Problem 14 - Intermediate Algebra

Question

A. Find \(a, h,\) and \(k: y=6(x-5)^{2}-9\) B. Find \(a, h,\) and \(k: x=-3(y+2)^{2}+1\)

Step-by-Step Solution

Answer

A: \(a=6, h=5, k=-9\); B: \(a=-3, h=1, k=-2\).

Step 1: Identify the standard form of the parabola equation

To solve for \(a, h,\) and \(k\), we need to identify the equation form. The standard form of a parabola is \( y = a(x-h)^2 + k \) for a vertically oriented parabola and \( x = a(y-k)^2 + h \) for a horizontally oriented parabola.

Step 2: Extract parameters for part A

For the equation \( y = 6(x-5)^2 - 9 \), compare it with the standard form \( y = a(x-h)^2 + k \). Here, \(a = 6\), \(h = 5\), and \(k = -9\).

Step 3: Extract parameters for part B

For the equation \( x = -3(y+2)^2 + 1 \), compare it with the standard form for a horizontal parabola \( x = a(y-k)^2 + h \). Here, \(a = -3\), \(h = 1\), and \(k = -2\).

Key Concepts

Vertex Form

The vertex form of a parabola is a powerful tool for quickly understanding its graph. It's written as \( y = a(x-h)^2 + k \) for vertical parabolas and \( x = a(y-k)^2 + h \) for horizontal ones. This format helps identify key features of the parabola:

• "a": The coefficient that determines the width and direction of the parabola. If \(a > 0\), the parabola opens upwards or to the right, while \(a < 0\) opens downwards or to the left. Larger values of \(|a|\) result in narrower parabolas, whereas smaller values result in wider ones.
• "(h, k)": Represents the vertex of the parabola. The vertex is a significant point because it's the highest or lowest in a vertical parabola, and the leftmost or rightmost in a horizontal one. By setting \(x = h\) or \(y = k\), we find the axis of symmetry for vertical and horizontal parabolas, respectively.

Using the vertex form, students can easily graph or reconfigure the equation of a parabola to see shifts or reflections without needing to complete the square each time.

Horizontal Parabola

When a parabola opens sideways, it is called a horizontal parabola. The general form is \( x = a(y-k)^2 + h \). This kind of parabola is unique because:

• Direction of opening: Determined by the sign of \(a\). If \(a > 0\), the parabola opens to the right. If \(a < 0\), it opens to the left.
• Vertex: Given by the coordinates \((h, k)\). This is the point where the parabola changes direction and lies perfectly centered on the axis of symmetry.
• Focus and directrix: For horizontal parabolas, the focus lies at \((h + \frac{1}{4a}, k)\), while the directrix is the line \(x = h - \frac{1}{4a}\).

Understanding horizontal parabolas is important, as they behave differently from their vertical counterparts. The shift in orientation requires a different perspective on how these equations are interpreted and graphed.

Vertical Parabola

Vertical parabolas are probably the most familiar type. They open upwards or downwards based on the equation \( y = a(x-h)^2 + k \). Here are their main characteristics:

• Direction and Shape: The sign and value of \(a\) determine how the parabola looks. A positive \(a\) indicates that the parabola opens upwards, while a negative \(a\) means it opens downwards. Larger absolute values of \(a\) give a narrow shape, while smaller ones let the parabola spread wide.
• Vertex: Detailed by \((h, k)\), the vertex is either the lowest or highest point on the graph, depending on whether the parabola opens upwards or downwards.
• Focus and directrix: Identifying these helps in understanding the reflective property of parabolas. The focus for a vertical parabola is at \((h, k + \frac{1}{4a})\), and the directrix is the line \(y = k - \frac{1}{4a}\).

By analyzing vertical parabolas in their vertex form, students can easily sketch them and predict their behavior without delving into complicated algebraic manipulations.