Problem 106 - 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. $$x=-2(y+3)^{2}$$

Step-by-Step Solution

Answer

Vertex: (0, -3). Axis: y = -3. Domain: (-∞, 0]. Range: (-∞, ∞).

Step 1: Identify the Form of the Equation

The given equation is \( x = -2(y+3)^2 \). This is in the form \( x = a(y-k)^2 + h \), where \( a = -2 \), \( k = -3 \), and \( h = 0 \). This is a parabola that opens horizontally.

Step 2: Find the Vertex

The vertex form \( x = a(y-k)^2 + h \) makes it simple to identify the vertex of the parabola. The vertex is \((h, k) \). Substituting the identified values gives the vertex as \((0, -3)\).

Step 3: Determine the Axis of Symmetry

For a horizontally opening parabola, the axis of symmetry is a horizontal line at the y-coordinate of the vertex. Therefore, the equation for the axis of symmetry is \( y = -3 \).

Step 4: Find the Domain

Since the parabola opens horizontally to the left (as indicated by \( a = -2 \), which is negative), the domain consists of all x-values that the parabola will cover. Here, that's all points to the left of and including the vertex. Hence, the domain is \(( -\infty, 0 ]\).

Step 5: Find the Range

The range of a horizontally opening parabola is determined by its possible y-values. Since the parabola doesn't open vertically, it covers all real y-values. Hence, the range is \(( -\infty, \infty )\).

Step 6: Graph the Parabola by Hand

Draw the y-axis and x-axis on a graph paper. Plot the vertex at \((0, -3)\). Since it opens to the left, draw a horizontal parabola by marking points symmetrically around and moving leftwards from the vertex.

Step 7: Verify Using a Graphing Calculator

Input the equation \( x = -2(y+3)^2 \) into a graphing calculator. Check that the graph matches your hand-drawn sketch, with the vertex at \((0, -3)\), opening to the left.

Key Concepts

Vertex of Parabola

The vertex of a parabola is a significant point that denotes its highest or lowest position, depending on the orientation. In the equation \( x = -2(y+3)^2 \) given in vertex form \( x = a(y-k)^2 + h \), the vertex can be directly identified because it is indicated by the terms \( (h, k) \). Here, the values \( h = 0 \) and \( k = -3 \) give us a vertex at the point \( (0, -3) \).

• The vertex is the turning point of the parabola, where it changes direction.
• For a horizontally opening parabola, the vertex is a point on the x-axis plane corresponding to \( x = h \).

Understanding the vertex is crucial as it also provides the highest or lowest x-value for vertically opening parabolas and the highest or lowest y-value for horizontally opening ones. Here, since our parabola opens horizontally, the vertex helps us to determine other features like the axis of symmetry and the precise direction it opens towards.

Axis of Symmetry

The axis of symmetry of a parabola is an imaginary line that serves as a mirror, dividing the parabola into two equal halves. For horizontally opening parabolas, like in our equation \( x = -2(y+3)^2 \), the axis of symmetry is a horizontal line.
You can find this axis by simply looking at the y-coordinate of the vertex. In this case, with the vertex at \( (0, -3) \), the axis of symmetry is the line \( y = -3 \). Essentially, this means every point on the parabola has a mirror point equally distant from this line.

• The axis of symmetry helps in drawing the parabola and ensuring that it is symmetrical.
• For vertically opening parabolas, the axis of symmetry is vertical, represented by \( x = h \).
• For horizontally opening parabolas, the axis is horizontal, represented by \( y = k \).

This line is critical in determining the balance of the parabola, as it ensures the graph is evenly distributed on either side.

Domain and Range

In mathematics, understanding the domain and range of a function is key to knowing which values will be produced as outputs. For the given horizontal parabola \( x = -2(y+3)^2 \), we explore both the domain and the range:
Domain:
Since the parabola opens to the left and the vertex is at \( x = 0 \), the domain includes all x-values \( x \leq 0 \). This means the parabola stretches infinitely towards the negative x-axis, covering values \((-\infty, 0] \).
Range:
For horizontal parabolas, the range includes all real y-values, as there is no restriction on y. Thus, the range for this parabola is \((-\infty, \infty) \).

• Domain: A set of x-values that the parabola touches or covers. For horizontal parabolas, it is determined by movement towards either the right or left from the vertex.
• Range: A set of y-values that the parabola spans, covering all possible y-values between its upward or downward extent.

This understanding ensures you can safely predict and interpret the limits of a parabola on a graph.