Problem 21 - Beginning and Intermediate Algebra

Question

Sketch the graph of \(f(x) .\) Then, graph \(g(x)\) on the same axes using the transformation techniques. $$\begin{aligned}&f(x)=|x|\\\&g(x)=|x|+3\end{aligned}$$

Step-by-Step Solution

Answer

To sketch the graphs of the functions f(x) and g(x), follow these steps: 1. Understand f(x) = |x| is an absolute value function that forms a V shape with vertex at (0, 0). 2. Plot basic points for f(x), such as (-3, 3), (-2, 2), (-1, 1), (0, 0), (1, 1), (2, 2), and (3, 3). 3. Understand g(x) = |x| + 3 is a vertical translation of f(x), shifted upwards 3 units. 4. Plot basic points for g(x) by adding 3 to the y-coordinates of the points in f(x), such as (-3, 6), (-2, 5), (-1, 4), (0, 3), (1, 4), (2, 5), and (3, 6). 5. Sketch both graphs on the same axes: f(x) as a V shape, and g(x) as a V shape shifted 3 units above f(x). Both graphs are symmetrical about the y-axis.

Step 1: Understand the function f(x)

f(x) = |x| is the absolute value function. It represents the distance of the input value (x) from 0. The graph of this function forms a V shape, with its vertex at the origin (0,0). The function is symmetrical about the y-axis.

Step 2: Plot basic points for f(x)

For f(x) = |x|, plot some basic points: 1. f(-3) = |-3| = 3 2. f(-2) = |-2| = 2 3. f(-1) = |-1| = 1 4. f(0) = |0| = 0 5. f(1) = |1| = 1 6. f(2) = |2| = 2 7. f(3) = |3| = 3 Now connect these points to form a V shape for f(x).

Step 3: Understand the function g(x)

g(x) = |x| + 3 is a transformation of the function f(x). Specifically, it is a vertical translation of the f(x) graph. The "+3" indicates that the graph will be shifted 3 units upwards.

Step 4: Plot basic points for g(x) on the same axes

As g(x) is a vertical transformation (+3) of f(x), plot the same basic points for f(x), but with the coordinates shifted by 3 units upwards: 1. g(-3) = |-3| + 3 = 3 + 3 = 6 2. g(-2) = |-2| + 3 = 2 + 3 = 5 3. g(-1) = |-1| + 3 = 1 + 3 = 4 4. g(0) = |0| + 3 = 0 + 3 = 3 5. g(1) = |1| + 3 = 1 + 3 = 4 6. g(2) = |2| + 3 = 2 + 3 = 5 7. g(3) = |3| + 3 = 3 + 3 = 6 Now connect these points to form a V shape for g(x) that is 3 units above the graph of f(x).

Step 5: Finished sketch

Both the graphs of f(x) = |x| and g(x) = |x| + 3 should be sketched on the same axes. The graph of g(x) will be a V shape that is shifted 3 units vertically above the graph of f(x). Both graphs should be symmetrical about the y-axis.

Key Concepts

Vertical Transformations

Vertical transformations involve shifting a graph up or down on a coordinate plane. When we look at \(g(x) = |x| + 3\), it reflects a vertical transformation. The "+3" indicates that every point on the graph of \(|x|\) will move three units upward. So, if the original \(f(x) = |x|\) had a point at \((1, 1)\), then \(g(x)\) will have the point \((1, 4)\).

This process doesn't change the shape of the graph; it maintains the same V shape but positioned higher.

• To perform a vertical transformation, add or subtract a constant to the function's equation.
• If you add a number, the graph moves up; if you subtract, it moves down.
• The distance moved equals the value of the transformation.

Understanding vertical transformations helps to quickly graph transformed functions without recalculating from scratch.

Absolute Value Function

The absolute value function \(f(x) = |x|\) is one of the fundamental mathematical functions. It expresses the distance of any number \(x\) from zero on the number line. Therefore, it always returns a non-negative value. The graph of an absolute value function is characterized by its V shape, with the vertex often located at the origin (0,0).

Here's why it's essential:

• The formula \(|x|\) jumps to its minimum value of 0 when \(x = 0\).
• For any positive \(x\), the graph ascends at a constant rate.
• Similarly, for any negative \(x\), the graph mirrors this ascent on the opposite side.

Knowing that the resulting V-shape is central to tackling problems involving absolute values, as it gives a visual idea of symmetry and transformations.

Symmetry About the Y-Axis

Symmetry about the y-axis is a crucial feature of the absolute value function. This symmetry means that for every point \((x, y)\), there is a corresponding point \((-x, y)\). Visually, this symmetry produces a V shape centered on the y-axis.

Here's why it's practical:

• The graph on the right (positive x-values) mirrors exactly the graph on the left (negative x-values).
• If you're plotting one side of the graph, the other can be easily deduced by simply reflecting over the y-axis.
• This symmetry aids in predicting the behavior of transformations, ensuring that any vertical shifts do not affect the symmetry about the y-axis.

Understanding this concept ensures that when you apply changes to the function, such as vertical translations, the fundamental symmetric property remains consistent across its graph.