Problem 56 - Precalculus with Limits

Question

In Exercises 51-60, find the standard form of the equation of the parabola with the given characteristics. Vertex: \((-1, 2) \quad\) focus: \((-1, 0)\)

Step-by-Step Solution

Answer

The standard form of the equation of the parabola with the given characteristics is \( y = 1/8 * (x+1)^2 +2 \).

Step 1: Calculate the distance from vertex to focus

First, we compute the distance, \( p \), from the vertex to the focus. In this case, \( p = k - f_y \), where \( f_y \) is the y-coordinate of the focus. So \( p = 2 - 0 = 2 \).

Step 2: Calculate the value of \( a \)

Next, we calculate the value of \( a \) using \( 1 = 4a(p) \). Substituting \( p= 2 \) into this equation gives \( 1 = 4a(2) \). Therefore, \( a = 1/8 \). You get the value of a as \( a= 1/8 \) by rearranging and solving the equation.

Step 3: Substitute the coordinates of the vertex and the value of \( a \) into the equation

Finally, we substitute the values of \( h, k \), and \( a \) into the equation: \( y=a(x-h)^2+ k \). We find: \( y = 1/8 * (x+1)^2 +2 \). This is the standard form of the parabola with the given characteristics.

Key Concepts

Vertex Form of a Parabola

Understanding the vertex form of a parabola is a vital part of graphing and working with these curves. The vertex form is an equation written as:\[ y = a(x - h)^2 + k \] This equation allows us to easily identify the vertex of the parabola, which is signified by the point \((h, k)\). This form is particularly useful because it shows the direction and width of the parabola easily through the coefficient \(a\).

• If \(a > 0\), the parabola opens upwards, like a smiling face.
• If \(a < 0\), it opens downwards, like a frowning face.

By substituting different values for \(a\), \(h\), and \(k\), you can adjust the scale and location of the parabola on a graph. Let's look at an example from the exercise: "The vertex form of the parabola is given as \( y = \frac{1}{8} (x + 1)^2 + 2 \)." Here, the vertex is at \((-1, 2)\), which tells us exactly where the parabola's peak (or lowest point if it opens upwards) is located. The coefficient \(\frac{1}{8}\) indicates how wide or narrow the parabola is.

Focus of a Parabola

The focus of a parabola is a special point located inside its curve. It is closely tied to the definition of a parabola itself. By definition, a parabola consists of points that are equidistant from this focus and a directrix, a line outside the curve.
The significance of the focus is its relationship to the parabolic shape. In our original problem, the focus is at \((-1, 0)\), which is directly "below" the vertex at \((-1, 2)\) in terms of y-values.
Generally, the focus helps in determining properties and behaviors of parabolas related to light reflection or signal paths, where signals reflecting off the parabola's surface have a path through the focus; a concept widely utilized in telescopes and satellite dishes. Depending on the coordinates of the parabola, the focus can shift, and its position may also indicate how you've constructed the parabolic curve.

Distance from Vertex to Focus

In a parabola, the distance from the vertex to the focus is a key component in determining the curve's properties and features. This distance is often symbolized and referred to as "\(p\)".

• An important insight from calculating \(p\) is understanding the parabola's "tightness" or "openness."
• Bigger \(p\) values mean the focus is further from the vertex, resulting in a wider parabola. Smaller \(p\) values bring a more closed-off curve closer to the vertex.

For our exercise: the vertex \((-1, 2)\) is 2 units away from the focus \((-1, 0)\), giving \(p = 2\).
Understanding \(p\) is essential when shifting from vertex form to another form or adjusting the parabola by transforming vertex and focus positions. Mathematically, this distance defines the geometric symmetry and helps calculate the parameter \(a\) in the vertex form equation, as seen in the formula \(1 = 4a(p)\). Here, substituting \(p = 2\) helped deduce \(a = 1/8\). Since \(p\) directly affects the coefficient \(a\), it plays a crucial role in sketching and comprehending the graph's exact shape.