quadratic equation have a significant impact on the orientation and shape of the parabola.
For a horizontal parabola in the form \(x = a y^2 + b y + c\):ewline

• The coefficient \(a\) determines whether the parabola opens to the right or left.
• The coefficients \(b\) and \(c\) affect the position of the vertex and the y-intercept.

In our given example, ewlineewline\(x = \frac{1}{2} y^2 + 10 y - 5\)ewline

• The coefficient \(a = \frac{1}{2}\) indicates that the parabola opens to the right and is wider than \(y = x^2\).
• The coefficients \(b = 10\) and \(c = -5\) help us calculate and complete the square, putting the equation into vertex form.

By analyzing the coefficients, one can understand key attributes about the parabola's shape and orientation.