Problem 76 - Intermediate Algebra

Question

Write each equation in standard form, if it is not already so, and graph it. If the graph is a circle, give the coordinates of its center and its radius. If the graph is a parabola, give the coordinates of its vertex. $$ x=-6(y+1)^{2}-4 $$

Step-by-Step Solution

Answer

The equation is a parabola with vertex at \((-4, -1)\). Graph it accordingly.

Step 1: Identify the Equation Type

Given the equation \(x = -6(y + 1)^2 - 4\), we notice that it fits the general form of a parabola, \(x = a(y - k)^2 + h\), where \((h, k)\) is the vertex. The negative coefficient indicates that the parabola opens horizontally to the left.

Step 2: Reformat the Equation into Standard Form

The current equation \(x = -6(y + 1)^2 - 4\) is nearly in standard form \(x = a(y - k)^2 + h\), which is already suitable for identifying the vertex and graphing.

Step 3: Identify the Vertex of the Parabola

Using the standard form of a parabola, \(x = a(y - k)^2 + h\), the vertex \((h, k)\) corresponds to the transformation. From the equation \(x = -6(y + 1)^2 - 4\), we identify the vertex as \((-4, -1)\).

Step 4: Graph the Parabola

Plot the vertex \((-4, -1)\) on the coordinate plane. Since \(a = -6\), the parabola opens horizontally to the left. Sketch the parabola, ensuring that it remains symmetric with respect to its axis of symmetry, which is horizontal at \(y = -1\). The parabola will stretch more vertically due to the absolute value of \(a\) being greater than 1, which makes it narrower.

Key Concepts

Vertex of a Parabola

The vertex of a parabola is a crucial concept to grasp when studying parabolic equations. It is essentially the point where the parabola changes direction. In graphing terms, it is the peak or the lowest (or highest) point if the parabola opens upwards or downwards, and the leftmost or rightmost point if it opens horizontally.
For a parabola in the form of \[ x = a(y - k)^2 + h \], the vertex is denoted as \((h, k)\). The vertex provides valuable information about the graph:

• It indicates the turning point or the extremum of the parabola.
• It helps in determining the symmetry and direction of the parabola.
• The coordinates \((h, k)\) directly give you a critical point to plot on the graph.

In our example with the equation \[ x = -6(y + 1)^2 - 4 \], we immediately identify the vertex as \((-4, -1)\). This point is vital as it centers the graph of the parabola on this point, allowing us to plot the rest of the curve correctly.

Graphing Parabolas

Graphing a parabola involves plotting a U-shaped curve on a coordinate plane. Parabolas can open upwards, downwards, left, or right, depending on the orientation of the equation.
For parabolas in the form \[ x = a(y - k)^2 + h \], the curve will open horizontally. Whether it opens to the left or right depends on the sign of \(a\):

• If \(a > 0\), the parabola opens to the right.
• If \(a < 0\), as in our exercise, it opens to the left.

After identifying the vertex from the equation \[ x = -6(y + 1)^2 - 4 \] as \((-4, -1)\), you plot this point first.
This parabola opens to the left and stretches more steeply than usual because the absolute value of \(a\) is larger than 1, indicating a narrower curve.
Understanding the vertex's role in graphing makes it easier to plot the rest of the parabola accurately, maintaining symmetry around the horizontal axis at \(y = -1\). Always remember, the form of the equation hints at how the graph should look, providing a guide for sketching the parabola.

Standard Form of a Parabola

The standard form of a parabola is a way of writing its equation that readily reveals its key features, especially the vertex. There are two main forms, depending on the axis of symmetry:

• For vertical parabolas: \[ y = a(x - h)^2 + k \].
• For horizontal parabolas: \[ x = a(y - k)^2 + h \].

The parameters \((h, k)\) represent the vertex of the parabola.
The value \(a\) affects the direction and the width of the parabola:

• When \(|a| > 1\), the parabola is narrower, and if \(0 < |a| < 1\), it is wider.
• A positive \(a\) means the parabola opens upwards (vertical) or right (horizontal), and a negative \(a\) means it opens downwards (vertical) or left (horizontal).

In the given equation \[ x = -6(y + 1)^2 - 4 \], it fits the standard form of a horizontally opening parabola, indicating the vertex at \((-4, -1)\) and a leftward opening because of the negative \(a\). Understanding the standard form helps in graphing the parabola more efficiently, by knowing exactly how and where to position it on the plot.