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Chapter 2: Problem 29

Graph the function in each of the given viewing rectangles, and select the one that produces the most appropriate graph of the function. \(f(x)=8 x-x^{2}\) (a) \([-5,5]\) by \([-5,5]\) (b) \([-10,10]\) by \([-10,10]\) (c) \([-2,10]\) by \([-5,20]\) (d) \([-10,10]\) by \([-100,100]\)

Short Answer

Expert verified
Rectangle (c), \([-2, 10]\) by \([-5, 20]\), is most appropriate.

Step by step solution

01

Understand the Function

Understand the function to be graphed: \( f(x) = 8x - x^2 \). It is a quadratic function, which is a downward-opening parabola because the coefficient of \( x^2 \) is negative.
02

Identify Key Points of the Function

Find the key features of the parabola, such as the vertex. The vertex form of a parabola is \( f(x) = -(x^2 - 8x) \). Completing the square, we find the vertex is at \( x = 4 \), so the vertex is \( (4, 16) \). The parabola opens downwards.
03

Determine the Range of the Function

For \( f(x) = 8x - x^2 \), the maximum value is at the vertex. The range is from \(-\infty\) to \(16\). Knowing this helps in deciding the appropriate viewing rectangle.
04

Consider Each Viewing Rectangle

Evaluate each rectangle:- (a) Range \([-5,5]\) for x and y doesn't include many points above the vertex (4, 16).- (b) Range \([-10,10]\) captures the parabola's vertex and a symmetrical portion around it.- (c) This range \([-2,10]\) by \([-5,20]\) will include the vertex and fit the entire parabola peak.- (d) Although this includes the peak, \([-100, 100]\) y-range makes the important features too small.
05

Select the Best Viewing Rectangle

Rectangle (c), \([-2, 10]\) by \([-5, 20]\), is the most appropriate as it includes the vertex and covers the majority of the function's positive y-values effectively.

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Key Concepts

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

Parabola
A parabola is a curve where any point is at an equal distance from a fixed point, known as the focus, and a fixed straight line, called the directrix. In the case of quadratic functions like \( f(x) = 8x - x^2 \), the graph takes the shape of a parabola.
This specific function is a downward-opening parabola because the coefficient of \( x^2 \) is negative. Understanding the shape of a parabola helps us determine the behavior and direction of the quadratic function. This shape means that the parabola goes upwards to a maximum point, called the vertex, and then opens downwards.
  • Downward-opening due to the negative coefficient of \( x^2 \).
  • Symmetrical shape around its vertex.
Knowing these properties allows us to predict and graph quadratic functions smoothly.
Vertex Form
The vertex form of a quadratic function is incredibly useful for graphing because it makes it simple to identify the vertex, or the peak, of the parabola. The vertex form is typically written as \( f(x) = a(x-h)^2 + k \), where \((h, k)\) represents the vertex.
In our function \( f(x) = 8x - x^2 \), by completing the square, we can rewrite it in vertex form. This involves rearranging and grouping terms so it's easier to see the quadratic's core features. In this process, the vertex form becomes \( -(x^2 - 8x) \), showing the vertex at \((4, 16)\).
  • Vertex is the peak or the bottom of the parabola depending on opening direction.
  • Helps in identifying the maximum or minimum points of the function.
Through vertex form, understanding the maximum values reached by the function becomes straightforward, aiding efficient graphing.
Graphing Quadratic Functions
Graphing quadratic functions can seem daunting, but understanding its core components simplifies the process. Start by identifying the function's general shape and key features like the vertex. For \( f(x) = 8x - x^2 \), the shape is a downward-opening parabola.
When graphing, it's crucial to pick points to plot that reflect the parabola's path, such as intercepts and the vertex. The intercepts are points where the graph crosses the axes:
  • x-intercept: Solved by setting \( f(x) = 0 \) to find \( x \) values.
  • y-intercept: Found by evaluating \( f(0) \).
Upon identifying these points, plot them, and sketch the curved path of the parabola centered around the vertex. The graph offers a visual representation of how the function behaves over different ranges.
Viewing Rectangle Selection
Choosing an appropriate viewing rectangle is essential for effective visualization of a quadratic function's graph. A viewing rectangle refers to the window on the coordinate plane where the function is displayed. The aim is to select dimensions that highlight the key features, especially the vertex.
For the function \( f(x) = 8x - x^2 \), several options are provided:
  • Option (a) \([-5,5]\) by \([-5,5]\) misses most parts above the vertex.
  • Option (b) \([-10,10]\) captures a symmetrical view but not fully detailed.
  • Option (c) \([-2,10]\) by \([-5,20]\) best includes the vertex and function's peak efficiently.
  • Option (d) \([-10,10]\) by \([-100,100]\) shows everything but compresses details too much.
Ultimately, option (c) is the best selection as it frames the vertex well and includes the essential parts of the parabola efficiently, ensuring that the top of the parabolic peak and the curve's direction are clearly visible.

