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Chapter 1: Problem 4

Use a graphing utility to generate the graph of the function \(f(x)=-12-x^{2}\) in the given viewing windows, and specify the window that you think gives the best view of the graph. (a) \(-1 \leq x \leq 1,-15 \leq y \leq-13\) (b) \(-2 \leq x \leq 2,-15 \leq y \leq-11\) (c) \(-4 \leq x \leq 4,-28 \leq y \leq-10\) (d) A window of your choice

Short Answer

Expert verified
Window (c) provides the best view of the graph. Window choice (d) also offers a comprehensive view.

Step by step solution

01

Understand the Function

We are working with the function \( f(x) = -12 - x^2 \). This is a quadratic function with a downward-opening parabola, as indicated by the negative sign before \( x^2 \). The vertex of this parabola is at the point where \( x = 0 \), which gives \( f(0) = -12 \).
02

Analyze Window (a)

For the window \(-1 \leq x \leq 1\) and \(-15 \leq y \leq -13\), the x-range is very narrow, only spanning from \(-1\) to \( 1 \). The y-range captures values around the vertex \( y = -12 \). The view will show the vertex but not much of the parabola's arms because the x-range is too small.
03

Analyze Window (b)

For the window \(-2 \leq x \leq 2\) and \(-15 \leq y \leq -11\), we have doubled the x-range compared to (a), allowing us to see more of the parabola's shape, specifically its arms. The y-range captures the vertex and more of the parabola's left and right sections, providing a clearer view than (a).
04

Analyze Window (c)

For the window \(-4 \leq x \leq 4\) and \(-28 \leq y \leq -10\), the x-range shows a complete view of the parabola's shape. The y-range is wide enough to capture more of the downward curve, allowing all important features and the overall structure of the parabola to be visible.
05

Choosing a Better Window (d)

Select a window that best shows the parabola's shape clearly: \(-5 \leq x \leq 5\) and \(-30 \leq y \leq 0\). This window encompasses the quadratic function's downward opening clearly, presenting an effective view of both the vertex and the arms.
06

Conclusion: Best Window Choice

Out of the given windows, window (c) \(-4 \leq x \leq 4, -28 \leq y \leq -10\) is the best choice as it provides the clearest view of the parabola and its features. The window from step 5 (d) is even better for more visualization, particularly showing more of the arms.

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Key Concepts

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

Viewing Window
When graphing a quadratic function, the choice of a viewing window is crucial to get an accurate and complete picture of the graph. The viewing window defines the range over which we can see the function on our screen, specified by boundaries for both the x-axis and y-axis.
  • The x-range: This determines how much of the parabola's width we can see. A wider range often means we can observe both "arms" of the parabola.
  • The y-range: This allows us to see the depth or height of the parabola, especially the vertex if it's not set too narrow or too limited to only a part of the graph.
Picking the right window size can enhance our understanding of the function's behavior. For instance, in our exercise with the function \(f(x) = -12 - x^2\), using a window like \(-4 \leq x \leq 4\) and \(-28 \leq y \leq -10\) can effectively show all significant features of the graph. It's important to adjust these ranges based on what part of the graph you are most interested in visualizing.
Vertex of a Parabola
The vertex of a parabola is a key point that represents the peak or the lowest point, depending on the parabola's orientation. For a function given by \(f(x) = ax^2 + bx + c\), the vertex can be calculated, but sometimes it is immediately visible from the function if it is in vertex form.
For the function \(f(x) = -12 - x^2\), the vertex occurs where the derivative is zero or, more simply, where \(x = 0\) because there is no linear \(x\) term. This makes finding the vertex straightforward:
  • **x-coordinate:** Here it is 0, making calculations simple.
  • **y-coordinate:** Plugging \(x = 0\) into the function gives \(f(0) = -12\).
Thus, the vertex is at (0, -12), serving as a useful reference point when setting up a viewing window for graph analysis.
Graphing Utility
To aid in visualizing functions like quadratics, a graphing utility serves as a powerful tool. These utilities can be online graph platforms, calculator features, or software programs that allow you to input a function and see its graphal representation.
  • **Simplifies Visualization:** Instead of calculating numerous points manually, graphing utilities plot the entire graph effortlessly.
  • **Adjustable Viewing Windows:** Most graphing tools allow you to set custom x and y ranges to better fit the function on your screen.
  • **Dynamic Interaction:** You can explore the graph by zooming in or out, moving around, or adjusting window sizes quickly to see various graph sections in better detail.
When using a graphing utility for a function like \(f(x) = -12 - x^2\), it instantly handles the parabola's properties, like its vertex and direction, showing these features on the graph based on your input settings.
Function Analysis
Function analysis involves breaking down the properties and characteristics of a function to understand its behavior fully. For a quadratic function like\(f(x) = -12 - x^2\), some analysis points are key:
  • **Direction of Opening:** This function's parabola opens downward due to the negative coefficient of \(x^2\).
  • **Vertex:** Occurring at (0, -12), it's the highest point of this downward parabola.
  • **Symmetry:** The graph is symmetric about the line \(x = 0\), which also passes through the vertex.
  • **Intercepts:** A y-intercept is at \(-12\), and this is the same as the vertex value for x = 0.
Conducting function analysis helps you predict and visualize graph behavior without even using a graphing utility. Coupled with a good choice of viewing window, it ensures a thorough understanding of the function's graphical properties and how best to view them.

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Most popular questions from this chapter

Thermometers are calibrated using the so-called "triple point" of water, which is \(273.16 \mathrm{K}\) on the Kelvin scale and \(0.01^{\circ} \mathrm{C}\) on the Celsius scale. A one-degree difference on the Celsius scale is the same as a one-degree difference on the Kelvin scale, so there is a linear relationship between the temperature \(T_{C}\) in degrees Celsius and the temperature \(T_{K}\) in kelvins. (a) Find an equation that relates \(T_{C}\) and \(T_{K}\) (b) Absolute zero ( \(0 \mathrm{K}\) on the Kelvin scale) is the temperature below which a body's temperature cannot be lowered. Express absolute zero in \(^{\circ} \mathrm{C}\).

A student must decide between buying one of two used cars: car \(A\) for 4000 dollars or car \(B\) for 5500 dollars. Car \(A\) gets 20 miles per gallon of gas, and car \(B\) gets 30 miles per gallon. The student estimates that gas will run 1.25 dollars per gallon. Both cars are in excellent condition, so the student feels that repair costs should be negligible for the foreseeable future. How many miles would the student have to drive before car \(B\) becomes the better buy?

Find the amplitude, period, and phase shift. and sketch at least two periods of the graph by hand. Check your work with a graphing utility. (a) \(y=-1-4 \sin 2 x\) (b) \(y=\frac{1}{2} \cos (3 x-\pi)\) (c) \(y=-4 \sin \left(\frac{x}{3}+2 \pi\right)\)

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