Problem 40 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the equation of the given conic. Parabola with focus \((2,5)\) and vertex \((2,6)\)

Step-by-Step Solution

Answer

The equation of the parabola is \((x-2)^2 = -4(y-6)\).

Step 1: Identify the Orientation of the Parabola

Since the x-coordinate of both the focus and vertex is the same (2), the parabola opens vertically.

Step 2: Determine the Direction of Opening

The vertex \(2, 6\) is above the focus \(2, 5\), so the parabola opens downwards.

Step 3: Calculate the Distance p

The distance from the vertex to the focus is \(|6 - 5| = 1\). Therefore, \(p = -1\) since the parabola opens downwards.

Step 4: Write the Equation of the Parabola

For a parabola that opens vertically, the equation is \((x-h)^2 = 4p(y-k)\). Here, \(h = 2\), \(k = 6\), and \(p = -1\). Substitute these into the formula to get: \((x-2)^2 = -4(y-6)\).

Step 5: Simplify the Equation if Necessary

The final equation of the parabola after plugging in the values is already simplified: \((x-2)^2 = -4(y-6)\).

Key Concepts

Vertex of a Parabola

In the language of parabolas, the vertex is like the main anchor point. It's the point where the parabola changes direction. - For vertically oriented parabolas, the vertex is either the highest or lowest point. - The vertex acts as a kind of midpoint between the parabola’s arms. - It is a crucial reference point in deriving the equation of a parabola.In our problem, the vertex is given at \((2, 6)\). This information helps to set the 'center' of the parabola for solving equations and graphing. The vertex impacts how we write our equation. Just like in the equation \((x-h)^2 = 4p(y-k)\), where \(h\) and \(k\) are the coordinates of the vertex. So, in our specific case, \(h = 2\) and \(k = 6\).
Understanding the vertex is key to understanding many other parts of a parabola!

Focus of a Parabola

The focus of a parabola is another important component, acting like the point towards which the parabola curves. Always living inside the parabola, the focus helps to guide how the parabola is shaped.- Located at \((2, 5)\) for our equation, the focus falls right on the axis of symmetry.- This position is critical because it helps determine the parabola's steepness and curvature.- Think of it as the heart of the parabola, dictating its defining shape.The relationship from the vertex to the focus, known as \(p\), identifies the distance and direction. In this exercise, we found \(p=-1\) since the parabola opens downwards and the focus is one unit below its vertex. It essentially tells us how far or close the focus is, influencing the formula used to write the equation.

Conic Sections

Parabolas are part of a family called conic sections, which are shapes formed by slicing through a cone at different angles. - Besides parabolas, conic sections include ellipses, circles, and hyperbolas. - Each of these shapes has different properties and applications. Parabolas specifically are obtained when the plane cuts parallel to the side of the cone. This results in a curve with a unique set of opening characteristics, and one focal point making it отличитьable from others. In many practical applications, such as satellite dishes and car headlights, the reflective properties of parabolas play a major role. Recognizing parabolas as conic sections helps in understanding their geometric significance and applications beyond simple equations.

Orientation of Parabolas

The orientation of a parabola defines the direction it opens—either upward, downward, left, or right.- In the problem, since the x-coordinates of both the focus and vertex were consistent (2), it implies a vertical orientation.- Based on the relative positions of focus and vertex, it is determined to open downwards.These orientations are crucial for solving equations. A vertically opening parabola has an equation of the form \((x-h)^2 = 4p(y-k)\), while horizontal parabolas follow: \((y-k)^2 = 4p(x-h)\).Knowing the orientation and direction guides us to use the correct formula and understand the nature of its curve, which in real-world terms can be inspiring for designing ramps, archways, or analyzing projectile paths.