Problem 8 - Algebra 2 and Trigonometry

Question

In \(7-10,\) the domain of each function is the set of real numbers. a. Sketch the graph of each function. b. What is the range of each function? $$ y=|3 x+9| $$

Step-by-Step Solution

Answer

The range of the function is \( y \geq 0 \).

Step 1: Understanding the Absolute Value Function

The function given is \( y = |3x + 9| \). The absolute value function \( y = |x| \) reflects all negative input values to their positive counterparts. Thus, \( y = |3x + 9| \) will take any value \( 3x + 9 \) and ensure the output is non-negative.

Step 2: Finding the Vertex of the Absolute Value Graph

The graph of \( y = |3x + 9| \) is a V-shaped graph with its vertex where the expression inside the absolute value equals zero. Set \( 3x + 9 = 0 \) and solve for \( x \). \( 3x + 9 = 0 \Rightarrow 3x = -9 \Rightarrow x = -3 \). The vertex is at \( (-3, 0) \).

Step 3: Sketching the Graph

Using the vertex \( (-3, 0) \), plot this point on the graph. Because the graph is V-shaped, the left side of the V descends from the top left toward this vertex, and the right side ascends from this vertex to the top right. This reflects the graph's symmetry about the vertical line \( x = -3 \).

Step 4: Determining the Range of the Function

The range is the set of all possible output values of the function. Since absolute values are always non-negative, \( y \geq 0 \) for any \( x \). Thus, the range of \( y = |3x + 9| \) is all real numbers \( y \) such that \( y \geq 0 \).

Key Concepts

Graph Sketching

Graph sketching is an essential skill in mathematics as it helps visualize the behavior of functions. When sketching the graph of an absolute value function like \( y = |3x + 9| \), we focus on its characteristic V-shape. The absolute value graph transforms the line \( y = 3x + 9 \) such that all values become non-negative.

To start sketching, find the critical point, known as the vertex. This is where the expression inside the absolute value, \( 3x + 9 \), equals zero. Solving \( 3x + 9 = 0 \) gives \( x = -3 \), making our vertex at \((-3, 0)\).

From the vertex:

• The left arm of the V descends from infinity, reaching \((-3, 0)\).
• The right arm ascends onward to infinity.

This symmetry balances at \( x = -3 \), sketching evenly on both sides.

Vertex of a V-Shaped Graph

The vertex is a pivotal point in understanding absolute value functions, serving as the graph's turning point where direction changes. For the function \( y = |3x + 9| \), we compute the vertex by setting the inside of the absolute value, \( 3x + 9 \), to zero, as follows:

\[ 3x + 9 = 0 \Rightarrow 3x = -9 \Rightarrow x = -3 \]
This results in a vertex at \((-3, 0)\).

At this point:

• The graph changes from descending to ascending.
• It serves as the minimum point for the graph in terms of the y-value, which is zero.
• Visually, the vertex represents the sharpest part of the V shape, denoting symmetry.

Identifying the vertex helps in graph plotting and understanding the function's behavior.

Range of a Function

The range of a function is defined by all possible output values, typically concerning the y-axis. For absolute value functions like \( y = |3x + 9| \), the range is dictated by the property that absolute values cannot be negative.

This gives us a range that begins at the lowest point on the graph, which coincidentally is the y-value of the vertex, zero. Thus, every function value \( y \) satisfies \( y \geq 0 \).

Analyzing the range:

• Y-values start from zero and extend upwards to infinity.
• This illustrates that no matter the value of \( x \), \( y \) will never be negative.
• Graphs of absolute value functions typically have their range simply characterized as \([0, \infty)\).

Understanding the range is critical in defining the limits and behavior of the function across the y-axis.