Problem 53 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to approximate any solutions of the equation. [Remember to write the equation in the form \(f(x)=0.1\) $$-|x+1|=-3$$

Step-by-Step Solution

Answer

Solving \(-|x+1|=-3\)\ using a graphing utility yields the solutions \(x=-4\) and \(x=2\).

Step 1: Transform Equation to Required Format

The equation \(-|x+1|=-3\) can be re-written as \(|x+1|=3\). This implies that the expression under absolute value symbol (the value of \(|x+1|\)) needs to be 3. To represent it in the form of \(f(x)=0.1\), subtract 3 from both sides to get the new equation \( |x+1|-3=0 \). This can be written as\(f(x)=|x+1|-3\)

Step 2: Graph the Function

Next, use a graphing utility to plot this function. The function \(f(x)=|x+1|-3\) describes a vertical shift of 3 units downwards from the graph of the absolute value function \(|x+1|\). Therefore, the graph will look like a V but 3 units below the usual 'V'. The vertex of the 'V' will lie on the x-axis.

Step 3: Identify Solutions

After drawing the function, note the x-values at which the graph crosses the x-axis. These x-values are the solutions of the equation. Whenever \(f(x)\) intersects the x-axis, it means \(f(x) = 0\), thus solving the equation we started with.

Key Concepts

Graphing Utility

Graphing utilities are indispensable tools in the realm of mathematics, especially when it comes to understanding complex equations and functions. They allow us to visually represent mathematical expressions and thereby gain insights that might be difficult to ascertain through algebra alone. When it comes to solving absolute value equations like \( -|x+1|=-3 \), a graphing utility transforms the abstract into something concrete.

Using a graphing utility, we can plot the equation once it is in the format \( f(x) = 0 \), such as \( f(x)=|x+1|-3 \). This visualization showcases how the function behaves across different values of \( x \), and, crucially, allows us to identify where the graph intersects the x-axis. These points of intersection represent the solutions to our equation. Such utilities are user-friendly and can handle a variety of functions, enabling students not only to confirm their solutions but also to explore the effects of each transformation applied to the function.

Absolute Value Function

An absolute value function is fundamental in algebra and presents a unique graph, usually in the shape of a 'V'. The absolute value of a number is its distance from zero on the number line, regardless of direction, which is why it's always non-negative. The general form is \( f(x) = |x| \), but this can be modified with additional transformations.

In the given exercise, the absolute value function \( |x+1| \) is altered by a subtraction of 3, changing its graph. The 'V' shape now shifts 3 units downward, altering the position of its vertex but not its symmetry or slope. The graph helps us understand how the output of the function, or \( f(x) \), varies with different inputs (the values of \( x \)) and is an excellent example of how absolute value functions encode distance in a geometrical context.

Algebraic Transformations

Algebraic transformations are operations applied to functions that change their appearance or position on a graph without altering their core characteristics. These include shifting, stretching, compressing, and reflecting the graph of a function. Understanding these transformations helps students manipulate and solve equations more effectively.

Considering our initial equation \( -|x+1|=-3 \), we apply the transformation by isolating the absolute value on one side resulting in \( |x+1|=3 \). Then, to express it as \( f(x)=0 \), we subtract 3 from both sides, revealing the downward shift in the graph. Whenever we solve for \( f(x)=0 \), we're looking for the input values \( x \) that return the function to its 'origin', which after transformation is represented as the x-axis intersection points. These operations are fundamental in the study of algebra, providing a visual and analytical understanding of equations and their solutions.