Problem 16 - Algebra 2 and Trigonometry

Question

a. Sketch the graph of \(y=|x| .\) b. Sketch the graph of \(y=2|x|\) c. Sketch the graph of \(y=\frac{1}{2}|x|\) d. Describe the graph of \(y=a|x|\) in terms of the graph of \(y=|x|\)

Step-by-Step Solution

Answer

Vertical scaling modifies the graph's width; larger a stretches it vertically, smaller a compresses it.

Step 1: Graph of y = |x|

Start by sketching the graph of the basic absolute value function, \( y = |x| \). This graph forms a "V" shape, with the vertex at the origin (0,0). For \( x \geq 0 \), the graph is the line \( y = x \). For \( x < 0 \), the graph is the line \( y = -x \). The slope of both lines is 1, and the graph is symmetrical with respect to the y-axis.

Step 2: Graph of y = 2|x|

To sketch \( y = 2|x| \), notice that it is a vertical stretching of the graph \( y = |x| \) by a factor of 2. For \( x \geq 0 \), the graph is the line \( y = 2x \); for \( x < 0 \), the graph is the line \( y = -2x \). Both parts of the graph have a slope of 2, making it sharper or narrower than \( y = |x| \). The vertex remains at the origin.

Step 3: Graph of y = (1/2)|x|

Sketch \( y = \frac{1}{2}|x| \) by considering this as a vertical compression of \( y = |x| \) by a factor of 1/2. For \( x \geq 0 \), the graph is the line \( y = \frac{1}{2}x \); for \( x < 0 \), it's \( y = -\frac{1}{2}x \). This makes the graph wider than \( y = |x| \), with a vertex remaining at the origin.

Step 4: Describe the graph of y = a|x|

The graph of \( y = a|x| \) is a vertical scaling of \( y = |x| \). If \( a > 1 \), the graph is vertically stretched, making it narrower. If \( 0 < a < 1 \), the graph is vertically compressed, making it wider. If \( a < 0 \), the graph inverts vertically, flipping the vertex to reflect over the x-axis while still forming a |shape. The vertex of the graph remains at (0,0), and the graph is always symmetrical about the y-axis.

Key Concepts

Vertical Stretching

Vertical stretching of an absolute value graph occurs when we multiply the absolute value function by a factor that is greater than 1. Visualize the basic function, \( y = |x| \), which forms a 'V' shape with its vertex at (0, 0). This 'V' stretches vertically when we apply such a factor.

For example:

• In \( y = 2|x| \), the graph stretches vertically due to the multiplication by 2.
• This stretching makes the graph narrower compared to the basic \( y = |x| \).
• The slopes of the lines forming the 'V' become steeper. In this case, the slopes are ±2, indicating a sharper ascent and descent.

Vertical stretching does not affect the vertex's position; it remains at the origin. However, the overall shape becomes more pointed, and the 'V' appears to climb more aggressively.

Vertical Compression

Vertical compression is essentially the opposite of vertical stretching. It happens when an absolute value function is multiplied by a number between 0 and 1. This reduces the steepness of the graph, making it appear wider.

Consider the function \( y = \frac{1}{2}|x| \):

• Here, the graph is compressed vertically by the factor of \( \frac{1}{2} \).
• This results in the 'V' shape being less steep compared to \( y = |x| \).
• The slopes of the lines that form the 'V' are softened, in this instance being ±\( \frac{1}{2} \).

This vertical compression does not alter the position of the vertex; it stays at (0, 0). The graph now spreads out over a wider area, showing a gentler rise and fall.

Vertex

The vertex of an absolute value graph is the point where the 'V' shape meets and changes direction. For the basic absolute value function, \( y = |x| \), the vertex is located at the origin, (0,0).

Key characteristics of the vertex:

• It's a pivotal point, marking the minimum or maximum in transformed cases.
• Remains unchanged at the origin in basic vertical stretching and compression scenarios.
• Represents the point of symmetry for the graph since both sides mirror each other from this point.

The vertex's stability in location during transformations like stretching and compressing simplifies understanding the graph's essential structure. Even if more complex transformations like shifting occur, the vertex remains a crucial indicator of the graph's symmetry.

Symmetry About the Y-Axis

Every absolute value graph, including \( y = |x| \) and its transformations, possesses symmetry about the y-axis. This symmetry means that for every point \((x, y)\) on the graph, there is a corresponding point \((-x, y)\).

Understanding this symmetry is helpful:

• It ensures that modifications like stretching, compressing, or reflecting maintain a consistent structure around the y-axis.
• Enhances predictability of how changes on one side of the graph will mirror on the opposite side.

This quality of symmetry not only makes sketching the graph simpler but ensures transformations maintain the balanced aesthetic inherent to absolute value graphs.