Problem 37 - Algebra A Combined Function

Question

Identify whether each equation, when graphed, will be a parabola, circle,ellipse, or hyperbola. Sketch the graph of each equation. If a parabola, label the vertex. If a circle, label the center and note the radius. If an ellipse, label the center. If a hyperbola, label the \(x\) - or \(y\) -intercepts. $$ x=y^{2}+4 y-1 $$

Step-by-Step Solution

Answer

The equation represents a parabola with the vertex at \((-5, -2)\).

Step 1: Analyze Equation Type

The given equation is \(x = y^2 + 4y - 1\). We need to determine what type of conic section this equation represents.This equation has the form \(x = Ay^2 + By + C\), which is a quadratic in \(y\). Hence, it represents a parabola oriented along the \(x\)-axis.

Step 2: Complete the Square

To find the vertex of the parabola, let's complete the square. Start with the term \(y^2 + 4y\):1. Take half the coefficient of \(y\), which is \(4\). Half of \(4\) is \(2\).2. Square this value: \(2^2 = 4\).3. Add and subtract this square inside the equation.The equation becomes: \[ x = (y^2 + 4y + 4) - 4 - 1 = (y + 2)^2 - 5 \].

Step 3: Identify Vertex

The completed square form of the equation is \(x = (y + 2)^2 - 5\). This is a parabola with the vertex at \((-5, -2)\).

Step 4: Sketch the Parabola

Given the equation \(x = (y + 2)^2 - 5\), we know this is an upward-opening parabola along the \(y\)-axis shifted left by 5 units and down by 2 units. Sketch it on a graph with the vertex labeled at \((-5, -2)\).

Key Concepts

Parabola

A parabola is a type of conic section that looks like a U-shaped curve. This curve is symmetrical, meaning one side mirrors the other. Parabolas can open upwards, downwards, left, or right, depending on their equations. They are formed by the graph of a quadratic equation.
A parabola has some distinct features, including its vertex, axis of symmetry, and directrix. In simple terms:

• The vertex is the turning point of the parabola.
• The axis of symmetry is a vertical or horizontal line that splits the parabola into two identical halves.
• The directrix is a line that lies outside of the parabola, helping to define it geometrically.

In this exercise, the equation given is already identified as a parabola aligned along the x-axis, represented as an equation of the form \(x = Ay^2 + By + C\). Unlike the default \(y = Ax^2 + Bx + C\), in this case, the parabola opens horizontally.

Vertex

The vertex of a parabola is its peak or lowest point, depending on its orientation. This is a crucial feature because it gives us insights into the parabola's position and shape. Finding the vertex involves a method called 'completing the square'. In this exercise, we started from the original quadratic expression \(y^2 + 4y\).
To complete the square:

• First, halve the coefficient of \(y\). Here, it's 4, and half of 4 is 2.
• Square it: \(2^2 = 4\).
• Add and subtract the squared term inside the quadratic expression.

This transformation turns the expression into \((y + 2)^2 - 5\), signifying that the parabola's vertex for the equation \(x = (y + 2)^2 - 5\) is located at \((-5, -2)\). This point represents the minimum point of the parabola along the x-axis.

Quadratic Equation

A quadratic equation is a polynomial equation of degree two, which generally takes the form \(ax^2 + bx + c = 0\). These equations graph as parabolas. Depending on the orientation (whether aligned with the x-axis or y-axis), the parabolas can open either vertically or horizontally.
Quadratic equations are instrumental in various scientific computations and everyday problem-solving. They exhibit the following pivotal features in graphing:

• The coefficient \(a\) determines the parabola's direction (upwards if positive, downwards if negative for vertical alignment).
• The vertex form of a quadratic equation, \(a(x-h)^2+k\) or \(a(y-k)^2+h\), shows the vertex \((h, k)\).
• The axis of symmetry ensures the parabola is a mirror image on both sides of this line.

In this exercise, the quadratic equation in terms of \(y\) \((x = y^2 + 4y - 1)\) specifies that the parabola is oriented along the x-axis due to its horizontal arrangement.