Problem 23 - Beginning and Intermediate Algebra

Question

Given a quadratic equation of the form \(x=a(y-k)^{2}+h,\) answer the following. If \(a\) is negative, which way does the parabola open?

Step-by-Step Solution

Answer

When \(a\) is negative, the parabola in the given quadratic equation, \(x=a(y-k)^{2}+h\), opens to the left.

Step 1: Direction of parabola based on the value of 'a'

A parabola can open either left or right. If the value of 'a' is positive, the parabola opens to the right (away from the vertex). If the value of 'a' is negative, the parabola opens to the left (towards the vertex). In this specific exercise, we are asked the direction of the parabola when 'a' is negative.

Step 2: Conclusion

Since the value of 'a' in this exercise is negative, the parabola in the given quadratic equation, \(x=a(y-k)^{2}+h\), will open to the left.

Key Concepts

Understanding Quadratic Equations

Quadratic equations are a type of polynomial equation that can be written in the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(x\) represents an unknown variable. These equations are called "quadratic" because of the presence of \(x^2\), which is the highest degree term in the equation. Quadratics produce parabolas when graphed, forming a U-shaped curve.
In certain situations, quadratic equations are expressed differently to emphasize their geometric properties.
For instance, in the exercise given, the equation \(x=a(y-k)^2+h\) emphasizes the orientation and position of the parabola in relation to the axes.

Parabola Orientation Basics

The orientation of a parabola is determined by the coefficient \(a\) found in the quadratic equation. This coefficient dictates whether the parabola opens upwards, downwards, left, or right.

• If \(a\) is positive in a vertical parabola, it opens upwards, like a regular smiley face.
• If \(a\) is negative, it opens downwards, like a frown.
• In our horizontal equation \(x=a(y-k)^2+h\), a positive \(a\) means the parabola opens to the right, while a negative \(a\) indicates it opens to the left.

Understanding this can help in graphing parabolas and predicting the direction in which they will extend. Since the exercise specifies \(a\) as negative, it confirms that the parabola will open to the left.

Vertex Form of a Parabola

The vertex form of a parabola makes it easy to recognize the vertex and orientation of the curve. It's written as \((y-k)^2 = 4p(x-h)\) for horizontal, or \((x-h)^2 = 4p(y-k)\) for vertical parabolas. In this format:

• The point \((h, k)\) is the vertex of the parabola.
• The value of \(p\) determines the distance from the vertex to the focus or directrix and influences how "wide" or "narrow" the parabola is.
• In the exercise example \(x=a(y-k)^2+h\), the presence of \((y-k)^2\) signifies a horizontal orientation.

The vertex form clearly indicates the shift in position from the origin, making it easier to graph parabolas without needing to complete the square or use other transformations.