Problem 25 - Precalculus with Limits

Question

In Exercises 23-26, use the given value of \(k\) to complete the table for the direct variation model \(y = kx^2\) Plot the points on a rectangular coordinate system. \(k = \frac{1}{2}\)

Step-by-Step Solution

Answer

By completing the table with \(x\) values ranging from -2 to 2, the corresponding \(y\) values range from 0 to 2. When these points are plotted, they form a parabola on the graph.

Step 1: Understanding Direct Variation Model

The given model is a quadratic direct variation model, where \(y\) is varying directly as the square of \(x\) and \(k\) is the constant of variation. This model can be expressed as \(y = kx^2\), where \(k\) is given as \(\frac{1}{2}\). So, the model becomes \(y = \frac{1}{2} x^2\).

Step 2: Completing the Table

First, choose a range of values for \(x\). For example, -2, -1, 0, 1, and 2. Then, substitute these \(x\) values into the equation \(y = \frac{1}{2} x^2\) to find the corresponding \(y\) values. Only the absolute values are taken here because the square of any real number is non-negative.

Step 3: Plotting Points

Plot these points on a rectangular coordinate system. It is observed from the graph that the points represent a parabolic curve which is the graph of a quadratic function.

Key Concepts

Quadratic Functions

Quadratic functions are an essential concept in algebra and are represented by the general form: \[ y = ax^2 + bx + c \] where \(a\), \(b\), and \(c\) are constants, and \(x\) is the variable. In this case, our quadratic function is a simpler form because it doesn't have the \(b\) or \(c\) terms, resulting in a direct variation model: \[ y = kx^2 \] where \(k\) is a constant that determines the rate of change or variation. The presence of the squared term, \(x^2\), results in a curve called a parabola. This function demonstrates how \(y\) changes proportionally to the square of \(x\).

• The graph of any quadratic function is a U-shaped curve known as a parabola.
• The vertex of the parabola is the highest or lowest point of the curve, depending on the sign of \(a\).
• The axis of symmetry is a vertical line that passes through the vertex.

Understanding these characteristics helps us analyze the behavior of quadratic functions in various mathematical problems.

Coordinate System

When working with functions, plotting points on a coordinate system is crucial for visualizing how equations behave. A rectangular coordinate system, or Cartesian plane, consists of two perpendicular lines: the x-axis, which is horizontal, and the y-axis, which is vertical.

• Each point on the plane is identified by a pair of numbers known as coordinates, written as \((x, y)\).
• The point where the axes intersect is called the origin, denoted as \((0,0)\).
• You can place the graph of a function by plotting its calculated points onto this system.

In this exercise, for each \(x\) value, we used the equation \(y = \frac{1}{2} x^2\) to compute the \(y\) values. After finding the coordinates, we plotted these points on the plane to illustrate the course of action of the function. By doing this, we could see how the function forms a parabolic curve on the coordinate system.

Parabola

A parabola is the graphical representation of any quadratic function, such as the equation \(y = kx^2\). Parabolas have some distinct features that help us understand their structure:

• The vertex is the point at which the parabola changes direction, either a minimum or maximum.
• The parabola is symmetric with respect to its axis, which is a vertical line through the vertex.
• When the constant \(a\) is positive, the parabola opens upwards; if negative, it opens downwards.

In this exercise, the equation \(y = \frac{1}{2} x^2\) creates an upward-opening parabola because the constant \(k = \frac{1}{2}\) is positive. This means as \(x\) moves away from zero in either direction, \(y\) increases, resulting in the distinguished U-shape of the parabola. This visual tool is key to understanding the behavior of quadratic functions and their applications in various fields, from physics to economics.