Problem 3 - Contemporary Precalculus

Question

In Exercises \(1-6,\) graph the equation by hand by plotting no more than six points and filling in the rest of the graph as best you can. Then use the calculator to graph the equation and compare the results. $$y=x^{2}-x$$

Step-by-Step Solution

Answer

Question: Graph the equation \(y = x^2 - x\) by hand and describe the step-by-step process used to create the graph. Answer: To graph the equation \(y = x^2 - x\) by hand, we first created a table of values for x from -2 to 3, calculated the corresponding y values using the given equation, and then plotted these points on the graph. After connecting the points, we observed the curve forms a parabola shape. Finally, we compared the hand-drawn graph to a calculator-generated graph to ensure both graphs have the same curve shape and pass through the same points.

Step 1: Set up the table of values

First, create a table of values for x to find the corresponding values of y using the given equation: \(y = x^2 - x\). We only need six points for this exercise, so we'll choose x values from -2 to 3.

Step 2: Calculate the y values

Using the equation \(y = x^2 - x\), calculate the corresponding y values for each x value picked in Step 1: - For x = -2: \(y = (-2)^2 - (-2) = 4+2 = 6\) - For x = -1: \(y = (-1)^2 - (-1) = 1+1 = 2\) - For x = 0: \(y = (0)^2 - (0) = 0-0 = 0\) - For x = 1: \(y = (1)^2 - (1) = 1-1 = 0\) - For x = 2: \(y = (2)^2 - (2) = 4-2 = 2\) - For x = 3: \(y = (3)^2 - (3) = 9-3 = 6\)

Step 3: Plot the points on the graph

Now that we have our table of values, plot these points on the graph: 1. (-2, 6) 2. (-1, 2) 3. (0, 0) 4. (1, 0) 5. (2, 2) 6. (3, 6)

Step 4: Connect the points and sketch the graph

After plotting these six points, you will notice that they form a parabola curve. Draw a smooth curve through these points, making sure it passes through all of them, to complete the graph.

Step 5: Use the calculator to graph the equation and compare the results

Now that we have our hand-drawn graph, you can use a graphing calculator or an online tool to graph the equation \(y = x^2 - x\). When comparing the hand-drawn graph to the calculator-generated graph, make sure they both have the same curve shape and pass through the same points. By following these steps, we have successfully graphed the equation \(y = x^2 - x\) by hand and compared it with a calculator-generated graph.

Key Concepts

Plotting Points

Understanding the basics of plotting points is essential for graphing quadratic equations like the one given, where the equation is in the form of \(y=x^{2}-x\). Plotting points effectively translates algebraic expressions into a visual representation on a coordinate plane.

To begin, establish a range of \(x\) values that will give us a clear view of the graph's behavior. For this quadratic equation, we can choose a small set of values, such as from -2 to 3. The next step is calculating the corresponding \(y\) values for each \(x\) by substituting the \(x\) values into the equation. You end up with pairs of coordinates that you can then plot on a graph.

Once plotted, these points serve as a foundational guide for drawing the curve. How accurate your plotted curve becomes will depend on the fineness of the points chosen and the parity in their distribution across the graph. In the classroom or on homework, it is often recommended to include the vertex of the parabola and points where the graph intersects the axes, which are key features of the graph's shape and position.

Parabola Sketching

Sketching a parabola is the natural continuation after plotting individual points for a quadratic equation. A parabola is a U-shaped curve that can either open upwards or downwards depending on the coefficient of the \(x^{2}\) term.

When you've got the points plotted, you can start sketching the parabola by gently drawing a curve that connects these points as smoothly as possible. The goal is to mirror the parabola's symmetry, which means the left and right sides of the graph should be mirror images of each other emanating from the vertex.

Important characteristics to capture in your sketch include the vertex, the axis of symmetry (a vertical line that runs through the vertex), and the direction the parabola opens. By ensuring that your parabola passes through all your plotted points, especially the vertex and intercepts, you'll create a graph that gives a true working model of the equation's graph.

Graphing Calculator Comparison

Once you have drawn your parabola by hand, a graphing calculator or online tool can offer a digital and often more precise representation of the equation. Today, numerous graphing calculators are available, as well as online graphing utilities, each with their own set of features and interfaces.

Using these tools, you can enter the equation \(y=x^{2}-x\) and have the software plot a perfectly curved parabola for you. Students can compare their hand-drawn graph with the graphing calculator's output. This practice helps identify errors in plotting points or misconceptions about the graph's shape.

Comparisons should focus on the key features: vertex location, axis of symmetry, and where the graph crosses the axes. Effective use of graphing calculators not only confirms the manually-drawn graph but also instills confidence in understanding the link between the algebraic formula and its graphical output.