Problem 69 - Intermediate Algebra

Question

Write each equation in standard form, if it is not already so, and graph it. If the graph is a circle, give the coordinates of its center and its radius. If the graph is a parabola, give the coordinates of its vertex. $$ x=-y^{2}+1 $$

Step-by-Step Solution

Answer

The vertex of the parabola is (1, 0). The parabola opens to the left.

Step 1: Identify the Type of Equation

The given equation is \( x = -y^2 + 1 \). Notice that it is quadratic in \( y \), but linear in \( x \). Therefore, it represents a parabola.

Step 2: Rewrite the Equation in Standard Form

The standard form of a parabola that opens either left or right is \((y - k)^2 = 4p(x - h)\). Let's rewrite the given equation:Start with: \[ x = -y^2 + 1 \]Reorder to: \[ -y^2 = x - 1 \]Multiply through by \(-1\) to match the standard form:\[ y^2 = - (x - 1) \]This is now in the form \((y - 0)^2 = -1(x - 1)\), where \(k = 0\), \(h = 1\), and \(4p = -1\).

Step 3: Determine the Vertex of the Parabola

From the standard form \((y - k)^2 = 4p(x - h)\), the vertex of the parabola is \((h, k)\). For our equation \( y^2 = -(x - 1) \), we see that:\(h = 1\) and \(k = 0\).So the vertex of the parabola is \((1, 0)\).

Step 4: Graph the Equation

The equation \( y^2 = -(x - 1) \) represents a parabola opening to the left because the term with \(x\) is negative. With vertex \((1, 0)\), plot a point here and sketch the parabola opening towards the left from the vertex.

Key Concepts

Standard Form

The concept of the standard form for parabolas is essential in understanding their orientation and position on a graph. When dealing with a parabola that opens left or right, its standard form is expressed as:
\[(y - k)^2 = 4p(x - h)\]
Here,

• \(h\) and \(k\) are the coordinates of the vertex of the parabola. \(h\) represents the horizontal shift, while \(k\) represents the vertical shift of the parabola's vertex from the origin.
• \(4p\) determines the "width" and direction of the parabola. If \(p\) is positive, the parabola opens to the right; if \(p\) is negative, it opens to the left.

To convert an equation into this standard form, like in the given equation \(x = -y^2 + 1\), rearrange terms to isolate the \(y^2\) term on one side.
For example, you might start with rearranging to get:
\[y^2 = -(x - 1)\]
This format fulfills the standard form requirements, helping to easily identify the parabola's characteristics.

Vertex

The vertex of a parabola is a crucial point that represents the "peak" or "lowest point" depending on the orientation. For horizontally oriented parabolas, the vertex signifies where the parabola stops changing direction laterally.
In the standard form \((y - k)^2 = 4p(x - h)\), the vertex is given directly by the coordinates \((h, k)\).

• This point \((h, k)\) is essential because it provides a reference from which the parabola extends either leftward or rightward.
• Understanding the vertex clarifies how the graph is situated relative to the axes, which is especially useful when sketching.

For the equation \(y^2 = -(x - 1)\), the vertex is found directly at the point \((1, 0)\). This location tells us that the graph is centered horizontally at \(x = 1\) and rests along the \(y = 0\) axis.
Knowing the vertex assists in establishing accurate plots and predicting the parabola’s behavior.

Graphing a Parabola

Graphing a parabola involves understanding its shape, direction, and key points like the vertex. After transforming an equation to its standard form, plotting becomes straightforward.
For the equation \(y^2 = -(x - 1)\), we observe:

• The negative sign before \((x - 1)\) confirms the parabola opens to the left, distinctive for parabolas expressed in terms of \(y^2\) instead of \(x^2\).
• The vertex at \((1, 0)\) serves as a central point. From here, we can sketch the curve that extends leftward, as determined by the orientation hint provided by the negative \(4p\).

In sketching, the vertex is plotted first, followed by plotting additional points symmetrically around it to ensure graph accuracy.
Consider using a table of values to find other points along the parabola, which can help visualize how sharply it curves.
By comprehending these constituents and the influence of each part of the equation, sketching becomes an approachable challenge, creating an accurate visual representation of the mathematical concept.