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Magazine Chapter 12: Problem 20
In Problems 17-22, sketch the level curve \(z=k\) for the indicated values of \(k\). $$ z=x^{2}+y, k=-4,-1,0,1,4 $$
Short Answer
Expert verified
Graph parabolas \(y = k - x^2\) for each \(k\).
Step by step solution
01
Understand the Expression
We begin with the equation for a level curve, given by \(z = x^2 + y\). The task is to sketch the curve by setting \(z\) equal to each value of \(k\) in \(-4,-1,0,1,4\). This will provide you with separate level curves for these \(k\) values.
02
Substitute and Simplify for Each k
For each \(k\), we substitute into the equation and simplify:- For \(k = -4\): \(-4 = x^2 + y\) becomes \(y = -4 - x^2\).- For \(k = -1\): \(-1 = x^2 + y\) becomes \(y = -1 - x^2\).- For \(k = 0\): \(0 = x^2 + y\) becomes \(y = -x^2\).- For \(k = 1\): \(1 = x^2 + y\) becomes \(y = 1 - x^2\).- For \(k = 4\): \(4 = x^2 + y\) becomes \(y = 4 - x^2\).
03
Analyze the Form of Each Curve
Each equation is of the form \(y = c - x^2\), which describes a parabola opening downward. The constant \(c\) shifts the vertex up or down the y-axis. Thus:- \(y = -4 - x^2\) has a vertex at \((0, -4)\).- \(y = -1 - x^2\) has a vertex at \((0, -1)\).- \(y = -x^2\) has a vertex at \((0, 0)\).- \(y = 1 - x^2\) has a vertex at \((0, 1)\).- \(y = 4 - x^2\) has a vertex at \((0, 4)\).
04
Graph Each Level Curve
With the vertices determined, graph each parabola on a coordinate plane:- The curve for \(k = -4\) is a downward-opening parabola with vertex at \((0, -4)\).- The curve for \(k = -1\) is a downward-opening parabola with vertex at \((0, -1)\).- The curve for \(k = 0\) is a downward-opening parabola with vertex at \((0, 0)\).- The curve for \(k = 1\) is a downward-opening parabola with vertex at \((0, 1)\).- The curve for \(k = 4\) is a downward-opening parabola with vertex at \((0, 4)\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Understanding Parabolas
Parabolas are fascinating shapes in the coordinate plane. They have a unique, symmetrical structure that resembles a U or an upside-down U shape. Parabolas are defined by quadratic equations, usually of the form \(y = ax^2 + bx + c\). In our specific example, the equations are simplified to \(y = c - x^2\), meaning each parabola opens downward rather than upward.
- The orientation of the parabola (opening up or down) is determined by the coefficient of the \(x^2\) term. If the coefficient is negative, like in \(y = -x^2\), the parabola opens downward.
- The vertex is the highest or lowest point of the parabola, depending on its orientation. For the downward-opening parabolas in our exercise, the vertex is the highest point of the curve.
- To find the vertex, we look at how the constant term shifts the position along the y-axis. In the form \(y = c - x^2\), the vertex is at \((0, c)\).
Coordinate Geometry: A Framework for Understanding
Coordinate geometry provides a structured way to understand shapes and equations on the plane using coordinates. With this method, every shape or curve can be described using equations and placed on a cartesian grid.
- The horizontal axis, or x-axis, and the vertical axis, or y-axis, divide the plane into four quadrants. This structure allows us to locate points using coordinates \((x, y)\).
- In the context of level curves, the coordinates help us track how changes in \(x\) lead to specific results in \(y\). For the given equations \(y = c - x^2\), the x-coordinate determines how far the point is from the origin horizontally, while the function computes the vertical position or y-coordinate.
- Coordinates are essential in graphing techniques as they offer a universal language for interpreting mathematical relationships visually.
Effective Graphing Techniques
Graphing is a primary technique used to visualize mathematical equations. It transforms abstract algebraic concepts into visual shapes, aiding comprehension. Here's how you can effectively graph the level curves given in this exercise:
- Start by identifying the vertex for each parabola from the equation \(y = c - x^2\). The vertex tells you where the curve reaches its highest point (since these are downward-opening).
- Next, plot the vertex on the coordinate plane. This serves as the anchor point for sketching the parabola.
- From there, determine other specific points by selecting values for \(x\) and calculating the corresponding \(y\) values. This helps you understand how wide or narrow the parabola is.
- Draw smooth, symmetric curves through these points for a precise graph. Symmetry is crucial in parabolas. Ensure that points equidistant from the vertex on the x-axis have equal y-values.
