Problem 28 - College Algebra

Question

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

Step-by-Step Solution

Answer

Graph a V-shape with vertex (-3, 2) opening upwards.

Step 1: Understand the Absolute Value Function

The function given is of the form \( f(x) = |ax + b| + c \). For the specific function \( f(x) = |3x + 9| + 2 \), this means that \( f(x) \) will change at the point where \( 3x + 9 = 0 \). This is due to the properties of the absolute value function, which changes direction at the point where the expression inside the absolute value is zero.

Step 2: Find the Vertex

To find the 'vertex' or the turning point of the function, set the inside of the absolute value to zero: \( 3x + 9 = 0 \). Solving this gives \( x = -3 \). Evaluate the function at \( x = -3 \) to find the y-coordinate of the vertex: \( f(-3) = |3(-3) + 9| + 2 = |0| + 2 = 2 \). So the vertex of the function is \((-3, 2)\).

Step 3: Determine the Shape and Direction

Since this is an absolute value function with a positive coefficient in front of \( x \), the graph will have a V-shape with the vertex being the lowest point (a minimum at \((-3, 2)\)). The parts of the graph will be linear, reflecting symmetrically upwards from this vertex.

Step 4: Identify Additional Points

To draw the graph, identify additional points on either side of the vertex. For example, calculate \( f(x) \) for \( x = -2 \) and \( x = -4 \). At \( x = -2 \), \( f(-2) = |3(-2) + 9| + 2 = |3| + 2 = 5 \). At \( x = -4 \), \( f(-4) = |3(-4) + 9| + 2 = |-3| + 2 = 5 \). These points, \((-2, 5)\) and \((-4, 5)\), are equidistant and symmetric about the vertex.

Step 5: Sketch the Graph

Start by plotting the vertex \((-3, 2)\) on your graph. Then plot the additional points identified: \((-2, 5)\) and \((-4, 5)\). Draw straight lines connecting these points, forming a V-shape that opens upwards. The line should have two linear segments, one from \((-\infty, -3)\) to \((-3, 2)\) and another from \((-3, 2)\) to \((\infty, 5)\).

Key Concepts

Absolute Value Function

An absolute value function is a fundamental concept where the function takes the form \( f(x) = |ax + b| + c \). It creates a characteristic V-shaped graph. The absolute value essentially means the distance from zero on a number line, so it is always non-negative. This is why the graph of an absolute value function typically has two linear pieces meeting at a point called the vertex.

The function changes its direction when the expression inside the absolute value becomes zero. For example, in the function \( f(x) = |3x + 9| + 2 \), the change occurs at \( x = -3 \) because it is where \( 3x + 9 = 0 \). At this point, the expression inside the absolute value flips in sign, creating the graph's vertices and changing its slope.

• Absolute values can be thought of as converting negative values to positive ones.
• Graphically, they create a V-shape, opening upwards or downwards, depending on the context.

Graphing Functions

Graphing functions is a crucial skill in algebra, helping visualize the set of all possible y-values a function can take for its corresponding x-values. For an absolute value function, understanding its V-shaped graph is key.

To graph \( f(x) = |3x + 9| + 2 \):
1. Start by calculating the vertex, where the graph changes direction.
2. Find points on either side of the vertex to establish the symmetry and slope of the lines extending from the vertex.
3. Plot these three main points and draw the linear segments. This results in the V-shape of the absolute value function.

These steps form the basis of plotting more complex functions and understanding their behavior visually. Using these techniques can help deduce the impact of different variables in the equation.

Vertex of a Function

The vertex of a function is the point where a change occurs in the direction of the graph, especially for absolute value and quadratic functions. The vertex gives us the minimum or maximum value of these functions.

In our example, the function \( f(x) = |3x + 9| + 2 \) has its vertex at \( (-3, 2) \). This vertex is derived by setting the inside of the absolute value (\( 3x + 9 \)) to zero and solving for \( x \).

After finding \( x = -3 \), the function's value at that point is evaluated to yield the full vertex coordinate \((x, y)\), as \( f(-3) = 2 \). Thus, the vertex is \((-3, 2)\), indicating the lowest point on the graph because it opens upwards.

Linear Equations

Linear equations represent the simplest form of algebraic functions, usually with graphs that are straight lines. In the context of an absolute value function like \( f(x) = |3x + 9| + 2 \), linear equations describe each arm of the V-shape.

Each side of the vertex forms a linear equation. For points to the left of the vertex \((-3, 2)\), the slope is negative compared to points on the right where the slope is positive. Calculating these involves understanding the absolute value's impact, which converts any negative solutions to positive at the vertex, thereby changing the equation's slope.

Linear segments convey the immediate growth rate of the function as it moves away from or toward the vertex, and these are identified by plugging values on either side of the vertex into the function.