Problem 109 - A Graphical Approach to College Algebra

Question

Graph each parabola by hand, and check using a graphing calculator. Give the vertex, axis, domain, and range. $$x=-4 y^{2}-4 y-3$$

Step-by-Step Solution

Answer

Vertex: (-2, -0.5); Axis: y = -0.5; Domain: (-∞, ∞); Range: (-∞, ∞).

Step 1: Rewrite the Equation

First, rearrange the given equation to make it easier to graph. We have the equation \( x = -4y^2 - 4y - 3 \). Factor out \(-4\) from the terms involving \(y\): \( x = -4(y^2 + y) - 3 \).

Step 2: Complete the Square

To complete the square, take the expression inside the parenthesis: \( y^2 + y \). Take half of the coefficient of \(y\) (which is \(\frac{1}{2}\) ), square it \(\left(\frac{1}{2}\right)^2 = \frac{1}{4}\), and add and subtract \(\frac{1}{4}\) within the parenthesis: \( y^2 + y + \frac{1}{4} - \frac{1}{4} \). Thus, \( x = -4\left((y + \frac{1}{2})^2 - \frac{1}{4}\right) - 3 \).

Step 3: Simplify the Equation

Distribute \(-4\) into the completed square form: \( x = -4(y + \frac{1}{2})^2 + 1 - 3 \). Which simplifies to \( x = -4(y + \frac{1}{2})^2 - 2 \). This equation is now in the form \( x = a(y - k)^2 + h \).

Step 4: Identify the Vertex

The vertex form \( x = a(y - k)^2 + h \) shows that the vertex is \((h, k)\). Here \(h = -2\) and \(k = -\frac{1}{2}\), so the vertex is \((-2, -\frac{1}{2})\).

Step 5: Determine the Axis of Symmetry

Since the parabola is in the form \( x = a(y - k)^2 + h \), the axis of symmetry is a horizontal line through \( k \). In this case, it is \( y = -\frac{1}{2} \).

Step 6: Find the Domain

For any quadratic equation where \( x \) is expressed in terms of \( y \), the domain is the set of all possible \( x \)-values. Here, all real numbers are possible for \( x \), giving us the domain: \((-\infty, \infty)\).

Step 7: Determine the Range

The parabola opens towards the left as evidenced by the negative coefficient \(-4\) in front of \( (y + \frac{1}{2})^2 \). Since it opens leftwards, the range of possible \( y \)-values is from \(-\infty\) to \(\infty\).

Step 8: Graph the Parabola

Plot the vertex \((-2, -\frac{1}{2})\) on a coordinate plane, draw the axis of symmetry at \( y = -\frac{1}{2} \), and sketch the parabola opening leftward. Use a graphing calculator to verify the sketch.

Key Concepts

Vertex

In a parabola, the vertex is a special point where the curve changes direction. Think of it as the peak or the lowest dip when looking at the curve. For the given equation, by rearranging and simplifying, we put it in a form that clearly shows the vertex: \( x = -4(y + \frac{1}{2})^2 - 2 \). Here, the vertex is located at the point \((-2, -\frac{1}{2})\). This point is crucial because it tells us the position on the graph where the change in direction occurs.

• If the parabola opens upwards or downwards, this vertex will represent the minimum or maximum point, respectively.
• In our case, the parabola opens leftwards, showing that the vertex is the most right-leaning point of the curve.

A visual representation helps in understanding this concept better. One way to remember the vertex's role is that it acts like a balancing point of the parabola, where it transitions direction.

Axis of Symmetry

The axis of symmetry of a parabola is a line that splits the curve into two perfect, mirror-image halves. It's particularly useful because it helps in understanding the symmetrical nature of the parabola.In our exercise, which is in the form \( x = a(y - k)^2 + h \), the axis of symmetry is a horizontal line defined by the value of \( y = k \). For our equation, this line is \( y = -\frac{1}{2} \). You can imagine this line as a mirror running horizontally right through the vertex:

• It allows you to reflect one side of the parabola to get the other side.
• It guarantees that if you know how the parabola behaves on one side, the other side will behave identically once flipped across this axis.

Understanding and drawing this line provides a powerful tool to form an accurate graph of the parabola, simplifying predictions about its behavior.

Domain and Range

The domain and range of a parabola tell us about the span of the graph along the horizontal and vertical directions, essentially marking the boundaries of the curve.

• The **domain** of a parabola, expressed in the form \( x = a(y - k)^2 + h \), refers to the set of all possible \( x \)-values. For our problem, since no restrictions are placed on \( x \), it covers all real numbers: \((-\infty, \infty)\).
• The **range** describes all the potential \( y \)-values the parabola can take. Due to the negative coefficient of \(-4\), indicating the parabola opens to the left, \( y \) can freely vary from \(-\infty \) to \( \infty \).

While the domain here is always the entire real line, different parabolas might restrict these values based on their opening direction. The range confirms the unbounded behavior vertically, encompassing every possible \( y \)-coordinate.