Problem 10 - Intermediate Algebra

Question

The graph of each equation is a parabola. Find the vertex of the parabola and then graph it. $$x=-4 y^{2}$$

Step-by-Step Solution

Answer

The vertex of the parabola is at (0, 0), and it opens left.

Step 1: Convert Equation to Vertex Form

The given equation is not in standard parabolic form. Let's rewrite it by comparing it with the standard form of a parabola that opens sideways, which is \((x-h)=a(y-k)^2\). The equation \(x = -4y^2\) implies \(x - 0 = -4(y-0)^2\). Here, \(h = 0\) and \(k = 0\).

Step 2: Identify the Vertex

Using the vertex form equation \((x-h)=a(y-k)^2\), we identified \(h = 0\) and \(k = 0\) from the previous step. Therefore, the vertex of the parabola is at \((h, k) = (0, 0)\).

Step 3: Determine Direction and Shape

Since the equation is \(x = -4y^2\), and \(a = -4 < 0\), the parabola opens to the left. The negative value indicates the leftward direction, and the coefficient \(-4\) tells us the parabola is relatively narrow.

Step 4: Graph the Parabola

To graph the parabola, plot the vertex at \((0, 0)\). Because it opens left, draw the curve starting from the vertex and extending leftward. The symmetry axis is the x-axis \(y = 0\). Use the knowledge of the parabola's width to guide the sketch; since \(a = -4\), it should be narrow.

Key Concepts

Understanding the Parabolic Form

A parabola is a curved shape that is defined by a quadratic equation. The standard parabolic form is essential for identifying and graphing parabolas efficiently. In the context of this exercise, we are dealing with a parabola that opens sideways, meaning it's horizontal, rather than vertical. This form is \[(x-h)=a(y-k)^2\].In this equation,

• \( h \) and \( k \) represent the coordinates of the vertex \((h, k)\), which is a crucial point as it provides the pivot or turning point of the parabola.
• \( a \) determines the direction and width of the parabola.

For the given equation \( x = -4y^2 \), we can see that it is in the form of \((x-h)=a(y-k)^2.\) By comparing, we find \(h = 0\), \(k = 0\) and \(a = -4\). Understanding parabolic form simplifies how we interpret and manipulate different parabolas.

Graphing Parabolas Made Simple

Graphing a parabola involves a few straightforward steps, once you have identified its form. Start by marking the vertex on a graph's coordinate plane, which in this case is \((0, 0)\) as we discovered from the parabolic form.Next, understand how the parabola broadens or narrows based on the absolute value of \(a\). A larger absolute value means a narrower shape, which is evident here as \(a = -4\), suggesting the parabola is quite narrow.

• Since the given equation is \( x = -4y^2 \), it indicates the parabola's symmetry along the x-axis or line \( y = 0 \).
• To graph, draw the curve extending from the vertex \((0, 0)\) out to both sides. This helps in capturing the complete shape of the parabola.

The simple steps of plotting the vertex, using the symmetry, and considering the shape allow for effective graphing of any parabola. This process highlights the graph's correct placement and curvature.

Direction of a Parabola Explained

Knowing the direction a parabola opens is vital for an accurate graph. The direction is primarily determined by the sign of \(a\) in the parabola's equation. In most common parabolic forms:

• If \(a\) is positive, for sideways parabolas \((x-h)=a(y-k)^2\), it opens to the right.
• If \(a\) is negative, it opens to the left.

For the vertical parabolas of the form \((y-k)=a(x-h)^2\):

• Positive \(a\) means opening upward.
• Negative \(a\) means opening downward.

In our equation \( x = -4y^2 \), the negative \(-4\) confirms the parabola opens to the left. This simple analysis of \(a\)'s sign directly guides the parabolic direction and is crucial in drawing an accurate graph. Understanding direction alongside shape provides a complete insight into what to expect when plotting.