Problem 45 - Algebra and Trigonometry

Question

33–48 ? Sketch the graph of the function, not by plotting points, but by starting with the graph of a standard function and applying transformations. $$ y=|x|-1 $$

Step-by-Step Solution

Answer

The graph is a V-shape with vertex at (0,-1), shifted down 1 unit from \( y = |x| \).

Step 1: Identify the Standard Function

The given function is \( y = |x| - 1 \), which is a transformation of the standard absolute value function \( y = |x| \). The graph of \( y = |x| \) is a V-shaped graph with its vertex at the origin (0, 0).

Step 2: Determine the Transformations

The function \( y = |x| - 1 \) indicates a vertical shift of the graph of \( y = |x| \). The \(-1\) means the entire graph of \( y = |x| \) will be shifted down by 1 unit.

Step 3: Apply the Transformations

To graph \( y = |x| - 1 \), start from the vertex of \( y = |x| \), which is (0, 0). Move this point down 1 unit to (0, -1). The graph will maintain its V-shape, with lines extending upwards and outwards, intersecting the x-axis at (1, 0) and (-1, 0).

Step 4: Sketch the Graph

Draw the transformed graph using the new vertex point at (0, -1), maintaining the V-shape. The left side from the vertex extends upwards with a slope of 1, indicating \( y = |x| - 1 \) for \( x < 0 \). The right side extends similarly for \( x > 0 \). The lowest point on the graph is at (0, -1), and the lines cross the x-axis at (1, 0) and (-1, 0).

Key Concepts

Absolute Value Function

The absolute value function is a fundamental piece of math that creates a well-known V-shaped graph. It is defined as \( y = |x| \). This function outputs the distance of a number from zero on the number line, which ensures that it is always non-negative. Because of this characteristic, the graph has a vertex at the origin, \((0, 0)\), and mirrors itself on either side of the y-axis.

There are a few key traits of the absolute value function to keep in mind:

• The graph is symmetrical around the y-axis.
• Both arms of the V-shaped graph have a slope of 1, meaning they rise 1 unit for every 1 unit they move horizontally away from the vertex.

Understanding the basic absolute value function is crucial for applying transformations and understanding how these changes shift and modify the graph's shape.

Vertical Shift

A vertical shift is a type of graph transformation that moves a function up or down on a graph. In the equation \( y = |x| - 1 \), the \(-1\) indicates a vertical shift.

To apply a vertical shift, you change the y-coordinate of each point on the graph by adding or subtracting a value.

• If the number is positive, the graph shifts up.
• If the number is negative, the graph shifts down.

For example, in \( y = |x| - 1 \), every point of the graph \( y = |x| \) moves down by one unit. This transforms the vertex of the V-shaped graph from \((0, 0)\) to \((0, -1)\).

While the position of the graph changes, the basic shape and symmetry of the function remain the same. Understanding vertical shifts helps you to see how graphs relate to their standard forms and predict their behavior in varied contexts.

V-shaped Graph

The V-shaped graph is a hallmark of the absolute value function. Its name comes from the distinctive V-like structure that appears when the absolute value graph is plotted. The most essential feature is its sharp vertex, which acts as a pivot or turning point.

Here's what you need to know about V-shaped graphs:

• The vertex is the lowest point if it opens upward, or the highest if it opens downward.
• In the case of \( y = |x| \), the V opens upward and contains only non-negative y-values.
• The slope of the lines forming the V is determined by the function. For \( y = |x| \), each side rises with a slope of 1 for each positive unit of movement in x.

This type of graph is not just limited to absolute value functions. Understanding V-shaped graphs and their characteristics allows for easier manipulation and interpretation of different functions and their transformations. In transformations like vertical shifts, this shape is preserved, though its position and orientation may change.