Problem 20 - Calculus Early Transcendentals: Pearson New International Edition

Question

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

Step-by-Step Solution

Answer

Parabolas for k=-4 to 4, varying vertically: down (-4 to 0), up (1 and 4).

Step 1: Understanding the Problem

We need to sketch the level curves of the function \(z = x^2 + y\) for different values of \(k\). A level curve \(z = k\) implies that \(x^2 + y = k\), so we set this for each value of \(k\).

Step 1: Set up Equation for k = -4

For \(k = -4\), the equation becomes \(x^2 + y = -4\). Solving for \(y\) gives \(y = -4 - x^2\). This is a downward opening parabola shifted down by 4 units from the origin.

Step 2: Set up Equation for k = -1

For \(k = -1\), the equation becomes \(x^2 + y = -1\). Solving for \(y\) gives \(y = -1 - x^2\). This is another downward opening parabola shifted down by 1 unit from the origin.

Step 3: Set up Equation for k = 0

For \(k = 0\), the equation becomes \(x^2 + y = 0\). Solving for \(y\) gives \(y = -x^2\). This is a downward opening parabola with its vertex at the origin.

Step 4: Set up Equation for k = 1

For \(k = 1\), the equation becomes \(x^2 + y = 1\). Solving for \(y\) gives \(y = 1 - x^2\). This is an upward opening parabola shifted up by 1 unit from the origin.

Step 5: Set up Equation for k = 4

For \(k = 4\), the equation becomes \(x^2 + y = 4\). Solving for \(y\) gives \(y = 4 - x^2\). This is an upward opening parabola shifted up by 4 units from the origin.

Step 7: Sketch the Curves

Using the information from the equations for each \(k\), sketch the curves on a coordinate system: downward opening parabolas for \(k=-4, -1, 0\) and upward opening parabolas for \(k=1, 4\). Each shift corresponds to the change in \(k\), which moves the vertex up or down on the \(y\)-axis.

Key Concepts

Coordinate System

The coordinate system acts as a framework for plotting points and graphing equations like level curves. It consists of two perpendicular lines labeled as the x-axis (horizontal) and y-axis (vertical). These axes divide the plane into four quadrants.

• In the context of level curves, the coordinate system helps visually represent how curves behave based on the values of the function.
• When graphing the equation \(x^2 + y = k\), each curve is drawn in relation to its position on the coordinate system, with \(x\) and \(y\) values marked accordingly.
• The intersection of the x-axis and y-axis is known as the origin (0,0).

Understanding how to use the coordinate system effectively allows one to precisely graph any mathematical function, revealing critical insights about its shape and shifts.

Parabolas

A parabola is a U-shaped curve that can open upwards or downwards depending on its equation. The function \(y = x^2\) is a classic example of a parabola opening upwards. However, when we manipulate this equation to \(y = -x^2\), the parabola then opens downwards.Parabolas are defined by their:

• Vertex: The peak or lowest point of the parabola. In \(z = x^2 + y\), this is altered as we change \(k\).
• Axis of Symmetry: A vertical line passing through the vertex, dividing the parabola into two mirror-image halves.
• Direction: Determined by the sign in front of \(x^2\); positive for upward and negative for downward parabolas.

The stunning simplicity of parabolas is perfect for understanding how algebraic and geometric concepts align.

Function Graphing

Graphing a function is all about plotting its level curves on a coordinate system. Each equation represents a distinct path, forming visually distinct curves for varied values of \(k\). This transforms abstract equations into comprehensible visuals.

• Level curves, like \(x^2 + y = k\), show how a change in \(k\) affects the parabola's position.
• For each \(k\) value, solving \(x^2 + y = k\) for \(y\) gives a specific equation representing the parabola for that level.
• Graph each resulting \(y = -x^2 + k\) on the coordinate system to see the transformation effect.

This approach to function graphing brings clarity and makes it easier to predict how shifts and transformations alter a graph.

Vertex Shifts

Vertex shifts occur when you modify the constant \(k\) in the equation \(x^2 + y = k\). This shift changes the parabola's vertex on the coordinate system, moving it up or down along the y-axis.

• A larger \(k\) value moves the vertex upwards, creating an upward shift in the parabola.
• A smaller or negative \(k\) shifts the vertex downwards, resulting in a downward parabola.
• This shift does not affect the parabola's general shape, only its vertical positioning.

Visualizing these vertex shifts is essential for comprehending the dynamic behavior of level curves and understanding the foundational concept of translation in graphing.