Problem 46 - 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

Shift the v-shape of \(y = |x|\) right by 1 unit for \(y = |x-1|\).

Step 1: Identify the Standard Function

The given function is \( y = |x-1| \). The standard function similar to this is \( y = |x| \), which is the graph of an absolute value function with a vertex at the origin (0,0) and v-shaped symmetry.

Step 2: Determine the Transformation Needed

The function \( y = |x-1| \) can be seen as a horizontal transformation of the standard function. A function of the form \( y = |x-a| \) means the graph of \( y = |x| \) is shifted to the right by \( a \) units. Here, \( a = 1 \), so the graph shifts 1 unit to the right.

Step 3: Sketch the Transformed Graph

Start by drawing the graph of \( y = |x| \), which is a v-shaped graph opening upwards with the vertex at the origin (0,0). For \( y = |x-1| \), move this entire graph 1 unit to the right. The new vertex will be at (1, 0), and the general v-shape remains the same.

Key Concepts

Absolute Value Functions

Absolute value functions are fundamental in mathematics, especially in algebra and calculus. These functions produce outputs that are always non-negative. The absolute value of a number represents its distance from zero on the number line, and this property is reflected in their mathematical definition.
The standard form of an absolute value function is expressed as \( y = |x| \). It is characterized by a V-shape, where the vertex, or the point where the direction changes, is located at the origin \((0,0)\).
Some key properties of absolute value functions include:

• They always yield positive results or zero.
• Symmetry around the y-axis, creating a mirrored effect.
• The V-shaped graph has two linear components, one increasing and one decreasing as you move away from the vertex.

In transformations, these functions often serve as a starting point for various manipulations, which shift or scale the basic graph in different dimensions.

Horizontal Shifts

Horizontal shifts are a type of transformation applied to functions, which moves the entire graph left or right on the coordinate plane. In the context of the exercise, horizontal shifts are used to move the graph of an absolute value function.
The function \( y = |x-a| \) describes a horizontal shift of the base function \( y = |x| \). Here, the shift is dictated by the value of \( a \):

• If \( a > 0 \), the graph shifts \( a \) units to the right.
• If \( a < 0 \), the graph shifts |\( a \)| units to the left.

In our example, the function \( y = |x-1| \) causes the graph of \( y = |x| \) to shift 1 unit to the right.
This horizontal movement changes the position of the vertex from \((0,0)\) to \((1,0)\), demonstrating the impact of the transformation. Horizontal shifts do not alter the shape or orientation of the graph, only its position along the x-axis.

V-Shaped Graphs

V-shaped graphs are a distinctive hallmark of absolute value functions. The sharp corner at the vertex, and the linear arms extending outwards create the characteristic V shape.
The graph of \( y = |x| \) opens upwards and is symmetric around the y-axis. This symmetry means each side of the graph is a mirror image of the other. The vertex is the point of turning, and in a standard absolute value function, this is located at the origin \((0,0)\).
With transformations, such as horizontal shifts, the V-shape remains constant. Whether the vertex moves to a different position on the plane, the angles and proportions of the V do not change.
Understanding this V-shape is crucial as it helps in predicting the graph’s behavior after transformations, and can aid in solving equations involving absolute values by visualizing how the graph intersects the x-axis or other lines.