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Chapter 1: Problem 16

In Exercises \(5-30,\) find an appropriate graphing software viewing window for the given function and use it to display its graph. The window should give a picture of the overall behavior of the function. There is more than one choice, but incorrect choices can miss important aspects of the function. $$ y=\left|x^{2}-x\right| $$

Short Answer

Expert verified
Use a viewing window with x: [-2, 3] and y: [0, 5] for best visualization.

Step by step solution

01

Understand the Function

The function given is \( y = |x^2 - x| \), which is composed of a quadratic term \( x^2 - x \) inside the absolute value. The absolute value function affects the graph by reflecting any part of the graph that would be below the x-axis to above the axis.
02

Analyze the Exponents and Coefficients

The quadratic expression \( x^2 - x \) has exponents and coefficients typical for a parabola. The highest degree term \( x^2 \) suggests that the parabola opens upwards.
03

Determine x-intercepts of the Quadratic

Set \( x^2 - x = 0 \) to find where the graph crosses the x-axis. Factoring, we have \( x(x-1) = 0 \), giving x-intercepts at \( x = 0 \) and \( x = 1 \). These are critical points for changes in the graph's direction within the \( | \cdot | \) operation.
04

Consider Important Points for Plotting

The graph changes at the intercepts (0,0) and (1,0) due to the absolute value. Test a point in each region (e.g., \( x = -1, 0.5, 2 \)) to see how the graph behaves between and outside these x-intercepts.
05

Decide on a Graphing Window

To capture the behavior between and beyond the intercepts, choose an x-range slightly wider than the intercepts. Suggested view for x: [-2, 3]. For y, since the function value is always non-negative, start from y: [0,5] to ensure capturing peak behavior as x increases.
06

Plot the Graph and Adjust as Needed

Using graphing software, plot the function within the chosen window. Check if all important aspects (like symmetry and reflections) are visible. Adjust the window as needed to better capture the shape and behavior.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Absolute Value Function
When considering the function \( y = |x^2 - x| \), the absolute value plays a significant role. The absolute value function transforms its argument such that any negative output is flipped to become positive. Here's how it affects the graph:
  • For values of \( x \) between the roots of \( x^2 - x \) (i.e., between \( x = 0 \) and \( x = 1 \)), the quadratic expression itself may attempt to dip below the x-axis. The absolute value will reflect this portion of the graph upwards, keeping it all above the x-axis.
  • The graph, thus, retains only non-negative y-values for all x, giving it an appearance similar to a parabola, but with a V-like structure wherever the original expression was negative.
Try visualizing it as a bouncy parabola where any parts that would dip below the ground (the x-axis) instead bounce back up into the air.
Quadratic Functions
The quadratic part of the function, \( x^2 - x \), resembles typical parabolic equations. Here's what you need to remember about quadratic functions:
  • The standard form of a quadratic function is \( ax^2 + bx + c \).
  • In our function, \( a = 1 \) (from \( x^2 \)), \( b = -1 \), and \( c = 0 \).
  • This hints at a parabola that naturally opens upwards with its vertex offering the lowest point on the curve in its unaltered state.
Understanding these components makes it easier to predict how the graph should behave. Quadratic functions are symmetrical, meaning the left and right sides of their vertex mirror each other鈥攁n important feature to consider while graphing.
X-Intercepts
To find the x-intercepts, which are points where the graph crosses the x-axis, you solve the equation \( x^2 - x = 0 \). These intercepts are crucial as they help in shaping the graph. Here's how you find them:
  • Factor the quadratic equation: \( x(x-1) = 0 \).
  • Solve for \( x \) to get the intercepts: \( x = 0 \) and \( x = 1 \).
These intercepts indicate where the "bouncing" caused by the absolute value begins and ends around certain values of x. Thus, they play a pivotal role in defining the segments of the graph.
Graphing Software
Graphing software is a powerful tool for visualizing functions like \( y = |x^2 - x| \). When setting up a view for graphing, here's what you should consider:
  • Select an appropriate window that captures important features of the function. For our function, the suggested x-range is [-2, 3], which covers a bit beyond the x-intercepts for a complete picture.
  • Since the output is always non-negative, start with a y-range from [0,5] to ensure peaks and valleys are visible.
  • Adjust the scale to maintain balance between clarity and comprehensiveness. It's crucial that the symmetry, peaks, and reflective behavior of the graph are clearly outlined.
Being flexible with the viewing window is key. It might take a few trials to capture all aspects of the graph effectively.

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Most popular questions from this chapter

Exercises \(59-68\) tell by what factor and direction the graphs of the given functions are to be stretched or compressed. Give an equation for the stretched or compressed graph. $$ y=\sqrt{4-x^{2}}, \quad \text { stretched horizontally by a factor of } 2 $$

Find the function values in Exercises \(47-50\) $$\sin ^{2} \frac{\pi}{12}$$

Use graphing software to graph the functions specified in Exercises \(31-36\) . Select a viewing window that reveals the key features of the function. Graph the upper branch of the hyperbola \(y^{2}-16 x^{2}=1\)

Graph the functions in Exercises \(37-56\) $$ y=\frac{1}{x^{2}}+1 $$

Use the addition formulas to derive the identities in Exercises \(31-36\) $$\sin (A-B)=\sin A \cos B-\cos A \sin B$$
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