Problem 21 - Algebra and Trigonometry

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 the equation \(y = 2|x|\) is a V shape that opens upwards. The points on the graph are \((-3,6), (-2,4), (-1,2), (0,0), (1,2),(2,4),(3,6)\)

Step 1: Calculate corresponding y values

First, calculate the corresponding y values for each x value by substituting the given x values in the equation \(y = 2|x|\). This means for any negative x values, turn them into positive before multiplying by 2. Calculate y values for \(x = -3, -2, -1, 0, 1, 2,\) and 3.

Step 2: List of coordinates

Now, pair each x-value with its corresponding y-value to get an ordered pair (coordinate). These coordinates represent points on the graph for the equation \(y = 2|x|\). Place x in the first position of the pair and y in the second.

Step 3: Plot the graph

Plot the graph of the equation \(y = 2|x|\) using the ordered pairs obtained in Step 2. Remember, the graph of an absolute value function is a V-shape that either opens up or down. In this case, since the coefficient of the absolute value is positive, the graph should open upwards.

Key Concepts

Absolute Value Graph

When we talk about graphing an absolute value function, such as y = 2|x|, we're referring to the visual representation of all the points that satisfy the equation. An absolute value graph is distinctive for its V-shaped curve, which can either open up or down. In our exercise, the coefficient of |x| is positive, therefore the 'arms' of the V open upwards.

The process begins by selecting a range of x-values - in this instance, x = -3, -2, -1, 0, 1, 2, 3. For each of these x-values, you plug them into the equation to find the corresponding y-value. This reflects the 'absolute' nature of the function, meaning it takes the positive magnitude of the x-value. Once you map these x-values to their respective y-values, you're ready for plotting coordinates on the graph. The graph's symmetry about the y-axis is an essential characteristic and often simplifies the plotting process because for every positive x, there's a mirroring negative x with the same absolute value.

Plotting Coordinates

Plotting coordinates is the act of marking points on a graph based on pairs of numbers which represent positions on two intersecting lines, commonly known as axes. In the context of our absolute value function, each pair (x, y) that you've obtained from your calculations represents a point in the coordinate plane.

To plot the coordinates, you start from the origin (0, 0), then move horizontally to the x-value and vertically to the y-value. For example, the coordinate (-2, 4) means you would move two units left and four units up from the origin. After plotting all the points, connect them to reveal the shape of the graph, which should be a V if plotted correctly. Remember to make your graph as precise as possible for it to be an accurate representation of the function.

Absolute Value Equations

Absolute value equations involve expressions within absolute value bars, such as |x|, which denote the distance of the number x from zero on a number line. To solve these equations, one typically considers two scenarios - what happens when the inside of the absolute value is positive and what happens when it's negative.

In our example, y = 2|x|, you would make x positive if it isn't already, effectively removing the absolute value bars and then multiply by 2 to find y. This simplicity comes from the definition of absolute value: it's always non-negative, regardless of the sign of the variable inside. These equations demonstrate the principle of 'positive distance', ensuring that the graph maintains its V shape and that the function's output y is always positive or zero.