Problem 17 - 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. $$y=3(x-2)-5$$

Step-by-Step Solution

Answer

From the graph, the x-intercept is approximately 2 and the y-intercept is approximately -5. Algebraically, setting y=0, we solve 0=3(x-2)-5, resulting in x=2 for the x-intercept. Setting x=0, we solve y=3(0-2)-5, yielding y=-5 for the y-intercept, hence verified.

Step 1: Graph the Equation

Using a graphing utility, plot the equation \(y=3(x-2)-5\). Once it is plotted, identify the points where the graph crosses the x-axis and y-axis.

Step 2: Estimate Intercepts

The x-intercepts are points where the graph intersects the x-axis which means at these points y=0. Similarly, y-intercepts are points where the graph intersects the y-axis (x=0). Estimate these points by observing the plotted graph.

Step 3: Algebraic Verification

Algebraically, for the x-intercept, set y=0 in the given equation and solve for x. For the y-intercept, set x=0. Solve 0=3(x-2)-5 to find x-intercept and y=3(0-2)-5 to find y-intercept.

Key Concepts

Understanding X-intercepts and Y-intercepts

The concepts of x-intercepts and y-intercepts are fundamental in graphing linear equations. In essence, an x-intercept is a point on the graph where the line crosses the x-axis, which reveals what x-values will make the equation equal to zero. Similarly, a y-intercept is where the line crosses the y-axis, showing the y-value when x is zero.

To find the x-intercept algebraically, you would set the y-value to zero and solve the equation for x. In our case with the equation \(y=3(x-2)-5\), setting y to 0 results in \(0=3(x-2)-5\). Solving for x will give us the exact x-intercept(s). Likewise, to find the y-intercept, we let x equal zero and solve for y, which here would be \(y=3(0-2)-5\).

It's crucial for students to understand that there can be multiple x and y-intercepts in more complex functions, but for linear equations, like the one in our exercise, there will be only one x-intercept and one y-intercept.

Leveraging a Graphing Utility

Graphing utilities are powerful tools that can help students visualize equations and their corresponding graphs. By inputting the equation \(y=3(x-2)-5\) into a graphing utility, it can swiftly plot the line, making it easier to identify the intercepts. Students, however, should not rely entirely on these tools without understanding the underlying concepts.

After plotting the graph, scrutinize it to estimate where the line crosses the x-axis and y-axis. These points are your intercepts. It's worth mentioning for students that while graphing utilities can give a quick picture, the actual points of intercept should be confirmed through algebraic calculation for precise answers.

Always cross-check the graphed intercepts with the algebraic solution. This exercise of transitioning from a graphical representation to algebraic verification strengthens a student's grasp of how equations represent lines on a graph.

The Process of Algebraic Verification

Algebraic verification compliments graphing by providing mathematical proof to what's observed on a graph. After estimating the x and y-intercepts using the graph, it's important to verify these findings with algebraic methods. By substituting the appropriate values in the equation \(y=3(x-2)-5\), we can solve for the precise coordinates of the intercepts.

Students should take this step seriously as it not only confirms their understanding of the graph but also ensures their skills in algebra are being applied correctly. For our equation, setting y to zero gives us the x-intercept by solving for x, and setting x to zero and solving for y gives us the y-intercept. This back-and-forth between the visual and numerical is a powerful approach to reinforce learning and understanding of linear equations.