Problem 37 - Contemporary Precalculus

Question

In Exercises \(37-42,\) obtain a complete graph of the equation by trying various viewing windows. List a viewing window that produces this complete graph. (Many correct answers are pos. sible; consider your answer to be correct if your window shows all the features in the window given in the answer section.) $$y=7 x^{3}+35 x+10$$

Step-by-Step Solution

Answer

Answer: The aim of this exercise is to find an appropriate viewing window to obtain a complete graph of the given equation \(y=7x^3+35x+10\). One possible suitable viewing window might be \(-4\le x \le 4\) and \(-40\le y \le 40\).

Step 1: Calculate the intercepts

To find the intercepts, we need to find the x-intercepts, which are the points where the graph crosses the x-axis, and the y-intercept, which is the point where the graph crosses the y-axis. For the x-intercepts, let \(y=0\), and solve for \(x\): $$0=7x^3+35x+10$$ This equation is a cubic equation and finding the roots can be difficult without a calculator. Since we are focusing on finding the viewing window, it's not necessary to find the exact values for the intercepts. Instead, we can use a graphing calculator or online graphing tool to find the x-intercepts approximately. For the y-intercept, let \(x=0\), and solve for \(y\): $$y=7(0)^3+35(0)+10$$ $$y=10$$ So, the y-intercept is at \((0, 10)\).

Step 2: Assess the steepness of the graph

Since the given equation is a cubic function, it will have an overall "S" shaped graph. Therefore, the slope will get steeper as we move away from the x-axis. Estimating a suitable range for the x-axis will require some trial and error with graphing tools, but the important thing is that the graph should show the steepness of the curve, so it doesn't appear "too flat" or "too steep" in the window.

Step 3: Select a suitable viewing window

The viewing window should show all the above-discussed features of the graph. Using a graphing tool or online software, you can begin by constructing a graph with a relatively large view, such as \(-10 \le x \le 10\) and \(-100 \le y \le 100\). Observe the graph features and determine if these ranges show the x-intercepts, the curve's steepness, and the y-intercept adequately. If necessary, adjust the viewing window proportions until all the main features can clearly be seen. There will be multiple correct answers for the appropriate viewing window. One possible suitable viewing window might be: $$-4\le x \le 4$$ $$-40\le y \le 40$$

Key Concepts

Understanding the Viewing Window

When graphing a function, the 'viewing window' is essentially the range of values you set for both the x-axis and y-axis on your graphing calculator or graphing tool. It's important to select a window that captures all the essential parts of the graph. For a cubic function, like the one given by \(y = 7x^3 + 35x + 10\), this might mean ensuring that the graph displays all intercepts and the characteristic cubic shape.
A good window range allows you to see the overall behavior of the graph without missing critical features like curves and intercepts. By experimenting with different window settings, such as starting with

• \(-10 \le x \le 10\)
• \(-100 \le y \le 100\)

you can adjust to find a view that clearly presents the whole graph. This process might require a few trials, but it's valuable for understanding how the graph changes with different view angles.

Finding the X-intercepts

X-intercepts are the points where the graph crosses the x-axis. These arise from setting \(y = 0\) and solving the resulting equation. With cubic functions like \(7x^3 + 35x + 10 = 0\), finding these exactly can be challenging.
Fortunately, graphing tools come to the rescue. They can approximate the roots and provide an idea of where the graph intersects the x-axis.

• Use graphing calculators to input the equation and explore where the graph meets the x-axis.
• Observe the graph to estimate values instead of solving manually.

By analyzing these intersections, you gain insights into the solution set of the equation, which is crucial for understanding the roots of the function.

Identifying the Y-intercept

The y-intercept is the point where the graph crosses the y-axis. It's obtained by setting \(x = 0\) and solving for \(y\).
For the given function \(y = 7x^3 + 35x + 10\), when \(x = 0\), the equation simplifies to \(y = 10\). Hence, the y-intercept is at \((0, 10)\).
This is an easy point to find and proves significant because it provides a fixed reference for where the graph starts on the vertical axis. Ensure that the chosen viewing window includes this point so that it displays clearly within your graph range.

Using Graphing Tools Effectively

Graphing tools are incredibly helpful when plotting complex functions such as cubics. They allow you to view the graph structure without solving the equation by hand.
Here’s how to make the most out of graphing tools:

• Input the full equation to get an initial graph.
• Adjust your viewing window to include main features: intercepts, turning points, and asymptotic behaviors.
• Use zoom options to better view precise areas for more detail if needed.

These tools offer more than just graph plotting. They help visualize mathematical functions and, by providing different window perspectives, support learning and understanding of complex equations more intuitively.