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Chapter 4: Problem 53

Sketch the graph of each power function by hand, using a calculator only to evaluate \(y\)-values for your chosen \(x\) -values. On the same axes, graph \(y=x^{2}\) for comparison. In each case, \(x \geq 0\) .$$f(x)=x^{2.5}$$

Short Answer

Expert verified
Plot y-values for chosen x-values for both functions, then compare the curves. f(x)=x^{2.5} rises faster than y=x^2.

Step by step solution

01

Choose x-values

Select a range of x-values to use for plotting the graph. Since we are given that \(x \geq 0\), choose non-negative values such as 0, 1, 2, 3, and 4.
02

Calculate y-values for f(x)=x^{2.5}

Using a calculator, compute the corresponding y-values for the function \(f(x)=x^{2.5}\) with the chosen x-values. For example, calculate \(f(0)=0^{2.5}=0\), \(f(1)=1^{2.5}=1\), \(f(2)=2^{2.5}=5.657\), \(f(3)=3^{2.5}=15.588\), and \(f(4)=4^{2.5}=32\).
03

Calculate y-values for y=x^2

Similarly, for comparison, calculate the y-values for the function \(y=x^{2}\) using the same x-values. For instance, \(y(x=0)=0^{2}=0\), \(y(x=1)=1^{2}=1\), \(y(x=2)=2^{2}=4\), \(y(x=3)=3^{2}=9\), and \(y(x=4)=4^{2}=16\).
04

Plot the points for f(x)=x^{2.5}

On the graph, plot the points obtained for \(f(x)=x^{2.5}\): (0,0), (1,1), (2,5.657), (3,15.588), and (4,32). Connect these points smoothly as the curve increases exponentially.
05

Plot the points for y=x^2

On the same graph, plot the points for \(y=x^{2}\): (0,0), (1,1), (2,4), (3,9), and (4,16). Draw the quadratic curve connecting these points.
06

Compare the graphs

Compare the two curves on the graph. Notice that the curve \(f(x)=x^{2.5}\) grows faster than \(y=x^{2}\) as \(x\) increases, indicating a steeper increase as \(x\) becomes larger.

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Key Concepts

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

Graphing
Graphing is a visual way to represent mathematical functions on a coordinate plane. It involves plotting points and connecting them to show how one variable affects another. In this context, we are looking at power functions like \(f(x) = x^{2.5}\) and quadratic functions like \(y = x^2\).

To graph effectively, follow these steps:
  • Choose a set of \(x\)-values: Since our constraint is \(x \geq 0\), focus on non-negative values. Common choices include 0, 1, 2, 3, and higher integers depending on the function.
  • Calculate the corresponding \(y\)-values for both functions. This step is crucial as it gives you the points you will plot.
  • Plot these points on the graph: For each \(x\)-value, there is a corresponding \(y\)-value for each function. Place these points accurately on your coordinate grid.
  • Connect the dots smoothly: Once the points are plotted, connect them with a smooth curve. This will show the overall shape and direction of the function.
This approach helps in understanding the behavior of different functions, such as how quickly they increase or the shapes they form. Graphing by hand gives a tactile understanding of how mathematical functions behave.
Exponents
Exponents are a way to represent repeated multiplication. In the function \(f(x) = x^{2.5}\), the term \(x^{2.5}\) means that \(x\) is raised to the power of 2.5. This is a non-integer exponent, which can be thought of as multiplying \(x\) by itself 2.5 times in a certain mathematical sense.

Understanding exponents is essential:
  • Whole number exponents: \(x^2\) means \(x \times x\). It's straightforward and easy to visualize.
  • Fractional exponents: These indicate roots. For example, \(x^{1/2}\) is the square root of \(x\).
  • Decimal exponents: These can be tricky but follow the same principles. \(x^{2.5}\) implies \(x^2\) times \(x^{0.5}\) or \(\sqrt{x}\).
Working with exponents requires familiarity with these rules. Calculators often simplify finding \(y\)-values when exponents are not whole numbers. Decimal exponents can lead to steeper curves compared to whole number exponents when graphing.
Quadratic function
Quadratic functions, represented by \(y = x^2\), are a specific type of polynomial function. They involve squaring the variable, and the graph of a quadratic function is always a parabola. This shape is symmetric about a vertical line down its center, which is known as the axis of symmetry. Understanding its properties helps in sketching accurate graphs.
  • Basic shape: The simplest quadratic function, \(y = x^2\), produces a parabola opening upwards.
  • Vertex: The highest or lowest point, depending on direction. For \(y = x^2\), the vertex is at (0, 0).
  • Axis of symmetry: For our function, it is the line \(x = 0\) or the \(y\)-axis itself.
  • Roots: These are the \(x\)-values where the function crosses the \(x\)-axis. For \(y = x^2\), the only root is \(x = 0\).
Quadratic functions are fundamental in algebra due to their simple yet rich structure. They're widely used to model situations where relationships between quantities form parabolic shapes. Graphically comparing them with other power functions helps illustrate differences in growth rates and curvatures.

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Most popular questions from this chapter

Solve each problem. Suppose the average number of vehicles arriving at the main gate of an amusement park is equal to 10 per minute, while the average number of vehicles being admitted through the gate per minute is equal to \(x .\) Then the average waiting time in minutes for each vehicle at the gate is given by $$f(x)=\frac{x-5}{x^{2}-10 x}$$ where \(x>10 .\) (Source: Mannering, \(\mathrm{F}\). and W. Kilareski, Principles of Highway Engineering and Traffic Analysis, 2d. ed., John Wiley and Sons.) (a) Estimate the admittance rate \(x\) that results in an average wait of 15 seconds. (b) If one attendant can serve 5 vehicles per minute, how many attendants are needed to keep the average wait to 15 seconds or less?

Solve each problem. For \(k>0,\) if \(y\) varies inversely with \(x,\) when \(x\) increases, \(y\) _______ and when \(x\) decreases, \(y\) _______.

Concept Check Use transformations to explain how the graph of the given function can be obtained from the graphs of the square root function or the cube root function. $$y=\sqrt{16 x+16}$$

In Exercises \(109-116\), describe the graph of the equation as either a circle or a parabola with a horizontal axis of symmetry. Then, determine two functions, designated by \(y_{1}\) and \(y_{2},\) such that their union will give the graph of the given equation. Finally, graph \(y_{1}\) and \(y_{2}\) in the given viewing window. $$\begin{aligned} &x=y^{2}-8 y+16\\\ &[-10,10] \text { by }[-10,10] \end{aligned}$$

Solve each rational inequality by hand. Do not use a calculator. $$\frac{3 x-3}{4-2 x} \geq 0$$
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