Question

Write a function \(g\) described by the given transformation of \(f(x)=|x|-5\). translation 2 units to the left

Short answer

The transformed function is \( g(x) = |x + 2| - 5 \).

Step-by-step solution

Identify the original function

The original function given is \( f(x) = |x| - 5 \). This is the function that is going to be transformed.

Determine the transformation

The transformation given is a shift 2 units to the left. This means \( h = 2 \). In general, a shift to the left by \( h \) units is represented by the function \( g(x) = f(x + h) \).

Apply the transformation to the original function

Substituting \( h = 2 \) in \( g(x) = f(x + h) \) gives \( g(x) = f(x + 2) \). Now replace \( f(x) \) in the equation with the form of the original function: \( g(x) = |x + 2| - 5 \).

Key concepts

Translations

In mathematics, especially in the realm of functions, a **translation** is a simple type of transformation that moves a figure or a graph to a different location without altering its shape, size, or orientation. Translations are very crucial when dealing with functions as they allow us to shift the graph along the axes.

When we talk about translating a function horizontally (along the x-axis), it can either be to the left or to the right. Here's how each works:

• A translation to the left is done by modifying the input of the function: replace every instance of "x" with "x + h", where "h" is the number of units you move it.
• A translation to the right translates the graph by replacing "x" with "x - h".

Essentially, these translations do not change the graph’s general appearance; they only move it around on the Cartesian plane. In the given exercise, the function was translated 2 units to the left, resulting in the function becoming \( g(x) = |x + 2| - 5 \). This signifies a horizontal shift leftwards while maintaining the same graph structure.

Absolute Value Functions

Absolute value functions possess an interesting property: they measure the distance of a number on the number line from zero, never considering the direction. This gives them a characteristic "V"-shape when graphed. The standard form of an absolute value function is \( f(x) = |x| \).

When additional constants are added, such as in \( f(x) = |x| - 5 \), it results in a vertical shift of the function’s graph. The constant "-5" in \( f(x) = |x| - 5 \) moves the whole "V"-shaped graph 5 units downward. The peak of the graph, which was originally at the origin \( (0,0) \), will now sit at \( (0,-5) \).

Absolute value functions are known for their piecewise nature. For \( f(x) = |x| \), it is equivalent to:

• \( f(x) = x \) for \( x \geq 0 \)
• \( f(x) = -x \) for \( x < 0 \)

This nature of breaking at the origin provides it the distinctive sharp turn, forming the "V"-shape on the graph.

Algebraic Functions

Algebraic functions encompass a broad category of functions formed using polynomial operations such as addition, subtraction, multiplication, division, and taking roots of polynomials. They are an essential component of algebra, providing a foundation for more complex mathematical concepts.

The basic algebraic functions include linear, quadratic, cubic, and generally any polynomial functions. In the exercise given, \( f(x) = |x| - 5 \) can be seen as an algebraic function. The transformation affected this algebraic function by modifying its input to \( x + 2 \), forming \( g(x) = |x + 2| - 5 \).

Understanding algebraic functions helps us appreciate how inputs are processed to obtain outputs, and how variables interact through arithmetic operations. For more complex forms, algebraic manipulation might involve factoring, expanding polynomials, or simplifying expressions, all of which form the basis for transformations and function behavior analysis.