Problem 108 - A Graphical Approach to Precalculus with Limits

Question

Graph each parabola by hand, and check using a graphing calculator. Give the vertex, axis, domain, and range. $$x=y^{2}+2 y-8$$

Step-by-Step Solution

Answer

Vertex: (-9, -1); Axis: y = -1; Domain: all real numbers; Range: all real numbers.

Step 1: Convert the Equation into Standard Form

To graph the parabola, we start by rearranging the equation to its standard form. The given equation is:\[ x = y^2 + 2y - 8 \]We can convert this into the standard form for a sideways parabola \( (y - k)^2 = 4p(x - h) \). First, we complete the square on the \( y \) terms.Rearrange terms related to \( y \):\[ x + 8 = y^2 + 2y \]Complete the square by adding and subtracting \( 1 \) (since \( (\frac{2}{2})^2 = 1 \)):\[ x + 8 = (y^2 + 2y + 1) - 1 \]This equation becomes:\[ x + 8 = (y + 1)^2 - 1 \] Now, move the \( -1 \) to the other side:\[ x + 9 = (y + 1)^2 \] Reorder it:\[ (y + 1)^2 = x + 9 \]

Step 2: Identify the Vertex

From the standard form \( (y - k)^2 = 4p(x - h) \), we identify the vertex of the parabola.The equation we have is:\[ (y + 1)^2 = x + 9 \]This implies the vertex \( (h, k) = (-9, -1) \).

Step 3: Determine the Axis of Symmetry

The axis of symmetry is a line that goes through the vertex and is parallel to the y-axis for a sideways parabola.Since the vertex is \((-9, -1)\), the axis of symmetry is the line:\[ y = -1 \]

Step 4: Determine Domain

The domain of a sideways parabola like \((y + 1)^2 = x + 9\) is defined by the \(x\)-values that the graph will cover. Since there is no restriction on \(x\), all real numbers are included in the domain.The domain is:\[ x \in (-\infty, \infty) \]

Step 5: Determine Range

The range of a sideways parabola is determined by the \(y\)-values it can take. The given equation indicates that \(y\) can be any real number, because the parabola does not open upwards or downwards.The range is:\[ y \in (-\infty, \infty) \]

Step 6: Graph the Parabola

On graph paper, plot the vertex at \((-9, -1)\). Since \((y + 1)^2 = x + 9\), the parabola opens to the right.Plot a few points by choosing values for \(y\) and solving for \(x\). Example:1. If \( y = 0 \), then \( x = 0 + 9 = 9 \), giving the point \((9, 0)\).2. If \( y = -2 \), then \( x = (-3)^2 - 9 = 0 \), giving the point \((0, -2)\).Plot these points, draw a smooth curve through them, and check using a graphing calculator to ensure accuracy.

Key Concepts

Completing the Square

Completing the square is a mathematical technique used to transform a quadratic equation into a perfect square trinomial. This makes it easier to analyze and graph parabolas. When dealing with equations like \( x = y^2 + 2y - 8 \), our goal is to rewrite it in a form that reveals the parabola's properties more clearly.

Here's how we do it:

• Rearrange the terms by isolating the quadratic and linear terms of \( y \). For our example, we have \( x + 8 = y^2 + 2y \).
• Next, determine the value needed to complete the square. This is done by taking half of the coefficient of the linear \( y \) term, squaring it, and then adding and subtracting it within the equation. So, \( (\frac{2}{2})^2 = 1 \), and we adjust the equation to \( x + 8 = (y^2 + 2y + 1) - 1 \).
• Simplify to get the perfect square trinomial: \( (y + 1)^2 \). Now, the equation looks like \( x + 9 = (y + 1)^2 \).

Completing the square helps to create the vertex form of the parabola, making graphing and analysis simpler.

Vertex Form

The vertex form of a parabola gives us direct information about the parabola's vertex, making it much easier to graph. For horizontal parabolas like ours, the vertex form is \((y - k)^2 = 4p(x - h)\). After completing the square, the equation \((y + 1)^2 = x + 9\) tells us the vertex directly.

• The vertex \((h, k)\) can be identified as \((-9, -1)\) from the formula. The \(h\) value comes from \(x + 9\) turning into \(x - (-9)\).
• Similarly, the \(k\) value comes from \( (y + 1) \), which rearranges to \( y - (-1) \).

The vertex form is particularly useful because it highlights symmetry, and transformations from the standard position make it easy to visualize shifts and orientations. Having the vertex means knowing exactly where to start plotting.

Axis of Symmetry

The axis of symmetry in a parabola is a critical line that shows the line where the parabola is mirrored. For sideways parabolas, like \((y + 1)^2 = x + 9\), this line is horizontal.

The axis of symmetry always passes through the vertex. For our case:

• The vertex is \((-9, -1)\), so the line of symmetry is a constant \(y\)-value where \(y = -1\).

Understanding the axis of symmetry is key to graphing because it ensures that points on one side of the parabola have corresponding points on the other. This symmetry helps simplify the process of plotting additional points after establishing the vertex.

Domain and Range

Identifying the domain and range for parabolas is crucial for understanding the extent of the graph. Let's break it down for our sideways parabola:

• The domain refers to all possible \(x\)-values that the parabola can cover. Since our equation does not have restrictions for \(x\) (e.g., no square root causing non-real numbers), the domain is \( x \in (-\infty, \infty) \).
• The range pertains to the possible \(y\)-values. Similar to the domain but looks at \(y\) instead, our equation shows \(y\) is unrestricted, giving a range of \( y \in (-\infty, \infty) \).

Understanding both these aspects ensures that when you graph, you know the complete span of the curve, helping in both manual sketching and digital verification.