Problem 15 - College Algebra

Question

For the following exercises, rewrite the given equation in standard form, and then determine the vertex \((V),\) focus \((F),\) and directrix \((d)\) of the parabola. $$ x=36 y^{2} $$

Step-by-Step Solution

Answer

The vertex is \((0,0)\), focus is \(\left(\frac{1}{144}, 0\right)\), and the directrix is \(x = -\frac{1}{144}\).

Step 1: Write Equation in Standard Form

Start with the given equation: \(x = 36y^2\). For a parabola that opens sideways, the standard form is \((y-k)^2 = 4p(x-h)\), where \((h, k)\) is the vertex, \(p\) is the distance from the vertex to the focus, and the directrix. Rewrite the given equation to match this form. Divide both sides by 36 to get \(\frac{x}{36} = y^2\). Next, rearrange terms to get \(y^2 = \frac{1}{36}x\). The equation is now in standard form with \(h = 0\), \(k = 0\), and \(4p = \frac{1}{36}\).

Step 2: Identify Vertex

In the standard form equation \((y-k)^2 = 4p(x-h)\), the vertex of the parabola is given by \((h, k)\). From Step 1, we have \((h, k) = (0, 0)\). This means the vertex \(V\) is at \((0, 0)\).

Step 3: Determine the Value of \(p\)

The term \(4p\) in the rearranged equation \(y^2 = \frac{1}{36}x\) is equal to \(\frac{1}{36}\). Therefore, solve \(4p = \frac{1}{36}\) to find \(p\). Divide both sides by 4 to get \(p = \frac{1}{144}\).

Step 4: Locate the Focus

The focus \((F)\) of a parabola that opens to the right (since \(4p > 0\)) is found at \((h + p, k)\). With \(h = 0\), \(k = 0\), and \(p = \frac{1}{144}\), the focus is at \(\left(\frac{1}{144}, 0\right)\).

Step 5: Determine the Equation of the Directrix

For a parabola that opens to the right, the directrix \((d)\) is a vertical line given by \(x = h - p\). Substitute \(h = 0\) and \(p = \frac{1}{144}\) into this equation to find \(x = 0 - \frac{1}{144}\). Therefore, the directrix is the line \(x = -\frac{1}{144}\).

Key Concepts

vertex

In a parabola, the vertex is the point where the curve makes its sharpest turn. It represents the minimum or maximum point of the parabola, depending on its orientation. For a sideways-opening parabola like the one in our exercise, the vertex is the starting point of the curve. To find the vertex using the standard form of the equation y^2 = 4p(x-h) we look at the values h and k. In the equation given, h = 0 and k = 0, which places the vertex of our parabola at the origin, (0,0). The vertex is important because it helps us understand the symmetry and direction of the parabola. It also serves as the midpoint when determining both the focus and the directrix.

focus

The focus of a parabola is a critical point that lies inside the curve. It is one of the defining features of the parabola's shape. The parabola is shaped in such a way that all the points on it are equidistant from the focus and the directrix.To calculate the focus in our equation, we use the formula for the parabola's focus, given by(h + p, k) for sideways-opening parabolas. Here, we determined h = 0, k = 0, and p = \(\frac{1}{144}\). Thus, the focus is located at \(\left(\frac{1}{144}, 0\right)\).This point helps in identifying the directional property of the parabola as it indicates towards which side the parabola "opens." When graphing, this point assists in drawing the curve with greater accuracy.

directrix

The directrix is a line that is used in conjunction with the focus to produce the specific shape of a parabola. It plays an equally important role as the focus when defining a parabola.The formula for the directrix of a sideways-opening parabola is given by the equation:x = h - p.For this exercise, h = 0 and p = \(\frac{1}{144}\). Substituting these values gives the equation for the directrix asx = -\(\frac{1}{144}\).The directrix is a vertical line, and every point on the parabola is equidistant to this line and the focus. Understanding the position of the directrix further aids in visualizing the correct and accurate orientation of the parabola. The balacing between the focus and the directrix helps to find how the curve bends and extends.