Problem 20 - Beginning and Intermediate Algebra

Question

Sketch the graph of \(f(x) .\) Then, graph \(g(x)\) on the same axes using the transformation techniques. $$\begin{aligned}&f(x)=|x|\\\&g(x)=|x|+1\end{aligned}$$

Step-by-Step Solution

Answer

To sketch the graph of \(f(x) = |x|\), plot a V-shaped graph with the vertex at the origin. The slopes of the two branches are -1 (negative domain) and 1 (positive domain). To graph \(g(x) = |x| + 1\), perform a vertical transformation, translating the graph of f(x) upward by one unit. The graph of g(x) is also a V-shaped graph, but with the vertex at the point (0,1). Both branches have the same slopes as f(x).

Step 1: Sketch the graph of f(x)

To sketch the graph of \(f(x) = |x|\), we need to analyze the function to understand how it behaves. The function \(|x|\) is the absolute value of x, which means that the function will always return a positive value for any real number input. Since f(x) measures the distance of x to 0, the graph starts at the point (0,0). The function will increase linearly as we move towards positive x and it will show a reflection in the positive direction when moving in negative x. You can sketch the graph of f(x) as a V shape with vertices at origin. The slopes of two branches will be -1 (in the negative domain) and 1 (in the positive domain).

Step 2: Perform the transformation on f(x)

Now, we need to graph \(g(x) = |x| + 1\) on the same axes as f(x). According to the equation, all the values with an absolute value of x greater or equal to 0 will have a value of +1. This means that the graph of g(x) is a translation of f(x) upward by 1 unit. To obtain the graph of \(g(x) = |x| + 1\), we simply take each point on the graph of f(x) and translate it upwards by one unit. This means that the new vertex of g(x) will be at (0,1), at the same point on the x-axis as f(x) but one unit higher on the y-axis.

Step 3: Connect the points

Finally, we connect the points for g(x) to obtain the graph. Just like the graph of \(f(x) = |x|\), this graph will also be a V-shaped graph but translated upward by one unit. This means the slopes of two branches in g(x) will remain the same; only the vertex will be at a different point. So, to summarize, the graph of \(g(x) = |x| + 1\) is a vertical transformation of the graph of \(f(x) = |x|\) where the graph of g(x) is shifted upward by one unit, and the vertex of g(x) lies at the point (0,1).

Key Concepts

Absolute Value Function

The absolute value function is one of the most fundamental functions in mathematics, represented by \( f(x) = |x| \). This function captures a simple concept: the distance of a number from zero on the number line. Regardless of whether \( x \) is positive or negative, the absolute value \(|x|\) will always be non-negative.
The graph of the absolute value function forms a V shape. This V shape is symmetric around the y-axis, creating a mirror image between the left and right branches.

• The vertex or the tip of the V is at the origin point (0, 0).
• To the right of the vertex, the graph rises with a slope of 1, representing the increase in value as \( x \) becomes more positive.
• To the left of the vertex, it rises with a slope of -1, as the absolute value negates any negatives to yield non-negative outputs.

Understanding this graph is crucial because many transformations are based on how it moves or reshapes, helping to explore more complex functions later.

Graphing Functions

Graphing functions allows us to visualize how functions behave across their domain. In this exercise, sketching the graph of \( f(x) = |x| \) is the first step. A good graph provides critical insights into the function's increasing or decreasing nature or fixed points.

• Start by identifying important points like the vertex, where the function changes its direction.
• Plot basic points that reflect the slope; for instance, (1,1) and (-1,1) on either side of the vertex for \( f(x) = |x| \).
• Draw straight lines that connect these points to form the shape of the graph.

When graphing, remember symmetry. The function \( f(x) = |x| \) has mirror symmetry about the y-axis. Knowing this helps in quickly sketching one side and deriving the other without recalculating every point. Practicing with these graphs enhances your intuition about how functions transform and relate to their algebraic expressions.

Vertical Translation

Vertical translation is a transformation that shifts a graph up or down along the y-axis, without affecting its shape or orientation. In the problem, \( g(x) = |x| + 1 \) is a perfect example of this transformation. From the base graph \( f(x) = |x| \), you add a constant (in this case, +1) to move every point one unit up.
Key characteristics of vertical translation:

• The entire graph moves in the same direction, preserving the shape.
• The vertex of the absolute value graph moves from (0,0) to (0,1), reflecting the shift upward by 1.
• The slopes of the branches remain the same; it is only the position that changes.

This transformation allows for adjusting functions to fit specific requirements or constraints, making it a powerful tool in graph analysis and equation solving. Understanding vertical translation opens the doors to explore more complicated transformations like horizontal shifts or reflections.