Question

Graph each function. Set the viewing window for \(x\) and \(y\) initially from -5 to 5 then resize if needed. $$y=x^{3}-x$$

Short answer

After setting the initial viewing window from -5 to 5, create a table of x and y values using the function, plot the points on a graph, connect them with a smooth curve, and adjust the viewing window if necessary to properly display the curve and its key features.

Step-by-step solution

Set Up the Graph

Begin by setting up a coordinate system with both the x-axis and y-axis ranging from -5 to 5. This will provide an initial viewing window to plot the points of the function.

Create a Table of Values

Choose several values of x within the viewing window (-5 to 5) to create a table of values. Compute the corresponding y-values using the function y = x^3 - x. For example, if x = 0, y = 0^3 - 0 = 0; if x = 1, y = 1^3 - 1 = 0; and if x = -1, y = (-1)^3 - (-1) = -2.

Plot the Points

Plot the points you have calculated on the graph. Make sure to plot enough points to accurately see the shape of the function.

Draw the Graph

Once all the points are plotted, draw a smooth curve that connects the points. If the function doesn't fit well within the initial viewing window, adjust the window appropriately to fully capture the behavior of the function.

Identify Key Features

Look for key features of the graph, such as intercepts and turning points. The function y = x^3 - x crosses the x-axis at x = -1, 0, and 1 (which can be seen from the factored form y = x(x - 1)(x + 1)), and there are turning points between these intercepts. Make sure these features are correctly represented on the graph.

Key concepts

Coordinate System

A coordinate system is a framework for plotting points, lines, curves, and shapes on a plane. Imagine the coordinate system as a grid where you can locate any point using two numbers: an x-value and a y-value. The place where the x-axis (horizontal) and the y-axis (vertical) intersect is called the origin, marked as (0,0). When graphing the function y = x^3 - x, start by setting up this system with a range that allows you to see the behavior of the function, typically from -5 to 5. If you can't see all important features of the graph within this range, adjust it by zooming in or out until the graph's shape becomes clear.

Table of Values

Creating a table of values is a systematic technique to find points that you can plot on your graph. For each x-value you choose, calculate the corresponding y-value using the given function y = x^3 - x. This process will give you pairs of (x, y) points, like (0,0), (1,0), and (-1, -2) from the example. It's often helpful to select x-values that are evenly spaced and include 0. This way, you build a reliable framework to base your graph on. Ensure that you evaluate y at enough x-values to capture the variations in the function's shape as it changes direction or crosses the axes.

X-Axis Intercepts

X-axis intercepts are points where the graph of a function crosses the x-axis. These are the values of x for which y is zero. Identifying these intercepts can tell you a lot about the behavior of a function. For the polynomial function y = x^3 - x, setting y to 0 and solving for x reveals the intercepts to be at -1, 0, and 1. Intercept points are crucial in graph sketching because they're fixed points that your curve will pass through, providing a scaffold for your graph. When you find these points, mark them on your coordinate system to guide the drawing of the curve.

Turning Points

The turning points of a graph are spots where the function changes direction, marking local maxima or minima. For polynomials, these can typically be found between the x-axis intercepts. The function y = x^3 - x has a characteristic 'S' shaped cubic curve, which implies it has two turning points. These points are located where the rate of increase or decrease of the function changes - technically, where the first derivative of the function is zero. Detecting these points gives you information about the peaks and valleys of the graph and aids in sketching an accurate representation of how the function behaves between the intercepts.