Question

Sketch the graph of the first function by plotting points if necessary. Then use transformation(s) to obtain the graph of the second function. \(y=|x|, \quad y=2|x+1|-1\)

Short answer

To sketch the graph of \(y = 2|x + 1| - 1\), start with the graph of \(y = |x|\) and apply the following transformations: vertical stretch by a factor of 2, horizontal shift of 1 unit to the left, and vertical shift of 1 unit downward. The resulting graph will be a V-shape with a steeper slope than \(y = |x|\), shifted 1 unit to the left and 1 unit downward.

Step-by-step solution

Graph of \(y = |x|\)

To sketch the graph of \(y = |x|\), let's find some key points by plugging in x-values and computing their corresponding y-values using the given function. - For x = -2, y = |-2| = 2 - For x = -1, y = |-1| = 1 - For x = 0, y = |0| = 0 - For x = 1, y = |1| = 1 - For x = 2, y = |2| = 2 Plot these points on the coordinate plane and connect them with a straight line for both sides, forming a V-shape for the graph of \(y = |x|\).

Identify transformations

Now, we need to identify the transformations that can be applied to the graph of \(y = |x|\) to obtain the graph of \(y = 2|x + 1| - 1\). Notice that the second function can be written in the general form of a transformed absolute value function: \(y = A|Bx + C| + D\), where A, B, C, and D determine the transformations. Based on this general form, we can clearly see the following transformations: - A = 2, which means a vertical stretch by a factor of 2 - B = 1, which means no horizontal stretch or compression - C = -1, which means a horizontal shift of 1 unit to the left - D = -1, which means a vertical shift of 1 unit downward

Apply transformations to obtain the graph of \(y = 2|x + 1| - 1\)

Based on the step 2, apply the identified transformations to the graph of \(y = |x|\): 1. Vertical Stretch: Multiply the y-coordinates of the key points that were found in step 1 by a factor of 2. 2. Horizontal Shift: Subtract 1 from the x-coordinates of the transformed points. 3. Vertical Shift: Subtract 1 from the y-coordinates of the transformed points. After applying these transformations, plot the resulting points and connect them with a straight line for both sides to obtain the graph of \(y = 2|x + 1| - 1\).

Key concepts

Graphing Absolute Value Functions

The graph of a basic absolute value function, represented as \(y=|x|\), is one of the foundational shapes in algebra. It forms a 'V' shape centered at the origin. To sketch it, you should start by finding key points where you substitute various x-values into the equation to find the corresponding y-values.

For instance, if you substitute \(x = -2\), \(x = -1\), \(x = 0\), \(x = 1\), and \(x = 2\) into the basic function, you'll end up with the points (-2, 2), (-1, 1), (0, 0), (1, 1), and (2, 2). These points help you visualize the V-shape as you plot them on a graph and draw straight lines between them. Remember that the V is symmetrical and has its vertex, the pointy bottom, right at the origin (0,0).

To enhance understanding, use graph paper for accurate placement of points, and always begin plotting from the vertex. The graph extends infinitely in both the positive and negative directions along the x-axis, but it never dips below the x-axis since absolute value measures distance, which is always non-negative.

Identifying Function Transformations

A transformation alters the appearance of the original function, either moving, stretching, or flipping it, while maintaining its basic shape. For an absolute value function in the form \(y = A|Bx + C| + D\), A, B, C, and D are the transformation parameters.

To identify these transformations:

• Vertical stretch/compression is controlled by A. If A is greater than 1, the function is stretched; if between 0 and 1, it is compressed.
• Horizontal stretch/compression is indicated by B. A value of B other than 1 will stretch or compress the graph horizontally.
• Horizontal shifts occur due to C. If C is positive, the graph shifts to the left; if negative, to the right.
• Vertical shifts are a result of D. Positive D moves the graph up, while negative D moves it down.

So, for the function \(y = 2|x + 1| - 1\), the graph undergoes a vertical stretch by a factor of 2, a horizontal shift to the left by 1 unit, and a downward vertical shift by 1 unit. Identifying these changes is crucial for understanding how to manipulate and predict the behavior of absolute value functions.

Applying Transformations to Functions

Once you identify the transformations that need to be applied to the graph of an absolute value function, it's time to apply them correctly. Let's tackle the transformations in the order which makes them easiest to understand.

First, perform a vertical stretch by multiplying the y-coordinates of your key points by factor A. Then, execute horizontal shifts by adding C to the x-values - remember, if C is negative, you're effectively subtracting. Finally, adjust vertically by adding D to the y-coordinates.

Applying Each Transformation Step-by-Step

For our exercise \(y = 2|x + 1| - 1\), here's how you'd proceed:

• Multiply the y-values from your originally plotted points by 2 to achieve the vertical stretch.
• Shift the stretched points 1 unit to the left, which means subtracting 1 from their x-coordinates.
• Finally, move the graph 1 unit down by subtracting 1 from the y-values.

Through these steps, we systematically transform the original graph of \(y=|x|\) into \(y=2|x + 1| - 1\). When applied correctly, these transformations help us draw the new function accurately without plotting many additional points.

Absolute Value Function Properties

Understanding the properties of absolute value functions is critical for analyzing and graphing them. Each absolute value function has a vertex, which is its lowest point when the function opens upwards, or its highest point when it opens downwards. The graph is symmetrical about the vertical line that passes through the vertex, known as the axis of symmetry.

The absolute value function will always produce a non-negative output, regardless of the input, as it represents the distance from zero on the number line. Thus, the graph will never dip below the x-axis. The slope of the lines forming the V is determined by the coefficient of x within the absolute value. If it's positive, the V opens upwards, and if it's negative, the V opens downwards.

These functions are piecewise functions, meaning they can be represented as two different linear functions connected at the vertex. Specifically, the function \(y = |x|\) can be thought of as the functions \(y = x\) for \(x \geq 0\) and \(y = -x\) for \(x < 0\). This piecewise nature allows us to analyze and graph each part of the function separately, providing a clear mathematical structure to its behavior on the graph.