Problem 24 - College Algebra

Question

Graph each equation .Let $x=-3,-2,-1,0,1,2, and 3. $$ y=|x|-1 $$

Step-by-Step Solution

Answer

The graph of the function \(y=|x|-1\) will start from the point \((-3,2)\), go down and touch the y-axis at point \(y=-1\) and then again start increasing.

Step 1: Understand the Function

Firstly, let's understand that the absolute value function, \(|x|\), is always non-negative. When \(x>0, |x|=x\), when \(x==0, |x|=0\), and when \(x<0, |x|=-x\). Considering this, our function \(y=|x|-1\) will comprise of two pieces: when \(x>=0, y = x -1\) and when \(x<0, y = -x - 1\).

Step 2: Calculate y values for given x values

Now we'll calculate the 'y' values for each of the given 'x' values. For \(x=-3, y=-(-3)-1=2\)For \(x=-2, y=-(-2)-1=1\)For \(x=-1, y=-(-1)-1=0\)For \(x=0, y=0-1=-1\)For \(x=1, y=1-1=0\)For \(x=2, y=2-1=1\)For \(x=3, y=3-1=2\)

Step 3: Graph the Function

The last step is to plot these points on a graph and also plot the function. The points we plot are \((-3,2), (-2,1), (-1,0), (0,-1), (1,0), (2,1), (3,2)\). When we connect these points, we obtain the graph of the function \(y=|x|-1\).

Key Concepts

Graphing Functions

Graphing functions is a visual way to understand their behavior. It helps to see how the function changes as the variable changes. The graph of a function represents all the solutions of an equation. In our case, we're working with the function \( y = |x| - 1 \).
To graph this function, we calculated the corresponding \( y \) values for each \( x \) from the given set \( \{-3, -2, -1, 0, 1, 2, 3\} \). Each solution \((x, y)\) is a point on the graph. In simpler terms, the function's graph is just a collection of points that we connect with lines. It allows us to see the shape and direction of the curve, which in this case is a "V" shaped due to the absolute value.

• Plot each \((x, y)\) point on a set of axes.
• Connect the points to visualize the entire function.

Understanding the plot helps to predict values, recognize patterns, and see function's symmetries or asymmetries clearly.

Piecewise Functions

Piecewise functions are functions defined by different expressions based on the input value (or "piece"). The absolute value function \( y = |x| - 1 \) is an example of a piecewise function because it operates differently based on whether \( x \) is negative, positive, or zero.
Here's how it works:

• For \( x \, \geq 0 \), the function is \( y = x - 1 \).
• For \( x < 0 \), the function becomes \( y = -x - 1 \).

The piecewise nature of the function gives rise to a graph that joins pieces together, each behaving differently. It is crucial to understand each piece to accurately graph and analyze the function.
In practical terms, think of piecewise functions as a set of rules, each applicable to specific conditions. This allows us to handle more complex real-world scenarios where a single equation may not suffice.

Algebraic Equations

An algebraic equation is a statement of equality containing variables and constants, related by algebraic operations such as addition, subtraction, multiplication, and division. In our exercise, we're dealing with the equation \( y = |x| - 1 \).
This equation tells us how to find \( y \) based on a given \( x \). The absolute value modifies the usual behavior you'd expect from a linear equation, introducing a point of reflection at \( x = 0 \). Recognizing this reinforces our understanding of how transformations can alter the graph's shape and position.
When solving or graphing algebraic equations:

• Identify the type of equation and any transformations (such as taking the absolute value).
• Calculate particular solutions to understand overall behavior.

By graphing, we can visualize these relationships, making algebraic understanding more intuitive and engaging. The key to mastering algebraic equations is practice, which strengthens the foundation for solving more complex problems later on.