Problem 89 - Intermediate Algebra

Question

Explain the error in each statement. The graph of \(x=5(y+5)^{2}+1\) is a parabola with vertex \((-5,1)\) and opening to the right.

Step-by-Step Solution

Answer

The vertex is at (1, -5), not (-5, 1). The graph opens rightward.

Step 1: Simplify and Reinterpret the Equation

The exercise provides the equation \(x=5(y+5)^{2}+1\). Let's rewrite it to relate it to a standard form. The equation can be written as \(x = 5(y + 5)^2 + 1\) which matches the form \(x = a(y - k)^2 + h\). In this form, \(a = 5\), \(h = 1\), and \(k = -5\). This suggests a horizontal parabola with the vertex at \((1, -5)\), not \((-5, 1)\). The equation is correct in identifying a parabola that opens right but misidentifies the vertex.

Step 2: Identify Parabola's Vertex

For the standard equation form of a horizontally opening parabola \(x = a(y - k)^2 + h\), the vertex is \((h, k)\). Here, \(h = 1\) and \(k = -5\). Therefore, the vertex should be \((1, -5)\) rather than \((-5, 1)\) as mentioned in the statement.

Step 3: Determine the Parabola's Orientation

Given the equation \(x = 5(y + 5)^2 + 1\), since \(a = 5\) (a positive value), the parabola opens to the right. The statement correctly identifies the orientation, but fails in determining the exact vertex.

Key Concepts

Vertex of a Parabola

Understanding the vertex of a parabola is crucial as it represents the point where the curve changes direction. In the context of a parabola, whether it opens vertically or horizontally, the vertex is the point

• at which the parabola is most curved,
• where the axis of symmetry intersects the parabola,
• often representing a minimum or maximum point depending on its orientation.

For horizontal parabolas like the one in the equation \(x = 5(y + 5)^2 + 1\), the vertex is determined using the form \((h, k)\).
The general formula for a horizontally opening parabola is \(x = a(y - k)^2 + h\), where \(h\) and \(k\) are constants that give the vertex its precise coordinates.
By examining these constants in our equation, we find \(h = 1\) and \(k = -5\), leading us to conclude the correct vertex is \((1, -5)\).
This makes the vertex easy to identify once you have recognized the standard form. Misinformation about the vertex often results from overlooking these constants.

Horizontal Opening Parabola

Unlike the more conventional vertical parabolas, a horizontal opening parabola has a different orientation. In a horizontal parabola, the graph either opens to the left or right, depending on the sign of the coefficient \(a\) in its standard form.
For instance, with a positive \(a\) such as in the equation \(x = 5(y + 5)^2 + 1\), the parabola opens to the right. A negative \(a\) would imply an opening to the left.
This change in orientation affects several key aspects of the parabola:

• Direction of the opening: Right for positive \(a\), Left for negative \(a\).
• Position of vertex, which remains unchanged but perceived differently due to direction.
• Symmetry axis, horizontally aligned, runs through the vertex.

By understanding whether the parabola opens right or left, we can predict the general shape and direction of the graph relative to its vertex.

Standard Form of a Parabola

The standard form of a parabola is crucial for identifying its properties. For horizontal parabolas, the standard form is \(x = a(y - k)^2 + h\). This differs from vertical parabolas, which are usually in the form \(y = a(x - h)^2 + k\).
The horizontal form specifically tells us several things about the parabola:

• The value \(a\) dictates the width and direction; a larger absolute value means a narrower parabola.
• Vertex can be directly read as \((h, k)\) in the horizontal form, contrary to the vertical form where it appears as \((h, k)\).
• Helps easily determine axis of symmetry, which will be parallel to the x-axis in a horizontal parabola.

Understanding the standard form allows us to transform from an equation to a tangible graph by considering these constants and realizing their practical implications on the graph's shape and orientation. It equips us to quickly sketch the parabola and pinpoint core features such as the vertex and direction of the opening.