Problem 3 - Algebra 2 and Trigonometry

Question

In \(3-6,\) find the coordinates of the ordered pair with the smallest value of \(y\) for each function. $$ y=|x| $$

Step-by-Step Solution

Answer

The ordered pair with the smallest value of \(y\) is \((0, 0)\).

Step 1: Analyze the Function

The given function is \( y = |x| \). This is an absolute value function, which is V-shaped and symmetric around the y-axis.

Step 2: Determine the Nature of the Function

Since \( y = |x| \) represents the absolute value, \( y \) can never be negative. The smallest possible value for \( y \) in the absolute value function is 0.

Step 3: Find the Corresponding x-values

For \( y = 0 \), we set the equation \( |x| = 0 \). Solving this, we find that \( x = 0 \) is the value that makes \( y = 0 \).

Step 4: Identify the Ordered Pair

The coordinates of the ordered pair that gives the smallest value for \( y \) are \((0, 0)\). This is the vertex of the graph of the absolute value function.

Key Concepts

Understanding the Coordinates of an Ordered Pair

When dealing with functions, we often talk about ordered pairs. An ordered pair \(x, y\) represents a point on a graph where \(x\) is the input value and \(y\) is the output value. For the absolute value function \(y = |x|\), each \(x\) input is processed through the absolute value operation to produce the corresponding \(y\) value. To find the coordinates of the ordered pair with the smallest \(y\), we set \(y\) to its minimum possible value, which in this case is 0. When the equation \(|x| = 0\) is solved, we get \(x = 0\). Therefore, the coordinates of this key ordered pair are \( (0, 0) \).

Finding the Smallest Value of y

The absolute value function \(y = |x|\) is unique because it only outputs non-negative values. Negative inputs \(x\) become positive, and positive inputs remain the same.So, for \(y\) to be at its smallest, \(y = 0\) is the goal. The absolute value of zero is zero itself, meaning we have to find where the graph touches or crosses the horizontal line \(y = 0\).In this exercise, setting \(|x| = 0\) simplifies directly to \(x = 0\), causing \(y\) to also be 0.Hence, the smallest value of \(y\) is at the coordinate \( (0, 0) \), which is the lowest point on the \(V\)-shaped graph.

Understanding the V-Shaped Graph of the Absolute Value Function

The graph of \(y = |x|\) has a distinct V-shape. This V-formation reflects how absolute value functions transform all input \(x\) values to non-negative \(y\) outputs.Key characteristics of this V-shaped graph include:

• Symmetry: It is perfectly symmetric around the \(y\)-axis, meaning for every \(x\) there's a \(-x\) with the same \(y\), making it mirror-like.
• Vertex: The vertex or lowest point of the graph occurs at \( (0, 0) \). This is because at this point, both the input and output are zero, the minimum value of \(y\).
• Two linear pieces: The graph is made of two line segments meeting at the vertex. One segment slopes upward for positive values of \(x\), and the other slopes symmetrically upward for negative values of \(x\).

This visualization helps in understanding why certain coordinates give the smallest \(y\) value and how the ordered pairs are distributed across the graph.