Problem 49 - Precalculus Essentials

Question

In each exercise, graph the functions in parts (a) and ( \(b\) ) in the same rectangular coordinate system. a. Graph \(f(x)=|x|\) using the ordered pairs \((-3, f(-3))\) \((-2, f(-2)),(-1, f(-1)),(0, f(0)),(1, f(1)),(2, f(2)), \quad\) and \((3, f(3))\). b. Subtract 4 from each \(y\) -coordinate of the ordered pairs in part (a). Then graph the ordered pairs and connect them with two linear pieces. c. Describe the relationship between the graph in part (b) and the graph in part (a).

Step-by-Step Solution

Answer

The graph of function \(f(x)=|x|\) forms a 'V' shape. When 4 is subtracted from every \(y\)-coordinate in part (a), it shifts the entire graph downwards by 4 units. Thus, the graph in part (b) is the same as in part (a), but it's located 4 units lower along the \(y\)-axis.

Step 1: Drawing the Graph for \(f(x)=|x|\)

First, plot the points given in the list of ordered pairs using \(f(x)=|x|\). Then join these points to draw the graph for the function. Remember, the graph for \(f(x)=|x|\) always produces a 'V' shape.

Step 2: Subtracting 4 from Each \(y\)-coordinate

Next, subtract 4 from the \(y\)-component of each ordered pair obtained from the first step. This will result in a set of new ordered pairs and graph them.

Step 3: Drawing the Transformed Graph

Graph the new ordered pairs obtained by subtracting 4 from each \(y\)-coordinate. As it is mentioned to connect them with two linear pieces, the graph would appear to be a translated version of original graph \(f(x)=|x|\) downwards along the \(y\)-axis by 4 units.

Step 4: Describe the Relationship

Finally, describe the relationship between the two graphs. This involves comparing the shapes, orientations, and positions of the graphs. Given that the operation performed was subtracting 4 from each \(y\)-coordinate, it can be stated that the graph obtained in part (b) is exactly the same as the graph in part (a), but translated down by 4 units along the \(y\)-axis.

Key Concepts

Function Transformation

Function transformation involves altering the position, size, or orientation of a graph. One common transformation is vertical translation, which means moving the graph up or down in a coordinate plane. In the exercise provided, the function transformation involves subtracting a constant from each \( y \)-coordinate.
This effectively lowers or translates the graph vertically downward by that constant's value. In part (b) of the exercise, subtracting 4 from each \( y \)-coordinate shifts the graph of \( f(x) = |x| \) down by 4 units without altering its shape.
Transformations can be studied systematically:

• **Shifting up or down**: Add or subtract a number to the function. This changes the vertical position.
• **Shifting left or right**: Modify the \( x \) values by adding or subtracting inside the function's argument. This adjusts the horizontal position.
• **Reflecting**: Multiply by \(-1\) impacts the function's orientation, either vertically or horizontally.
• **Stretching or compressing**: Multiply by a factor. Larger factors stretch the graph, while factors between \(0\) and \(1\) compress it.

Understanding these concepts helps in graphing transformed functions accurately.

Graphing

Graphing functions like \( f(x) = |x| \) involves plotting points derived from the function's equation. Each point is an ordered pair \((x, f(x))\). These points are then connected to reveal the shape of the graph.
The absolute value function \( f(x) = |x| \) creates a characteristic 'V' shape, with the vertex located at the origin \((0,0)\). Points derived by varying \(x\) explain the symmetry of the graph, with both branches sloping upwards equally.
In the exercise, you first plot the original graph of \( f(x) = |x| \). Then, after transforming it by subtracting 4 from each \( y \)-coordinate, you re-plot to get the new, translated function graph.
Steps in the graphing process:

• Choose a range of values for \( x \), such as \(-3\) to \(3\).
• Calculate corresponding \( f(x) \) values.
• Plot these points on the coordinate plane.
• Connect the points to model the graph.

This visual representation helps in understanding how the graph behaves and changes with transformations.

Coordinate System

The coordinate system is a framework for graphing functions. It consists of two perpendicular lines, known as the \( x \)-axis (horizontal) and the \( y \)-axis (vertical), intersecting at the origin. This forms a grid where each point is represented by an ordered pair \((x, y)\).
When graphing functions, the coordinate system helps to systematically organize and visualize the position of points.
For the function \( f(x) = |x| \), points such as \((-3, 3), (-2, 2), ... , (3, 3)\) are graphed to show where the function is defined.
The same system is used when graphing the transformed version, making it easy to compare the original and new positions.
Key characteristics of the coordinate system:

• **Origin**: The point \((0,0)\) where both axes intersect.
• **Quadrants**: The axes divide the plane into four quadrants, each with a unique combination of positive and negative \( x \) and \( y \) values.
• **Scale**: The axes are often scaled in consistent, equal increments to aid in precise plotting.
• **Axes and Labels**: Clearly labeling increases understanding and accuracy in graphing tasks.

Utilizing the coordinate system properly is crucial for examining function graphs and understanding transformations.