Problem 68 - Thomas Calculus

Question

Sketch the surfaces in Exercises \(13-76\) $$ z=4 x^{2}+y^{2}-4 $$

Step-by-Step Solution

Answer

Sketch an upward-opening paraboloid with a vertex at (0, 0, -4).

Step 1: Understanding the Equation

The equation given is \( z = 4x^2 + y^2 - 4 \). This is a quadratic equation in three variables, where \(x\) and \(y\) represent the horizontal axes on a 3D coordinate plane, and \(z\) represents the vertical axis. The form resembles \(z = ax^2 + by^2 + c\), indicating that this is an equation of a paraboloid.

Step 2: Identify Type and Orientation

Recognize that the equation \( z = 4x^2 + y^2 - 4 \) represents an elliptic paraboloid. Since \(z\) is a function of \(x^2\) and \(y^2\), it will open upwards. Here, the coefficients \(4\) and \(1\) suggest an elliptical cross-section, with different stretching in the \(x\) and \(y\) directions.

Step 3: Determine Vertex and Shifts

Write \(z = 4(x^2) + (y^2) - 4\). This shows the paraboloid is centered at the vertex \((0, 0, -4)\), since the quadratic part doesn't introduce any shifts away from the axis other than vertically by -4 in the \(z\)-direction.

Step 4: Describe Cross-section Shapes

To analyze cross-sections, set \(z=k\) and solve for \(x^2\) and \(y^2\). Cross-sections for constants \(z = k\) are ellipses given by \(4x^2 + y^2 = k + 4\). This shows the paraboloid as having elliptical cross-sections increasing as \(z\) increases.

Step 5: Sketch the Surface

Draw the 3D coordinate axes. Plot the vertex at \((0, 0, -4)\). Since the surface is an upward-opening paraboloid, sketch curves extending out and upwards from the vertex, becoming wider as \(z\) increases. The \(x\)-direction will be stretched more than the \(y\)-direction, reflecting the \(4x^2\) term.

Key Concepts

Quadratic Equation

A quadratic equation in its basic form is expressed as \( ax^2 + bx + c = 0 \). In this expression, \( a \), \( b \), and \( c \) are constants, with \( a eq 0 \). In three dimensions, the concept extends to quadratic surfaces, where the equation may involve two or more variables, such as \( z = ax^2 + by^2 + c \). Here, rather than a single curve, you visualize a surface.
This form can represent different types of surfaces, like a paraboloid, ellipsoid, or hyperboloid, depending on the coefficients and signs of \( a \) and \( b \).
In the given exercise, the quadratic equation is \( z = 4x^2 + y^2 - 4 \). This specific form outlines an elliptic paraboloid. Its quadratic parts \(4x^2\) and \(y^2\) suggest how the surface stretches in relation to the \(x\) and \(y\) axes.

• If both quadratic terms involve squares of variables, it's typically a paraboloid.
• If one or both of the coefficients of \(x^2\) or \(y^2\) are negative, the surface would differ, potentially forming a hyperboloid.

Grasping these aspects is key to identifying and sketching surfaces drawn from quadratic equations.

3D Coordinate Plane

The 3D coordinate plane is pivotal for visualizing equations like paraboloids. It comprises three axes: the \(x\)-axis, \(y\)-axis, and \(z\)-axis, all mutually perpendicular. These axes allow you to plot points and surfaces in three dimensions, expanding beyond the typical 2D graph that only uses the \(x\) and \(y\) axes.
Let's contextualize this with the equation \( z = 4x^2 + y^2 - 4 \). It involves:

• The \(x\) and \(y\) axes for horizontal dimensions. These represent the plane on which the elliptic paraboloid is grounded.
• The \(z\)-axis, which acts as the vertical line showing height or depth, crucial for understanding the paraboloid's direction of opening (in this case upwards).

Plotting points on a 3D coordinate plane requires determining the \(x\), \(y\), and \(z\) values. For instance, the vertex of our paraboloid appearing at \((0, 0, -4)\) translates directly to a location on this grid, serving as a pivotal starting point for sketching.
Understanding the 3D space is essential not just for accurate plotting but also for visualizing how different surfaces expand, skew, or stretch within this plane.

Vertex of Paraboloid

The vertex of a paraboloid is a point where it reaches its extremum point. For the equation \( z = 4x^2 + y^2 - 4 \), this vertex is at the coordinates \((0, 0, -4)\). Understanding this point's position is crucial for accurately sketching and interpreting the surface's characteristics.

• The vertex tells you where the curve begins extending upwards or downwards. For elliptic paraboloids, the region around this vertex is where the surface is the smallest or largest, depending on the orientation.
• The location of the vertex in this context shows no horizontal shift (as there are no linear \(x\) or \(y\) terms), but a vertical shift downward by 4 (the constant \(-4\) affecting the \(z\) value directly).

The significance of the vertex:
The paraboloid opens upwards from this vertex, delineating how cross-sections (ellipses for constant \(z\) values) widen as they move away from the vertex, creating the characteristic bowl shape of the elliptic paraboloid.
Overall, identifying and using the vertex helps you understand the scale, orientation, and central shape of the paraboloid on the 3D coordinate plane.