Problem 22 - College Algebra

Question

Graph each equation in Exercises \(13-28 .\) Let \(x=-3,-2,-1\) \(0,1,2,\) and 3. $$y=2|x|$$

Step-by-Step Solution

Answer

The graph of \(y=2|x|\) is a 'V' shaped curve. The points plotted from the given \(x\) values are (-3, 6), (-2, 4), (-1, 2), (0, 0), (1, 2), (2, 4), and (3, 6), and are joined by a line that forms the graph.

Step 1: Understand the Absolute Function

The absolute function \(|x|\) is always non-negative. That is, for any value of \(x\), \(|x|\) is positive or zero. This gives the graph of \(y = |x|\) a 'V' shape. For this equation \(y=2|x|\), the graph will have the same shape but will be stretched vertically.

Step 2: Substitution

We substitute the given \(x\) values (-3, -2, -1, 0, 1, 2, 3), into the equation \(y=2|x|\). For \(x=-3,-2,-1,\) \(|x|\) will be positive so we replace \(x\) with \(-x\). For \(x=0,1,2,3\), \(|x|\) equals \(x\). We get the pairs (-3, 6), (-2, 4), (-1, 2), (0, 0), (1, 2), (2, 4), (3, 6), respectively.

Step 3: Plotting the Points

Next, the ordered pairs derived from the substitution are plotted on the graph. Join the plotted points with a line to form the graph of the function \(y=2|x|\). Because of the nature of the absolute function, the graph will be V-shaped.

Step 4: Verify the Shape

Make sure the final graph has a 'V' shape, showing that for any \(x\), \(y\) is either zero or positive as \(y=2|x|\) cannot take negative values.

Key Concepts

Understanding Absolute Value Functions

An absolute value function is a type of function represented by \(y = |x|\). The absolute value of a number, \(|x|\), is the distance of that number from zero on the number line. This means it is always non-negative.
In graphing, the basic absolute value function \(y = |x|\) creates a distinctive 'V' shaped graph. This occurs because, for any negative value of \(-x\), \(|x|\) becomes positive, mirroring the positive side. For the equation given, \(y=2|x|\), the graph retains this 'V' shape and stretches it vertically. This vertical stretching happens because the entire graph is multiplied by 2, effectively doubling the distance each point has from the x-axis, making it steeper.

Plotting Points on the Graph

To graph a function like \(y=2|x|\), plotting points is a fundamental step. You start by choosing specific values of \(x\). In this exercise, the values are \(-3, -2, -1, 0, 1, 2,\) and \(3\).
For each \(x\), substitute into the function to find the corresponding \(y\). This results in pairs of coordinates:

• For \(x = -3\), \(y = 2|3| = 6\).
• For \(x = -2\), \(y = 2|2| = 4\).
• For \(x = -1\), \(y = 2|1| = 2\).
• For \(x = 0\), \(y = 0\).
• For \(x = 1\), \(y = 2|1| = 2\).
• For \(x = 2\), \(y = 2|2| = 4\).
• For \(x = 3\), \(y = 2|3| = 6\).

These ordered pairs are then plotted on a coordinate grid, marking the exact locations where each set of \(x\) and \(y\) meet.

Scaling Transformations in Graphs

Scaling transformations refer to changes in a graph's size or orientation. For absolute value functions, transformations can make the graph appear wider, narrower, or rotated.
In \(y=2|x|\), we're applying a vertical scaling transformation, multiplying each \(y\) value by 2. This change affects how steep or flat the graph appears. By stretching vertically, each point at \(y=|x|\) is moved further from the x-axis, doubling the \(y\) value. This gives the graph a sharper 'V' shape than the standard \(y=|x|\).
Understanding these transformations helps us get an accurate visual representation of the function and intuitively grasp the impact of multiplying by a coefficient like 2.

Coordinate Geometry Basics

Coordinate geometry is indispensable when graphing functions. It involves using coordinates to place points in a two-dimensional plane.
The \(x\)-axis is horizontal and the \(y\)-axis is vertical. Together, they form a grid where each point is defined by a pair known as an ordered pair \((x, y)\).
In graphing, you identify each point from the equations you're working with, starting from the origin \( (0, 0) \).
By accurately plotting these on the coordinate plane, you see the graphical behavior of equations like \(y=2|x|\).
Understanding the grid allows you to visualize how the functions transform and behave, linking algebraic expressions with their geometric counterparts on the graph. It provides a concrete way to analyze and interpret mathematical concepts.