Problem 71 - Intermediate Algebra

Question

Sketch the graph of each equation. If the graph is a parabola, find its vertex. If the graph is a circle, find its center and radius. $$x=y^{2}+8 y-4$$

Step-by-Step Solution

Answer

The graph is a parabola with vertex at (-20, -4), opening to the right.

Step 1: Identify the Equation Type

The given equation is \( x = y^2 + 8y - 4 \). To determine the graph type, identify the terms. The equation has a quadratic term \( y^2 \), indicating it represents a parabola. Additionally, since \( x \) is expressed in terms of \( y \), the parabola opens left or right.

Step 2: Complete the Square

Rearrange the equation to make it easier to read: \( x = y^2 + 8y - 4 \). Focus on the \( y \) terms: \( y^2 + 8y \). To complete the square, take half of the coefficient of \( y \), which is 8, giving 4. Square it to get 16. Add and subtract 16 inside the equation: \( x = (y^2 + 8y + 16) - 16 - 4 \).

Step 3: Simplify the Equation

Now, modify the equation with the square completion: \( x = (y+4)^2 - 20 \). This represents a horizontal parabola in the vertex form \( x = a(y-k)^2 + h \).

Step 4: Identify the Vertex

From the equation \( x = (y+4)^2 - 20 \), identify the vertex form \( x = a(y-k)^2 + h \). The vertex is at the point \((h, k)\), which is \((-20, -4)\).

Step 5: Sketch the Graph

Plot the vertex \((-20, -4)\) on a coordinate plane. Since the parabola is in the form \( x = (y+4)^2 - 20 \), it opens to the right (as the \( y^2 \) term is positive). Draw the parabolic curve starting from the vertex, opening to the right symmetrically.

Key Concepts

Completing the Square

Completing the square is a technique used in algebra to simplify quadratic expressions by transforming them into a perfect square trinomial plus or minus a constant number. This method is especially helpful for solving quadratic equations and for rewriting equations to easily find their properties, such as the vertex of a parabola.

To complete the square for the expression \( y^2 + 8y \), we identify the coefficient of the linear term \( y \), which is 8 in this case. Here are the steps to complete the square:

• Take half of 8 (the coefficient of \( y \)), which results in 4.
• Square 4 to get 16.
• Add and subtract this result (16) within the equation so it stays balanced: \( y^2 + 8y + 16 - 16 \).

This results in \( (y+4)^2 - 16 \). Now, the equation \( x = (y+4)^2 - 20 \) is easier to interpret, because it shows the parabola's features prominently.

Vertex of a Parabola

The vertex of a parabola is a crucial feature, as it is the point where the parabola changes direction. It is either the highest or the lowest point on the graph, depending on the parabola's orientation.

In the vertex form of a quadratic equation, which is either \( y = a(x-h)^2 + k \) or \( x = a(y-k)^2 + h \), the vertex is represented by the point \((h, k)\). Understanding this form is essential for quickly identifying the vertex.

• For the equation \( x = (y+4)^2 - 20 \), we see that it is in the form \( x = a(y-k)^2 + h \).
• Here, \( a = 1 \), \( k = -4 \), and \( h = -20 \).
• Thus, the vertex of this parabola is at \((-20, -4)\).

Knowing the vertex helps in sketching the curve accurately and understanding the nature of the parabolic graph.

Quadratic Equations

Quadratic equations are polynomial equations of degree two, often taking the form \( ax^2 + bx + c = 0 \). These equations can be expressed in various forms to reveal different features, such as the standard form, factored form, or vertex form.

A distinguishing property of parabolas derived from quadratic equations is their U-shape, which can open upwards, downwards, left, or right depending on the terms involved. Our focus equation, \( x = y^2 + 8y - 4 \), is a quadratic equation in terms of \( y \). This shows us that the parabola opens horizontally.

• When rearranged to \( x = (y+4)^2 - 20 \), it becomes clear that this is a horizontal parabola opening to the right because the \( y^2 \) term is positive.
• This characteristic is common in equations where the x-term is isolated (expressions involving \( y \) are squared).
• Understanding how these equations are structured enables students to predict the shape and orientation of the graph without needing to plot extensive points.

Quadratic equations in different forms facilitate solving, graph sketching, and analyzing the properties of the parabola they represent.