Problem 22 - Algebra 2

Question

Identify the type of function represented by each equation. Then graph the equation. \(y=|2 x|\)

Step-by-Step Solution

Answer

The function \(y = |2x|\) is an absolute value function, and its graph is a V-shaped parabola opening upwards, vertically stretched by a factor of 2.

Step 1: Identify Function Type

The equation given is \(y = |2x|\), which represents an absolute value function. Absolute value functions are typically characterized by their V-shaped graphs and transformations depending on the expression inside the absolute value.

Step 2: Rewrite for Simplicity

To better understand the function, rewrite \(y = |2x|\) as \(y = |2| \cdot |x| = 2|x|\). This shows that the function will have a vertical stretch by a factor of 2.

Step 3: Determine Key Points

Find key points of the function by evaluating the expression at various \(x\) values. For example: when \(x = 0\), \(y = 0\); when \(x = 1\), \(y = 2\); and when \(x = -1\), \(y = 2\). More generally, the graph will pass through the points \((0, 0)\), \((1, 2)\), and \((-1, 2)\).

Step 4: Sketch the Graph

Plot the key points on a coordinate plane. Start at (0,0), plot (1,2) and (-1,2). Draw lines connecting these points, ensuring the graph forms a V-shape, characteristic of absolute value functions, and is symmetric about the y-axis.

Key Concepts

Graphing Absolute Value Functions

Graphing absolute value functions can be a fun and straightforward process once you understand the unique features of these functions. Absolute value functions generally form a V-shaped graph. This is because the absolute value of a number is always non-negative, meaning it reflects all negative inputs above the x-axis, giving the graph its distinct shape.
The most basic form of an absolute value function is \(y = |x|\), which has a vertex, the point where the graph changes direction, at the origin (0,0).
To graph an absolute value function like \(y = |2x|\), you'll follow these steps:

• Determine the vertex of the graph, usually the point where \(x = 0\), resulting in \(y = 0\).
• Identify other key points by substituting different values of \(x\) into the equation and solving for \(y\).
• Plot these points on a graph and draw the V-shape, making sure it's symmetrical about the y-axis.

Absolute value functions are symmetrical, making them easy to graph if you correctly identify the key points.

Vertical Stretch

A vertical stretch changes the steepness of the V-shape in an absolute value graph. If you compare the basic absolute value function \(y = |x|\) to \(y = 2|x|\), you will notice that \(y = 2|x|\) stretches the graph vertically.
In the equation \(y = 2|x|\), the coefficient 2 indicates a vertical stretch by a factor of 2. This means that for any given \(x\), the value of \(y\) will be twice as large as it would be in the basic \(y = |x|\) function. This makes the V-shape of the graph steeper.

• When the coefficient is greater than 1, the graph stretches vertically.
• If the coefficient were between 0 and 1, it would result in a vertical compression, making the graph wider and less steep.

Understanding these changes helps in visualizing how different equations impact the absolute value graph's shape and size.

Key Points of Function

Identifying key points is crucial when graphing an absolute value function. These points provide a framework for sketching the graph accurately. For \(y = |2x|\), key points can be identified by substituting various values of \(x\) into the equation and solving for \(y\).
For instance:

• When \(x = 0\), \(y = |2 imes 0| = 0\).
• When \(x = 1\), \(y = |2 imes 1| = 2\).
• Similarly, at \(x = -1\), \(y = |2 imes -1| = 2\).

More generally, the graph passes through key points like (0,0), (1,2), and (-1,2). These points not only help plot the graph but also emphasize the symmetry about the y-axis, ensuring the V-shape is correctly formed. Identifying these key points simplifies the process of drawing and interpreting absolute value functions.