Problem 46 - Calculus with Concepts in Calculus

Question

Sketch the quadric surface. \(y=1-x^{2}\)

Step-by-Step Solution

Answer

It's a downward-opening parabola with vertex at (0,1).

Step 1: Identify the Surface Type

The given equation is \( y = 1 - x^2 \). This equation is in two variables, \( x \) and \( y \), and represents a parabola because it contains an \( x^2 \) term and no \( z \) variable.

Step 2: Determine the Axis

The equation \( y = 1 - x^2 \) is of the form \( y = a - bx^2 \), indicating that it is a parabola opening along the \( y \)-axis. The \( y \)-variable is not squared, while \( x \) is square, confirming its orientation along the \( y \)-axis.

Step 3: Identify the Vertex

The vertex of the parabola is at the point where the equation reaches its maximum or minimum value. The term \( -x^2 \) shows it is a downward-opening parabola. Thus, the vertex is at the highest point, which is \((0, 1)\).

Step 4: Sketch the Graph

Plot the parabola on the \( xy \)-plane. The vertex is at \((0, 1)\) on the graph. As \( x \) increases or decreases, \( y \) decreases, since \( y = 1 - x^2 \). Sketch a U-shaped curve opening downwards centering at \( (0, 1) \).

Key Concepts

Parabola

A parabola is a specific type of curve that you often see in algebra and geometry. It is defined by a quadratic equation that primarily includes a variable squared. In simple terms, it's the shape you get when you plot an equation with an \(x^2\) term on a coordinate plane. Parabolas have a distinct U-shape, which can either open upwards or downwards. The direction depends on the sign of the \(x^2\) term:

• If the term is positive, the parabola opens upwards.
• If the term is negative, the parabola opens downwards.

For the given exercise \(y = 1 - x^2\), the \(-x^2\) term indicates that the parabola opens downwards. This understanding will help when determining where the curve reaches its highest or lowest points.
The simplistic beauty of parabolas lies in their symmetry - a mirrored image about their axis, which in this case, is a line parallel to the y-axis.

Graph Sketching

Graph sketching is a helpful skill that allows you to visualize mathematical equations on a plane. The case with \(y = 1 - x^2\) involves sketching a parabola. When you start graph sketching, begin by determining the key features of the equation:

• Identify if the parabola opens upwards or downwards.
• Find the vertex, which represents the top or bottom of the U-shape.
• Plot some additional points on each side of the vertex to ensure the curve is accurate and symmetric.

For our equation, you'd plot the vertex at \( (0, 1) \). Next, for each side, as x increases or decreases, y decreases as well due to the negative sign before \(x^2\). Connect these points smoothly to form the downward-opening U-curve. Graph sketching is not just about precision; it's also about gaining insight into the equation's behavior.

Vertex Identification

The vertex is a crucial part of any parabola, acting as its highest or lowest point. To accurately identify the vertex in a parabola equation like \(y = 1 - x^2\), you generally look at the format to find the coordinates:

• The standard form \(y = a(x-h)^2 + k\) shows the vertex at \((h, k)\).
• For our equation, the vertex occurs when \(x = 0\), since there is no \(h\) to shift it right or left, and thus the highest point is when \(y = 1\), making the vertex \((0, 1)\).

Identifying the vertex is important as it tells you where the parabola reaches its maximum or minimum value. When graphing, this provides a central guideline to sketch around. Understanding the concept of a vertex helps set the stage for understanding higher dimensions of graphing and deeper mathematical concepts.