Problem 24 - 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. $$ f(x)=|x|-1 $$

Step-by-Step Solution

Answer

Graph is a V shape with the vertex at (0, -1).

Step 1: Identify the standard function

The given function is \( f(x) = |x| - 1 \). The standard function for this is the absolute value function, \( g(x) = |x| \), which is a V-shaped graph with the vertex at the origin (0,0).

Step 2: Analyze transformations

Our function has a transformation applied to the standard absolute value function. The term \(-1\) at the end of \( f(x) = |x| - 1 \) is a vertical shift downwards. This means we move the entire graph of \( g(x) = |x| \) down by 1 unit.

Step 3: Sketch the transformed graph

Begin with the graph of \( y = |x| \), which has a vertex at (0,0) and opens upwards. Apply the transformation by shifting the entire graph down by 1 unit. The new vertex of the graph \( f(x) = |x| - 1 \) is now at (0, -1), with the arms of the V shape opening upwards as before.

Key Concepts

Understanding the Absolute Value Function

The absolute value function, typically expressed as \( |x| \), is central to understanding many graph transformations. On a graph, it forms a distinctive V-shaped curve. The characteristic shape originates from the definition of absolute value: it gives the distance of a number from zero on the number line. Thus, any negative inputs become positive, leading to the symmetrical nature of the graph.

To visualize this, think of the graph as consisting of two parts:

• For \(x \geq 0\), the graph is a straight line with the equation \(y = x\).
• For \(x < 0\), the graph is also a straight line but reflects across the y-axis, following \(y = -x\).

The vertex of this V-shaped graph is at the origin, (0,0), where the two lines meet. This point acts like a hinge, giving the graph its unique form.

Exploring Vertical Shift in Functions

The vertical shift is one of the basic transformations in function graphing. When we talk about shifting a function vertically, we move the entire graph up or down on the Cartesian plane without altering its shape or orientation.

In the context of our exercise, the function \( f(x) = |x| - 1 \) involves shifting the graph of \( g(x) = |x| \) downward by one unit. The term \(-1\) indicates the direction and magnitude of this shift:

• If the number were positive, the graph would move upwards.
• Since it's negative, the graph moves downwards.

This transformation changes the position of the function's vertex. For \( g(x) = |x| \), the vertex is at (0,0). After the shift in \( f(x) = |x| - 1 \), the vertex adjusts to (0,-1). The entire graph maintains its V-shape, just in a new vertical location.

Mastering Function Graphing Through Transformations

Function graphing can seem daunting at first, but understanding transformations simplifies the process. Transformations allow us to alter the position, shape, or size of a graph based on certain modifications. By applying transformations, we can take a well-known function, like the absolute value function, and adapt it to suit specific conditions.

Graph transformations generally include shifts, stretches, reflections, and combinations thereof. In our exercise, we applied a vertical shift. To accurately graph functions using transformations, follow these steps:

• Identify the base or parent function, such as \( y = |x| \). Recognize common attributes, like the V-shape and vertex.
• Determine which transformations will be applied to this base graph. In this example, it was a vertical shift downward.
• Execute the transformation on the base graph. Adjust the vertex, and ensure to reflect changes across the entire graph.

These steps ensure that even complex functions become manageable, as you systematically apply each transformation to graph the function precisely.