Problem 78 - Algebra and Trigonometry Real Mathematics, Real People

Question

Graphical Analysis (a) use a graphing utility to graph the equation, (b) use the graph to approximate any \(x\) -intercepts of the graph, (c) set \(y=0\) and solve the resulting equation, and (d) compare the result of part (c) with the \(x\) -intercepts of the graph. $$y=|x-2|-3$$

Step-by-Step Solution

Answer

The solutions are \(x = 5\) and \(x = -1\). These are the x-intercepts where the graph of the equation \(y = |x - 2| - 3\) crosses the x-axis.

Step 1: Graph the Equation

Use a suitable graphing utility to sketch the graph of the equation \(y = |x-2| - 3\).

Step 2: Estimate x-intercepts

Observe the graph to approximate where the curve crosses the x-axis, these points are the x-intercepts.

Step 3: Set y equal to 0 and Solve

Set \(y = 0\) in the equation which gives \(0 = |x-2| - 3\). Solving this, we first isolate the absolute value which gives \(|x-2| = 3\). Now, the solutions will be \(x = 2 + 3\) and \(x = 2 - 3\), which gives us \(x = 5\) and \(x = -1\).

Step 4: Compare Calculated x-intercepts with the Graph

Compare the calculated x-intercepts (\(x = 5, -1\)) with those observed from the graphed equation. They should align closely with the approximations made in Step 2.

Key Concepts

Graphing Absolute Value Functions

Understanding how to graph absolute value functions is crucial for several branches of mathematics, including algebra. The graph of an absolute value function forms a distinctive 'V' shape. The cornerstone of sketching the graph of an absolute value equation like y = |x - 2| - 3 is identifying the vertex (the point where the graph takes a sharp turn), and the axis of symmetry (a vertical line that divides the graph into two mirror images).

For the given function, the vertex is obtained by setting the expression inside the absolute value to zero; in this instance, x - 2 = 0 implies the vertex is at (2, -3). To graph the function, plot the vertex and then create the 'V' shape by plotting points to the left and right of it, always considering the absolute value's affect on the sign of the 'y' values.
Remember, the graph is a visual representation of all the points that satisfy the equation, so plotting additional points can help to ensure accuracy before connecting them to form the graph.

Approximating X-intercepts

When you've drawn the 'V' shaped graph of an absolute value function, the next step is approximating the x-intercepts. These are the points where the graph crosses the x-axis, and they are crucial for understanding the function's behavior. For the equation y = |x - 2| - 3, visually inspect the graph to approximate these intercepts.

To get a better estimate, look for where the 'legs' of the 'V' touch the x-axis. If you're using a digital graphing tool, you can zoom in to get closer, or if you’re drawing by hand, ensure the graph is to scale as much as possible.

• Note any symmetry in the x-intercepts, as the absolute value function is usually symmetric about the vertex.
• Verify your approximations with a table of values if necessary, especially for more complex functions.

Approximating x-intercepts is a key skill for visualizing solutions to equations and understanding the function's roots.

Solving Absolute Value Equations

Solving absolute value equations algebraically involves isolating the absolute value expression and then considering the two possible cases for the contents inside the absolute brackets. With an equation such as y = |x - 2| - 3, you set y to zero and solve for x.

After isolating the absolute value expression, you would create two separate equations to account for the positive and negative scenarios; in our example, |x - 2| = 3 leads to x = 5 and x = -1. It's essential to solve both since the absolute value represents a distance from zero, which could be in either direction on the number line.
Once you have the possible x values, it’s good practice to plug them back into the original equation to check if they truly satisfy it. This process verifies that the solutions make sense in the context of the problem and avoids extraneous or incorrect answers.

Comparing Graphical and Algebraic Solutions

The beauty of mathematics lies in the interconnectedness of its parts; the solutions to absolute value equations can be analyzed both graphically and algebraically – and these methods should corroborate each other. After graphing the absolute value function and solving the equation by algebraic manipulations, it’s important to compare the x-intercepts from the graph with the solutions found algebraically.

For y = |x - 2| - 3, if graphically you approximated intercepts around x = 5 and x = -1, and algebraically you calculated the intercepts to be exactly those values, the comparison confirms the correctness of both methods.

• Such a comparison can point out any errors made in either process.
• If there’s a discrepancy, check calculations and graph plotting for accuracy.
• Learning to trust this relationship builds a deeper understanding of functions and their behaviors.

Through regular practice of solving and graphing absolute value equations, you'll develop a keen eye for these comparisons, making such problems much easier to tackle.