Problem 17 - Algebra and Trigonometry

Question

In Exercises 17–30, find the standard form of the equation of each parabola satisfying the given conditions. $$ \text { Focus: } \quad(7,0) ; \text { Directrix: } \quad x=-7 $$

Step-by-Step Solution

Answer

The standard form of the equation of the parabola is \(y^2=28x\)

Step 1: Identify the vertex

To find the vertex, we know that it is equidistant from both the focus and the directrix. So calculate the midpoint between the focus at (7,0) and the directrix at x=-7. Use the midpoint formula (mean of x-coordinates, mean of y-coordinates) -> \((\frac{7+(-7)}{2}, \frac{0+0}{2})\) which results in the vertex (0,0).

Step 2: Find the value of p

Now that we have the vertex, we can find the value of p, which is the distance from the vertex to either the focus or directrix. Given that our vertex is at the origin, the p value will be the x-coordinate of the focus which is 7.

Step 3: Plug the values into the standard equation of a parabola

Replace h, k and p into the standard equation of a horizontal parabola \((y-k)^2 = 4p(x-h)\) with the values h=0, k=0 and p=7. The equation simplifies to \(y^2=4*7*x\), which after further simplification gives the final equation is \(y^2=28x\)

Key Concepts

Vertex of a Parabola

Understanding the vertex of a parabola is fundamental when studying quadratic functions. The vertex is the point where the parabola either reaches its highest or lowest value, depending on whether it opens upwards or downwards. In the case of a horizontal parabola, like the one from our exercise, the vertex represents the turning point where the direction changes from left to right, or vice versa.

The procedure for finding the vertex is quite straightforward when you are given the focus and directrix. Remember, the vertex lies exactly midway between these two elements. This property holds true for all parabolas and helps significantly when drafting the graph. In the given problem, by calculating the midpoint between the focus at (7,0) and the directrix at x=-7, we determined that the vertex is (0,0).

Knowing the vertex doesn't just aid in sketching the graph. It is also integral in writing the equation of the parabola in its standard form. Once the vertex is determined, the rest of the process becomes much easier, as you’ll see in the sections concerning focus, directrix, and the equation's derivation.

Focus and Directrix of a Parabola

The focus and directrix of a parabola together define its shape and orientation. The focus is a fixed point within the parabola, and the directrix is a line outside it; both are perpendicular to the axis of symmetry. Every point on the parabola is equidistant to the focus and the directrix.

To elaborate, the focus is the point towards which the parabola ‘focalizes’ or converges. It plays a vital role in the reflection properties of a parabola. As for the directrix, it is a conceptually helpful guide, a line that helps in constructing and defining the curve.

In our specific exercise, the focus given is at (7,0) and the directrix is the line x=-7. The 'p' value, representing the distance of either the focus or directrix from the vertex, is essential. Here, it’s equal to 7, obtained from the position of the focus relative to the vertex at the origin. This 'p' value will be crucial when we derive the parabola's equation.

Equation of a Parabola Derivation

The final step is the derivation of the equation of a parabola in its standard form. Once you have the vertex and the 'p' value, you can plug these into the generic formula. For a horizontal parabola, like the one in our exercise, the standard form is \(y-k)^2=4p(x-h)\), where \(h,k)\) is the vertex, and \(p\) is the distance to the focus or directrix.

Using our derived values, with the vertex at (0,0) and \(p=7\), the equation simplifies to \(y^2=4(7)x\), which further simplifies to \(y^2=28x\). That’s the compact and powerful standard form of the parabola's equation. It is vital for calculations pertaining to the parabola’s properties and when graphing it.

The standard form equation is elegantly simple but conveys all the necessary geometric information about the parabola. With it, you can determine the general shape, orientation, and the specific features like the vertex, focus, and directrix with ease. The arithmetic involved isn’t complex; it's more about understanding what each term represents and how they relate to the geometry of the parabola.