Problem 40 - Algebra 2

Question

Write an equation of a parabola with a vertex at the origin. directrix \(x=9\)

Step-by-Step Solution

Answer

The equation of the parabola with a vertex at the origin and directrix \(x=9\) is \(x = 1/36 * y²\).

Step 1: Identify the known values

The vertex at the origin signifies that h=0 and k=0. The directrix is given as x=9. This means that the parabola opens to the left side.

Step 2: Determine the distance from vertex to directrix

The distance between the vertex and the directrix is equal to the absolute value of the x-coordinate of the directrix (because the y-coordinate of the vertex is 0), which is d=|9-0|=9.

Step 3: Determine the coefficient a

Use the formula \(a=1/(4d)\) to find the value for a. Plug in the value of d found in step 2: \(a=1/(4*9)=1/36\).

Step 4: Apply the values to the formula

Finally, substitute the known values into the parabolic equation. Since the parabola is opening to the left, we need to use the formula \(x = ay²\). Substituting a yields: \(x=1/36 * y²\).

Key Concepts

Understanding the Vertex of a Parabola

The vertex of a parabola is an essential concept when it comes to graphing and interpreting parabolic equations. It's essentially like the "tip" or the turning point of the parabola, and understanding its position is key to working with these equations. In the equation of a parabola that opens sideways, such as in this exercise, the standard form is \[ x = a(y-k)^2 + h \] where

• \( h \) is the x-coordinate of the vertex.
• \( k \) is the y-coordinate of the vertex.

The vertex is simply the point \( (h, k) \) . So, when a problem states that the vertex is at the origin, it means that both \( h \) and \( k \) are zero, simplifying the equation to \( x = ay^2 \) for parabolas that open sideways. Knowing the vertex helps in easily determining other features of the parabola, such as the focus and axis of symmetry.

Exploring the Directrix of a Parabola

The directrix of a parabola is a fixed line that, together with the focus, helps define the shape and position of the parabola. It's like an invisible guide that ensures the parabola maintains its curve accurately. For horizontal parabolas like the one in this problem, the directrix is parallel to the y-axis, and the equation for it is typically \( x = constant \). The significance of the directrix is in its distance from the vertex, denoted as \( d \).

• In this particular exercise, the directrix is at \( x = 9 \).
• Since the vertex is at the origin \( (0, 0) \), the distance \( d \) is simply \( 9 \), being the absolute difference between \( x = 9 \) and \( x = 0 \).

Understanding the position of the directrix helps in determining the *a* coefficient, which directly influences how "steep" the parabola appears.

Determining Parabola Orientation

Parabola orientation refers to the direction in which a parabola opens. This is influenced primarily by the placement of the directrix and vertex, and the sign of the coefficient \( a \) in a parabolic equation. In standard parabolic equations such as \( x = ay^2 \) or \( y = ax^2 \), the sign of \( a \) dictates if the parabola opens upwards, downwards, left, or right.For this exercise:

• Since the directrix \( x = 9 \) is to the right of the origin and the parabola must maintain its set distance opposite the directrix, the parabola opens to the left.
• In a horizontal parabola, like this one, if \( a \) is positive, it opens to the right; if negative, it opens to the left. However, because here we deal with \( x = ay^2 \) instead of \( ax^2 \), the directrix itself guides the direction precisely.

This oriented understanding allows more precision in plotting or modeling parabolas in graphical solutions.