Problem 8 - Intermediate Algebra

Question

Graph each of the functions. $$f(x)=|x|+3$$

Step-by-Step Solution

Answer

The graph of \( f(x) = |x| + 3 \) is a 'V' shape, shifted up 3 units from \( f(x) = |x| \).

Step 1: Understand the Function

The function given is \( f(x) = |x| + 3 \). This function is a transformation of the basic absolute value function \( f(x) = |x| \). The absolute value function creates a 'V' shape graph centered at the origin.

Step 2: Determine the Transformation

The function \( f(x) = |x| + 3 \) involves a vertical shift. The '+3' indicates that the entire graph of \( |x| \) is shifted 3 units upwards.

Step 3: Plot Key Points

Choose a few key values for \( x \) to plot: - For \( x = 0 \), \( f(x) = |0| + 3 = 3 \). So the point is (0, 3).- For \( x = 1 \), \( f(x) = |1| + 3 = 4 \). So the point is (1, 4).- For \( x = -1 \), \( f(x) = |-1| + 3 = 4 \). So the point is (-1, 4).- For \( x = 2 \), \( f(x) = |2| + 3 = 5 \). So the point is (2, 5).- For \( x = -2 \), \( f(x) = |-2| + 3 = 5 \). So the point is (-2, 5).

Step 4: Draw the Graph

Plot the points from Step 3 on a coordinate plane. Draw two lines: one line from the origin moving left, passing through (-1,4) and (-2,5), and the another line moving right, passing through (1,4) and (2,5). The graph should be a 'V' shape centered at the point (0, 3), indicating the graph has been shifted up by 3 units.

Key Concepts

Vertical Shifts

Vertical shifts are a key concept in graphing transformations, especially with functions like the absolute value function. When we talk about vertical shifts, we refer to moving the entire graph of a function up or down without altering its shape. This happens by adding or subtracting a constant from the function.
For example, in the function \(f(x) = |x| + 3\), the "+3" is responsible for the vertical shift. This means:

• Every point on the graph of \(f(x) = |x|\) is moved 3 units upwards.
• This does not affect the 'V' shape of the graph; it merely relocates it higher on the y-axis.

Understanding vertical shifts is crucial because it helps predict and quickly identify the position of the graph on the coordinate plane.
This simple transformation allows us to take a known graph and adjust its position, making it easier to handle different kinds of mathematical problems.

Coordinate Plane

The coordinate plane is the stage where all graphing takes place. It consists of two perpendicular axes intersecting at a point called the origin.
The x-axis is horizontal, and the y-axis is vertical. Each point on the coordinate plane is represented as an ordered pair \((x, y)\):

• The first number represents the location along the x-axis.
• The second number indicates the position along the y-axis.

When graphing a function, such as \(f(x) = |x| + 3\), we move across the coordinate plane to plot points that lie on the graph. Those points help to visualize the function in two dimensions by showing its height (y-value) for every input (x-value).
The coordinate plane is fundamental because it provides a visual representation of mathematical relationships, making complex concepts easier to understand.

Plotting Points

Plotting points is the process of selecting specific values of \(x\) and determining the corresponding \(y\) values (or \(f(x)\)). These points are then marked on the coordinate plane and help to form the shape of the function being graphed.
To plot points for the function \(f(x) = |x| + 3\), follow these simple steps:

• Choose convenient values for \(x\), such as -2, -1, 0, 1, and 2.
• Calculate \(f(x)\) for each chosen value of \(x\). For example, \(f(0) = 3\) and \(f(1) = 4\).
• Record the resulting points: (0, 3), (1, 4), (2, 5), etc.
• Plot these points on the coordinate plane.

Connecting these points gives the geometric shape of the graph. For \(f(x) = |x| + 3\), these points collectively form a 'V' shape, shifted upwards due to the vertical transformation.
Plotting points is not only about marking locations; it's about revealing what the function looks like visually, helping you understand its behavior across different values of \(x\).