Problem 60 - 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. $$x=-2(y+5)^{2}$$

Step-by-Step Solution

Answer

The equation represents a parabola with a vertex at (0, -5) that opens to the left.

Step 1: Recognize the Equation Type

The given equation is \(x = -2(y+5)^2\). This equation represents a parabola. Parabolas typically have the form \(x = a(y-k)^2 + h\) or \(y = a(x-h)^2 + k\). This helps us identify that this is a horizontal parabola because the squared term is in \((y+5)^2\) rather than \(x^2\).

Step 2: Reformat to Identify the Vertex

The given equation \(x = -2(y+5)^2\) can be compared to the form \(x = a(y-k)^2 + h\). From this, \(h = 0\) and \(k = -5\). Thus, the vertex is \((0, -5)\).

Step 3: Determine the Parabola's Direction

Since the coefficient \(-2\) of \((y+5)^2\) is negative, the parabola opens to the left. Parabolas with a negative coefficient in the \(x = a(y-k)^2 + h\) form open in the direction opposite to the coordinate of the squared term.

Step 4: Sketch the Parabola

To sketch the parabola, plot the vertex \((0, -5)\) on the Cartesian plane. Then, since it opens to the left, draw the parabola extending leftward from the vertex. The width of the parabola is influenced by the coefficient \(-2\), which means it is relatively narrower compared to a parabola with a coefficient closer to zero.

Key Concepts

Equation of a Parabola

Understanding the equation of a parabola is key when it comes to sketching or analyzing its graph. Parabolas are U-shaped curves that can open either upwards, downwards, to the left, or to the right. The general form of a vertical parabola is given by the equation \(y = a(x-h)^2 + k\), while a horizontal parabola can be expressed as \(x = a(y-k)^2 + h\).

In these equations:

• \(a\) determines the width and direction of the parabola. A positive \(a\) value opens upwards or to the right, whereas a negative \(a\) opens downwards or to the left.
• \(h\) and \(k\) represent the coordinates of the vertex, which is the point where the parabola changes direction.

For the given exercise, the equation \(x = -2(y+5)^2\) is a horizontal parabola. This means the opening is towards the left (since the coefficient, \(-2\), is negative) with the squared term involving \((y+5)\), not \(x\).

Recognizing these forms helps you understand which direction the parabola opens and provides the fundamentals for finding the vertex and sketching the graph.

Vertex of a Parabola

The vertex of a parabola is a crucial concept as it is the turning point of the curve. It marks the highest or lowest point on the graph, depending on the direction in which the parabola opens. For vertical parabolas, this is the topmost or bottommost point, whereas for horizontal parabolas, it's the furthest point left or right.

To find the vertex, compare the parabola's equation to its general form. For the given horizontal parabola \(x = -2(y+5)^2\), comparing with the general form \(x = a(y-k)^2 + h\), we can directly determine:

• \(h = 0\)
• \(k = -5\)

Thus, the vertex is the point \((0, -5)\). This point is where the parabola starts its leftward opening. Always remember that the vertex provides a key placement for sketching and understanding the parabola's position on the Cartesian plane.

Having the vertex helps set the base of your graph, making it easier to draw the correct curve.

Horizontal Parabolas

Horizontal parabolas, such as the one given by the equation \(x = -2(y+5)^2\), present an interesting variation from the more commonly seen vertical parabolas. The direction in which they open—left or right—depends on the sign of \(a\) in the equation.

Several characteristics define horizontal parabolas:

• When \(a\) is positive, the parabola opens to the right.
• When \(a\) is negative, like in our case with \(-2\), the parabola opens to the left.
• Horizontal parabolas have a central axis parallel to the x-axis, making the concept of expansion or contraction of the curve relate directly to its horizontal spread.
• The vertex still serves as the main reference point, determining the initial positioning of the parabola.

Drawing the parabola involves plotting the vertex and utilizing the coefficient \(-2\) to judge the curvature's width and direction. In this exercise, since \(-2\) contributes to the leftward opening, it naturally creates a narrow shape oriented towards the negative x-direction.

By understanding these features, you can effectively sketch and identify the graphs of horizontal parabolas in your studies.