Problem 81 - College Algebra

Question

Begin by graphing the absolute value function, \(f(x)-|x| .\) Then use transformations of this graph to graph the given function. $$ g(x)-|x|+4 $$

Step-by-Step Solution

Answer

The graph of the given function \(g(x) = |x| + 4\) is the graph of the absolute value function \(f(x)=|x|\) shifted upwards by 4 units.

Step 1: Graph the Base Function

Begin by graphing the absolute function, \(f(x) = |x|\). It forms a V-like shape, the vertex is at the point (0,0), with lines forming a 45 degree angle with the x axis on either side of the vertex.

Step 2: Identify Transformation

Examine the given function \(g(x) = |x| + 4\) to identify transformations. Here the +4 in the given function denotes a vertical transformation, or a shift.

Step 3: Apply the Transformation

Apply the transformation by shifting the graph of \(f(x) = |x|\) upwards by 4 units to get the graph of \(g(x)= |x|+4\). All the points in \(f(x)\) have y-coordinates increased by 4.

Step 4: Graph the Transformed Function

The transformed graph still has the same V-like shape but now it's vertex is at point (0,4) because of the upward shift.

Key Concepts

Graph Transformations

Graph transformations involve shifting, reflecting, stretching, or compressing a graph. When graphed, functions can take on different shapes based on these operations.

• Translation: Moves every point of the graph a certain distance in specified directions.
• Reflection: Flips the graph over a specific axis.
• Scaling: Stretches or compresses the graph.

In the exercise, we start with the absolute value function, which is a V-shaped graph. To modify its appearance, we use graph transformations to produce the desired function. These transformations help us see functions in different contexts and understand their characteristics deeper.

Vertical Shift

A vertical shift is a special type of graph transformation that involves moving a graph up or down on the coordinate plane. Imagine shifting your graph vertically without changing its shape.
The operation changes all the y-coordinates of the graph but leaves the x-coordinates untouched.
In our example with the absolute value function, we moved the graph of \( f(x) = |x| \) upward to create the function \( g(x) = |x| + 4 \).

• The '+4' indicates that each point on the graph of the base function moves up by 4 units.
• This operation is simple yet powerful, allowing us to quickly redraw the function in a new location without altering its shape.

Vertical shifts are common in function transformations and an important concept to grasp when learning about graphing functions.

Vertex of a Parabola

The vertex of a parabola is a crucial point that defines its shape and position. In the context of the absolute value function, the vertex marks the point where the graph changes direction.
For the basic absolute value function \( f(x) = |x| \), the vertex lies at \((0,0)\).
With the transformation we applied, the graph shifts, resulting in a new vertex.

• After applying a vertical shift upwards by 4 units, the vertex of \( g(x) = |x|+4 \) can be found at \((0,4)\).
• This new point marks the lowest or highest part of the parabola, often referred to as the *turning point*.

Understanding the vertex is beneficial for sketching parabolic graphs and identifying the graph's minimum or maximum value, based on the data presented. It's a foundational element that extends into many areas of mathematics.