Problem 50 - Algebra and Trigonometry

Question

49–52 ? Graph the functions on the same screen using the given viewing rectangle. How is each graph related to the graph in part (a)? Viewing rectangle \([-8,8]\) by \([-6,6]\) $$ \begin{array}{ll}{\text { (a) } y=|x|} & {\text { (b) } y=-|x|} \\ {\text { (c) } y=-3|x|} & {\text { (d) } y=-3|x-5|}\end{array} $$

Step-by-Step Solution

Answer

Graphs relate through reflection, stretching, and horizontal shifting.

Step 1: Graph Function (a)

Plot the basic function \( y = |x| \) within the viewing rectangle \([-8, 8]\) by \([-6, 6]\). This is a V-shaped graph with the vertex at the origin (0, 0), opening upwards.

Step 2: Graph Function (b)

Plot the function \( y = -|x| \) in the same viewing window. This graph is the reflection of \( y = |x| \) across the x-axis, resulting in an upside-down V-shaped graph with the vertex at the origin.

Step 3: Graph Function (c)

Plot the function \( y = -3|x| \) using the same viewing rectangle. This graph is a vertical stretch and reflection of \( y = |x| \) across the x-axis. It looks like an upside-down V, but with steeper lines due to the multiplier -3.

Step 4: Graph Function (d)

Plot \( y = -3|x-5| \) in the viewing rectangle \([-8, 8]\) by \([-6, 6]\). This graph is the same as \( y = -3|x| \) but shifted 5 units to the right. It retains its steepness and V-shape.

Step 5: Analyze Relationships

Function (b) is a reflection of function (a) across the x-axis. Function (c) is a vertically stretched and reflected version of (a). Function (d) is function (c) translated to the right by 5 units.

Key Concepts

Absolute Value Functions

Absolute value functions are basic building blocks in mathematics, especially for graphing purposes. An absolute value function is defined as \( y = |x| \). This function creates a "V" shaped graph. The vertex is located at the origin, (0,0), and it opens upwards. The graph is symmetric around the y-axis, which means both halves of the "V" are mirror images of each other.
When plotting \(y = |x|\), you primarily consider two parts:

• If \(x\) is positive (or zero), the absolute value \(|x|\) is just \(x\).
• However, if \(x\) is negative, \(|x| = -x\).

This creates the V shape because of the behavior change at \(x = 0\). Understanding this simple pattern helps in comprehending more complex transformations down the line.

Reflection across the x-axis

Reflections change the direction in which a graph opens. The function \(y = -|x|\) is a reflection of \(y = |x|\) across the x-axis. This means that every point on \(y = |x|\) is flipped to be equidistant but on the opposite side of the x-axis.
This transformation results in an upside-down V shape, with the vertex remaining at the origin, (0,0).
Useful to remember is:

• The reflection across the x-axis changes the sign of the function, making all y-values negative.

This shift doesn't alter the x-values or the width of the graph, just the direction it opens.

Vertical Stretch and Translation

A vertical stretch makes a V-shaped graph narrower or steeper. When the function \(y = |x|\) is multiplied by a factor, such as seen in \(y = -3|x|\), the graph is vertically stretched by an absolute value of the multiplier. The graph not only stretches vertically, becoming steeper, but flips if the multiplier is negative.

For example, the graph of \(y = -3|x|\) indicates both a vertical stretch by a factor of 3 and a reflection. The negative sign causes a reflection across the x-axis. The lines of the V become three times steeper.
Meanwhile, horizontal translations involve shifts along the x-axis. This is evident in the function \(y = -3|x-5|\). Here, \(-3|x|\) is shifted 5 units to the right because of the \(x-5\).
This shift doesn't affect the steepness or orientation of the V shape, just its position on the graph.

Function Transformation

Function transformation is a crucial tool for moving graphs around without changing their basic structure. Depending on the type of transformation, we can slide, flip, stretch, or compress a graph.
In the given functions, transformations include reflections, vertical stretches, and horizontal shifts.
For example:

• The basic graph \(y = |x|\) undergoes a reflection across the x-axis to become \(y = -|x|\).
• The vertical stretch in \(y = -3|x|\) changes the opening steepness.
• The translation \(x-5\) in \(y = -3|x-5|\) shifts the graph horizontally.

Function transformations help in graphically representing different situations or solutions effectively, impacting how a specific graph relates to others visually and dynamically.