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Most popular questions from this chapter

DISCUSS: Finding an Inverse "in Your Head" In the margin notes in this section we pointed out that the inverse of a function can be found by simply reversing the operations that make up the function. For instance, in Example 7 we saw that the inverse of $$f(x)=3 x-2 \quad \text { is } \quad f^{-1}(x)=\frac{x+2}{3}$$ because the "reverse" of "Multiply by 3 and subtract 2" is "Add 2 and divide by 3 ". Use the same procedure to find the inverse of the following functions. (a) \(f(x)=\frac{2 x+1}{5}\) (b) \(f(x)=3-\frac{1}{x}\) (c) \(f(x)=\sqrt{x^{3}+2}\) (d) \(f(x)=(2 x-5)^{3}\) Now consider another function: $$f(x)=x^{3}+2 x+6$$ Is it possible to use the same sort of simple reversal of operations to find the inverse of this function? If so, do it. If not, explain what is different about this function that makes this task difficult.

DISCOVER: Graph of the Absolute Value of a Function (a) Draw graphs of the functions $$f(x)=x^{2}+x-6$$ $$\text { and } \quad g(x)=\left|x^{2}+x-6\right|$$ How are the graphs of \(f\) and \(g\) related? (b) Draw graphs of the functions \(f(x)=x^{4}-6 x^{2}\) and \(g(x)=\left|x^{4}-6 x^{2}\right| .\) How are the graphs of \(f\) and \(g\) related? (c) In general, if \(g(x)=|f(x)|,\) how are the graphs of \(f\) and \(g\) related? Draw graphs to illustrate your answer.

When a bowl of hot soup is left in a room, the soup eventually cools down to room temperature. The temperature \(T\) of the soup is a function of time \(t .\) The table below gives the temperature (in "F) of a bowl of soup \(t\) minutes after it was set on the table. Find the average rate of change of the temperature of the soup over the first 20 minutes and over the next 20 minutes. During which interval did the soup cool off more quickly? $$\begin{array}{|c|c||c|c|} \hline t \text { (min) } & T\left(^{\circ} \mathrm{F}\right) & t \text { (min) } & T\left(^{\circ} \mathrm{F}\right) \\ \hline 0 & 200 & 35 & 94 \\ 5 & 172 & 40 & 89 \\ 10 & 150 & 50 & 81 \\ 15 & 133 & 60 & 77 \\ 20 & 119 & 90 & 72 \\ 25 & 108 & 120 & 70 \\ 30 & 100 & 150 & 70 \\ \hline \end{array}$$

Sketch a graph of the piecewise defined function. $$f(x)=\left\\{\begin{array}{ll} -1 & \text { if } x < -1 \\ 1 & \text { if }-1 \leq x \leq 1 \\ -1 & \text { if } x > 1 \end{array}\right.$$

As a weather balloon is inflated, the thickness \(T\) of its rubber skin is related to the radius of the balloon by $$T(r)=\frac{0.5}{r^{2}}$$ where \(T\) and \(r\) are measured in centimeters. Graph the function \(T\) for values of \(r\) between 10 and \(100 .\)
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