Problem 28 - College Algebra with Corequisite Support

Question

For the following exercises, graph the given functions by hand. \(f(x)=|3 x+9|+2\)

Step-by-Step Solution

Answer

The graph is a 'V' shape with vertex at (-3, 2) and additional points like (-4, 5) and (-2, 5).

Step 1: Identify the Absolute Value Function

The function given is \( f(x) = |3x + 9| + 2 \). This is an absolute value function of the form \( f(x) = |ax + b| + c \), where \( a = 3 \), \( b = 9 \), and \( c = 2 \).

Step 2: Determine the Vertex

The vertex of an absolute value function \( |ax + b| + c \) occurs where the expression inside the absolute value \( ax + b = 0 \). Solving \( 3x + 9 = 0 \) gives \( x = -3 \). So, the vertex is \((-3, 2)\).

Step 3: Sketch the V-Shaped Graph

An absolute value function forms a "V" shape. The vertex \((-3, 2)\) is the point where the graph changes direction. To the left of \(-3\), the graph has a negative slope, and to the right, a positive slope. This means the graph is linear with a slope of 3 when \( x < -3 \) and also \( x > -3 \) but changes direction at \( x = -3 \).

Step 4: Plot Additional Points

Choose values of \( x \) on each side of the vertex to plot additional points. For example, try \( x = -4 \) and \( x = -2 \). Calculate \( f(-4) = |3(-4) + 9| + 2 = | -3 | + 2 = 5 \) and \( f(-2) = |3(-2) + 9| + 2 = |3| + 2 = 5 \). Plot the points \((-4, 5)\) and \((-2, 5)\).

Step 5: Draw the Graph

Connect the points plotted with straight lines to form the "V" shape. Make sure the vertex at \((-3, 2)\) is the lowest point and that the graph extends linearly out from this point, with the slope changing at \((-3, 2)\).

Key Concepts

Vertex of Absolute Value Function

In an absolute value function, the vertex is a critical point where the graph shifts direction. For example, in the function \( f(x) = |3x + 9| + 2 \), the vertex can be found by setting the expression inside the absolute value to zero. This means solving the equation \( 3x + 9 = 0 \). By isolating \( x \), we derive \( x = -3 \). At this point, substitute back into the function to evaluate \( y \). Thus, the vertex is \((-3, 2)\).
Understanding the vertex is crucial because it represents the minimum or maximum point of the graph. The vertex is where the graph makes its "V" turn. Knowing this helps to sketch the graph more accurately, ensuring you have a clear start and turn-off point.

Plotting Points

Plotting points around the vertex is essential to fully graph an absolute value function accurately. Once the vertex is determined, we choose points on either side of it to better define the graph's shape. For our example function \( f(x) = |3x + 9| + 2 \), we find additional points like \((-4, 5)\) and \((-2, 5)\).
Here is how you can plot these points:

• Choose an \( x \) value slightly less than the vertex \(-3\), such as \( x = -4 \), and solve for \( f(x) \).
• Choose another \( x \) value slightly greater than \(-3\), such as \( x = -2 \), and again, compute \( f(x) \).

Plotting these points and connecting them with the vertex helps visualize the graph's "V" shape and ensures it isn't skewed or misplaced.

V-Shaped Graph

Absolute value functions have a distinct "V" shape which is incredibly useful to understand before graphing. The vertex, the pinpoint at \((-3, 2)\), serves as the cusp of this "V". Each arm of the "V" extends outward and is symmetric around the vertex.
When you look at the function \( f(x) = |3x + 9| + 2 \), this V-shaped pattern arises because of how absolute values affect linear equations. Before the vertex, the slope has a negative value, but afterwards, it becomes positive. Visualizing this "V" shape ahead of graphing will improve accuracy and comprehension, ensuring the absolute value function is portrayed correctly.

Slope of Linear Functions

An important part of understanding absolute value functions is knowing how the slope of the linear equation inside the absolute value contributes to the graph's shape. In the function \( f(x) = |3x + 9| + 2 \), the linear expression \( 3x + 9 \) contributes a slope of 3. This slope impacts both arms of the "V" shape.

• On the left of the vertex \((-3, 2)\), the function's slope is \(-3\).
• On the right of the vertex, the slope switches to \(3\).

Reminding yourself that the absolute value causes a flip at the vertex that transforms negative slopes into positive ones can aid in drawing each side of the V-shaped graph accurately. Understanding these slope changes will deepen your comprehension of graphing absolute value functions by defining how quickly or slowly each side of the "V" ascends or descends.