Question

Sketch the level curve \(z=k\) for the indicated values of \(k\). \(z=\frac{x^{2}}{y}, k=-4,-1,0,1,4\)

Short answer

Sketch parabolas of different widths and directions for each value of \(k\).

Step-by-step solution

Understand the Equation

The given equation is \(z = \frac{x^2}{y}\), and you are asked to sketch level curves where \(z = k\) for various values of \(k\). This means you set \(\frac{x^2}{y} = k\) and solve for \(y\) in terms of \(x\) and \(k\).

Solve the Equation for y

Re-arrange the equation \(\frac{x^2}{y} = k\) to solve for \(y\):\[y = \frac{x^2}{k}.\] This equation describes the level curves for different values of \(k\).

Determine the Level Curves for Each k

Substitute each value of \(k\) into the equation to find the corresponding level curve. This will give you:- For \(k = -4\): \(y = \frac{x^2}{-4}\)- For \(k = -1\): \(y = \frac{x^2}{-1} = -x^2\)- For \(k = 0\): The equation \(y = \frac{x^2}{0}\) is undefined because division by zero is not possible.- For \(k = 1\): \(y = x^2\)- For \(k = 4\): \(y = \frac{x^2}{4}\)

Sketch the Level Curves

On a Cartesian plane, sketch the curves:- For \(k = -4\): \(y = -\frac{x^2}{4}\), which is a downward-opening parabola wider than \(y = -x^2\).- For \(k = -1\): \(y = -x^2\), a standard downward-opening parabola.- \(k = 0\) does not produce a curve.- For \(k = 1\): \(y = x^2\), a standard upward-opening parabola.- For \(k = 4\): \(y = \frac{x^2}{4}\), which is an upward-opening parabola that is wider than \(y = x^2\).

Key concepts

Parametric Equations

Parametric equations are a way to define a group of quantities as functions of one or more independent variables, called parameters. In contrast to representing constants directly with one equation, parametric equations allow the description of curves and surfaces in a flexible manner. This method is particularly useful for curves where the relationship between variables is not a standard function.
For example, the equation \(z = \frac{x^2}{y}\), when rewritten to \(y = \frac{x^2}{k}\) for level curves, can be thought of as a parametric equation if we allow \(k\) to act as a parameter that shifts the shape and position of a parabola within the cartesian plane. Changing \(k\) changes the nature of the curve.

• When \(k = 1\): Produces a conventional upward opening parabola such as \(y = x^2\).
• For \(k > 1\) or \(k < -1\): Parabolas that either widen or compress, opening upwards or downwards based on the sign of \(k\).
• For complex curves or more dimensions, parametric equations can help visualize and analyze properties efficiently.

Cartesian Plane

The Cartesian plane, also known as the xy-plane, is a two-dimensional surface where we can graphically represent equations and functions. It consists of two axes: a horizontal axis, x, and a vertical axis, y, dividing the plane into four quadrants.
In the context of the exercise, we sketched various level curves on the Cartesian plane based on different values of \(k\). Each unique value of \(k\) represents a different horizontal slice through the three-dimensional surface of \(z = \frac{x^2}{y}\).

• For \(k = -4\) and \(k = -1\), the level curves appear as downward-opening parabolas.
• Positive values, such as \(k = 1\) and \(k = 4\), render upward-opening parabolas.
• The Cartesian plane allows us to visually distinguish these shapes and analyze their intersections or behaviors at particular points.

Parabolas

Parabolas are a type of curve defined as the set of all points in the plane that are equidistant from a fixed point (the focus) and a fixed straight line (the directrix). They can open up, down, left, or right depending on the equation.
In level curve analysis, parabolas are utilized to represent the given equation through different values of \(k\).

• For \(k = 1\): The equation \(y = x^2\) represents a standard parabola that opens upwards.
• For \(k = 4\): The equation transforms to \(y = \frac{x^2}{4}\), leading to a parabola that is wider due to division by 4.
• Negative values, such as \(k = -1\), invert the parabola to \(y = -x^2\), making it downward-opening.

Thus, understanding parabolas is essential in interpreting how curves change orientation and width, significantly affecting their representation in graphical analyses.

Equation Sketching

Equation sketching involves graphing mathematical equations on the Cartesian plane to visualize relationships and behaviors between variables. This is particularly important for equations like \(z = \frac{x^2}{y}\) interpreted for level curves associated with different \(k\) values.
While sketching, you first translate the equation into a familiar form like that of parabolas, \(y = \frac{x^2}{k}\), making it easier to graph.

• Begin by substituting values for \(k\) to generate specific equations: \(y = -\frac{x^2}{4}\) for \(k = -4\), and so on.
• Ensure the understanding of key transformational changes, such as the direction, width, and vertex placement.
• Combine graphical solutions with analytical checks to confirm behavior across all relevant \(k\).

Hence, equation sketching is a crucial skill in illustrating and validating theoretical equations' behaviors in a tangible way.