Problem 42 - Algebra and Trigonometry

Question

\(21-44\) . Sketch the graph of the function, not by plotting points, but by starting with the graph of a standard function and applying transformations. $$ y=2-|x| $$

Step-by-Step Solution

Answer

Start with \(y = |x|\), reflect over the x-axis, then shift up by 2 units.

Step 1: Identify the Base Function

The given function, \( y = 2 - |x| \), is based on the standard absolute value function \( y = |x| \). The graph of \( y = |x| \) is a V-shaped graph centered at the origin (0,0) and opens upwards.

Step 2: Transform the Base Function: Vertical Reflection

Our function is of the form \( -|x| \), meaning the \( |x| \) graph should be reflected downwards (over the x-axis). This transforms the graph into an upside-down V-shape that still has its vertex at the origin.

Step 3: Transform the Base Function: Vertical Shift

Next, look at the \( +2 \) in \( y = 2 - |x| \). This transformation is a vertical shift upwards by 2 units. Therefore, take the inverted V-shape from the previous step and shift it upwards so that the new vertex is at (0,2).

Key Concepts

Absolute Value Function

The Absolute Value Function, denoted as \( y = |x| \), is a fundamental mathematical function that creates a V-shaped graph. This graph is centered at the origin, \(0,0\), and it opens upwards.
The defining feature of the absolute value function is its ability to output only non-negative values, regardless of the input \(x\).
Here's how it behaves depending on the value of \(x\):

• If \( x \geq 0 \), then \( |x| = x \).
• If \( x < 0 \), then \( |x| = -x \).

This dual behavior results in the characteristic sharp vertex at the origin.Additionally, the absolute value function is often used as a parent function to understand more complex transformations.

Graphing Transformations

Graphing transformations involve altering a function's graph in various structured ways to create a new graph. For the function \( y = 2 - |x| \), it is essential to recognize the starting point is the standard absolute value graph.
Transformations can include:

• Translations: Shifting the entire graph up, down, left, or right.
• Reflections: Flipping the graph over a specific axis.
• Stretches and Compressions: Changing the steepness of the graph.

By combining these, we can reshape the base absolute value graph into more complex forms, maintaining a structured mathematical approach.

Vertical Reflection

A Vertical Reflection is a type of graph transformation where the entire graph is flipped over the x-axis. In the exercise, the base function \( y = |x| \) is reflected vertically, as evidenced by the \(-|x|\) within the function \( y = 2 - |x| \).
Here's how it works:

• The original graph, which opens upwards, is reflected to open downwards.
• The vertex remains at the origin \( (0,0) \) until further transformations are applied.

This transformation results in an upside-down V-shape, which is crucial before applying any additional vertical shifts or other transformations.

Vertical Shift

A Vertical Shift is where the entire graph moves up or down the y-axis without changing its shape. For \( y = 2 - |x| \), the vertical shift is indicated by the \(+2\), which moves the graph upwards by two units.
The steps for applying a vertical shift include:

• Identify the shift value, here it is \(+2.\)
• Move the vertex of the graph from its original position \( (0,0) \) to the new position \( (0,2) \).
• Each point on the graph follows this upward movement by the same amount.

After applying this shift, the inverted V-shape graph has its vertex at the new location, demonstrating how vertical shifts modify the overall position of the graph while keeping its structure intact.