Problem 5 - Algebra A Combined Function

Question

Sketch the graph of each function. $$ f(x)=|x-4| $$

Step-by-Step Solution

Answer

The graph is a V-shape with vertex at (4, 0) and opens upward.

Step 1: Understanding the Absolute Value Function

The function provided is \( f(x) = |x-4| \). This is an absolute value function, which means it creates a 'V' shape on the graph. The expression inside the absolute value, \(x-4\), indicates a horizontal shift of the basic absolute value graph \(f(x) = |x|\). The graph of \(f(x) = |x-4|\) will shift to the right by 4 units.

Step 2: Identify Important Features

The vertex of the graph \( f(x) = |x-4| \) occurs where the expression inside the absolute value is zero. Solve \( x - 4 = 0 \) to find that \( x = 4 \). The vertex gives the lowest point of the graph at coordinate \((4, 0)\). This means the graph will be at its minimum value (0) at \( x = 4 \). The graph is symmetric around the vertical line \( x = 4 \).

Step 3: Plotting the Vertex and Points

Firstly, plot the vertex at \((4, 0)\). Then, choose points around the vertex to get a sense of how the graph looks. For example, for \(x=3\), \(f(3) = |3-4| = 1\), so plot the point \((3, 1)\). Similarly, for \(x=5\), \(f(5) = |5-4| = 1\), plot the point \((5, 1)\). Additionally, for \(x=2\), \(f(2) = |2-4| = 2\), plot the point \((2, 2)\). Finally, for \(x=6\), \(f(6) = |6-4| = 2\), plot the point \((6, 2)\).

Step 4: Draw the Graph

Using the points plotted, draw a 'V' shape. The left arm of the V will pass through the points \((3, 1)\) and \((2, 2)\), and the right arm through the points \((5, 1)\) and \((6, 2)\). Each arm of the V continues infinitely because as \(x\) moves away from the vertex, the absolute value function continues to increase.

Key Concepts

Graphing

Graphing an absolute value function, such as \( f(x) = |x-4| \), involves understanding how the shape of 'V' is formed on the coordinate plane. The graph of an absolute value function has a distinct 'V' shape because the function outputs positive values regardless of whether the input is positive or negative. This characteristic creates two linear arms that intersect at a point called the vertex.
To begin graphing:

• Identify the vertex, which is the lowest point on the graph for absolute value functions not involving reflections.
• Determine the direction of the arms, which arise from the nature of the absolute value operation.

Plot several key points to accurately illustrate the graph's shape, focusing on how the graph changes near the vertex and expands outward.

Horizontal Shift

A horizontal shift refers to the movement of the graph along the x-axis. This occurs due to changes inside the absolute value expression. For the function \( f(x) = |x-4| \), the horizontal shift is caused by the \(x-4\) inside the absolute value.
Here's how it works:

• The graph of \(f(x) = |x|\) starts from the origin \((0,0)\).
• When the equation becomes \(f(x) = |x-4|\), it shifts 4 units to the right.

This shift is visible because the expression \(x-4\) equals zero when \(x=4\), moving the entire graph towards the positive x-axis. It's important to note this horizontal movement alters only the position along the x-axis and not the overall shape of the graph.

Vertex

The vertex is a vital feature of an absolute value graph, acting as the turning point where the two linear sections meet. For the function \( f(x) = |x-4| \), the vertex is located at \((4, 0)\). This point is where the expression within the absolute value becomes zero, indicating no vertical distance from the x-axis.
To find the vertex:

• Set the inside of the absolute value to zero: \(x-4=0\).
• Solve for \(x\), leading to \(x=4\).

The y-coordinate is always zero unless there is a vertical shift involved. Hence, for \(f(x) \), the vertex lies exactly on the x-axis, representing the minimum value of the function. The vertex serves as a critical reference point for graph development.

Symmetry

Symmetry in the graph of an absolute value function like \( f(x)=|x-4| \) signifies that the graph is mirrored on either side of a vertical line. For this function, the line of symmetry is at \( x=4 \), which passes through the vertex.
Understanding symmetry:

• The arms of the 'V' are identical in shape and size on both sides of this line.
• If a point is at \( (x, y) \) on one side, typically \( (4-x, y) \) or \( (4+x, y) \) will be mirrored on the opposite side.

Symmetry is a powerful concept in graphing as it allows you to easily predict the location of points and ensures the graph is balanced. This property helps in accurately sketching the function once the vertex and line of symmetry are known.