Problem 20 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to graph the equation and approximate any \(x\) - and \(y\) -intercepts. Verify your results algebraically. $$y=10+2(x-2)$$

Step-by-Step Solution

Answer

The approximated intercepts from the graph should be around \(x = 2\) for the x-intercept and around \(y = 6\) for the y-intercept. These are confirmed algebraically as the x intercept at \(x = 2\) and the y intercept at \(y = 6\).

Step 1: Graphing the equation

Use a graphing utility to plot the equation \(y = 10 + 2(x-2)\), then visually locate the x-intercept (where the line crosses the x-axis) and the y-intercept (where the line crosses the y-axis).

Step 2: Approximate the intercepts

On the graph, the x-intercept appears where \(y = 0\) and the y-intercept where \(x = 0\). Approximate these points.

Step 3: Algebraically calculating the x-intercept

Set \(y = 0\) in the equation and solve for \(x\). Doing so gives: \(0 = 10 + 2(x-2)\), which simplifies to \(x = 2\).

Step 4: Algebraically calculating the y-intercept

Set \(x = 0\) in the equation and solve for \(y\). Doing so gives: \(y = 10 + 2((0)-2)\), which simplifies to \(y = 6\).

Key Concepts

X-Intercepts and Y-Intercepts

Understanding where a line crosses the axes on a graph is crucial for graph analysis. X-intercepts are the points where the graph of an equation crosses the x-axis, which implies that the y-coordinate at these points is zero. Conversely, y-intercepts are where the graph crosses the y-axis, indicating that the x-coordinate is zero at these intercepts.

To find the x-intercept algebraically, we set the y value to zero and solve the equation for x. For the given equation, setting y = 0 translates to solving 0 = 10 + 2(x - 2). This can be simplified to finding that x = 2, indicating the graph crosses the x-axis at this point. For the y-intercept, we set x = 0 and solve for y, which gives us the point where the line crosses the y-axis. In this case, manipulating the equation yields y = 6.

These intercepts are fundamental in sketching graphs, as they provide two points through which the line must pass, enabling us to draw it accurately.

Using Graphing Utilities

In the age of technology, graphing utilities offer an efficient and effective way to visualize equations. These utilities can provide immediate visual representations of equations such as the linear equation in our exercise. By inputting y = 10 + 2(x - 2) into the graphing utility, we can observe the behavior of the line and pinpoint intercepts with precision.

Benefits of Graphing Utilities

• Quick visualization of equations
• Ability to analyze graph features such as slope and intercepts
• Can handle complex equations that are difficult to graph by hand

While technological aids are invaluable, it is also important to verify results algebraically to build a deeper understanding of the equations and underlying mathematical principles.

Algebraic Verification of Intercepts

After approximating the intercepts using a graphing utility, it is essential to verify these points through algebra. This not only reinforces our understanding of the equation but also ensures the graphical representation's accuracy.

To algebraically verify the x-intercept from our graphing utility, we set y = 0 in the original equation. By solving for x, we confirm the graph crosses the x-axis at x = 2. Similarly, setting x = 0 and solving for y gives us the exact y-intercept, y = 6, which must match our graphing utility's approximation.

Algebraic verification is a critical step that serves as a proof for our graphical findings. By incorporating this method, we ensure the results are not only visually observed but also mathematically sound.