Problem 50 - Intermediate Algebra

Question

Write each equation of a parabola in standard form and graph it. Give the coordinates of the vertex. $$ x=-2 y^{2}+4 y+1 $$

Step-by-Step Solution

Answer

The vertex is at (3, 1), and the parabola opens to the left in standard form: \( x = -2(y-1)^2 + 3 \).

Step 1: Rearrange the Equation

Start with the given equation in terms of x: \[ x = -2y^2 + 4y + 1 \]We aim to rearrange this equation into the standard form of a parabola that opens horizontally. To do this, we'll first separate the linear and constant terms from the quadratic term in y.

Step 2: Complete the Square

To rewrite the quadratic part in the form \( a(y-k)^2 \), we need to complete the square for the terms involving \( y \).1. Consider the expression: \( -2y^2 + 4y \).2. Factor out \( -2 \) from the quadratic and linear terms: \[ -2(y^2 - 2y) \]3. To complete the square inside the parenthesis, take the coefficient of \( y \) (which is -2), divide by 2 to get -1, and square it to add 1. So inside we have: \[ -2(y^2 - 2y + 1 - 1) = -2((y-1)^2 - 1) \]4. Simplify this to get: \[ -2(y-1)^2 + 2 \].5. Substitute it back into the equation: \[ x = -2(y-1)^2 + 2 + 1 = -2(y-1)^2 + 3 \].

Step 3: Write in Standard Form

The standard form of a parabola opening horizontally is \[ x = a(y-k)^2 + h \]. Our equation now matches this form:\[ x = -2(y-1)^2 + 3 \].Here, \( a = -2 \), \( h = 3 \), and \( k = 1 \).

Step 4: Determine the Vertex

The vertex form of the equation \( x = a(y-k)^2 + h \) reveals the coordinates of the vertex, which are \( (h, k) \).Substitute the values we've identified:\( h = 3 \), \( k = 1 \).Therefore, the vertex of the parabola is \( (3, 1) \).

Step 5: Understand the Parabola's Direction

Since \( a = -2 \), which is negative, the parabola opens to the left.

Key Concepts

Standard Form

The standard form of a parabola helps us express its equation in a way that easily reveals its shape and orientation. For a parabola opening sideways (horizontally), the standard form is written as

• \( x = a(y-k)^2 + h \)

Here, \( a \), \( k \), and \( h \) are constants that play specific roles:

• \( a \) determines the width and the direction of the opening.
• \( h \) and \( k \) are the coordinates of the vertex, the highest or lowest point depending on the orientation.

By putting an equation into this format, it becomes much easier to graph the parabola and predict its crucial features. This was demonstrated in the given exercise when the equation \( x = -2y^2 + 4y + 1 \) was converted to its equivalent standard form \( x = -2(y-1)^2 + 3 \). This format clearly shows that the parabola is horizontally oriented.

Vertex

Understanding the vertex of a parabola is important because it represents the turning point or the point of symmetry of the curve. For horizontally opening parabolas, the vertex is the point where the parabola changes direction. The standard form equation

• \( x = a(y-k)^2 + h \)

reveals this vertex with its coordinates as \( (h, k) \). In our exercise, with the formula transformed to

• \( x = -2(y-1)^2 + 3 \)

we identify \( h = 3 \) and \( k = 1 \), so the vertex is at \((3, 1)\). The vertex is crucial for graphing since it sets the central point of the parabola, and knowing it helps us sketch the graph accurately. It's essential to ensure \( h \) and \( k \) are correctly identified to situate the vertex properly.

Horizontal Opening

A horizontal opening means the arms of the parabola extend left and right rather than up and down. This type of orientation is distinguished by parabolas where the x-variable is isolated in their equations, such as

• \( x = a(y-k)^2 + h \)

In this format, the direction of opening is influenced by the sign of \( a \). If \( a \) is positive, the parabola opens to the right. If \( a \) is negative, as in our example with \( a = -2 \), the parabola opens to the left. Horizontal opening impacts the interpretation and visualization of the parabola, as different directions will change how the vertex and the axis of symmetry line up with the graph's axes. This orientation needs to be carefully considered when analyzing problems involving reflections or cutting between intersections.

Complete the Square

Completing the square is a mathematical process used to simplify quadratic expressions and it helps to convert a general quadratic equation into its standard (vertex) form. This technique enhances understanding and aids in graphing.
To complete the square for terms involving \( y \), you follow these steps:

• Isolate the quadratic and linear y terms.
• Factor out any coefficients of \( y^2 \).
• Add and subtract a number to transform the quadratic expression into a perfect square trinomial.

In our solution:

• From \( -2(y^2 - 2y) \), add and subtract \( 1 \) within the parentheses to create \( (y-1)^2 \).
• This adjustment rewrites the quadratic as \( -2((y-1)^2 - 1) \).

Substituting back gives the standard form:

• \( x = -2(y-1)^2 + 3 \)

This process not only beautifies the equation but reveals key features like the vertex and direction of opening, assisting in drawing accurate graphs.