Section 6.1: Powers and Proportions

In this section, we quickly review the numerical and graphical properties of the power functions, such as f(x) = x⁵ and g(x) = x^-2.  We also discuss the terminology and applications of proportions.

Power Functions

We can quickly see how the graphs of x⁰ = 1, x¹ = x, x², x³, etc. are related by looking at their plots in XFunctions:

Selecting the "Power Animation" example, you can see the plots of x^k for k = 0, 1, ..., 10.  Note: To see each plot clearly, it may be helpful to hit the "Stop" button and then step through each plot by clicking the "Next" button.  

Numerically, they all go through the points (0, 0) and (1, 1).  We can also see that two geometric trends:

This animation only focused on the plot for positive values of x.  

If we look at the entire plot, we know that the remainder of the graph is determined by whether the exponent is even or odd.  If the exponent is even, it will go through the point (-1, 1); if it is odd, it will go through the point (-1, -1):

The same geometric trends are still evident:

Note: For power functions with integer exponents, the exponent is also referred to as the degree of the power function.  If we now consider negative exponents, we will discover similar trends.  Looking at the plots of x^-k = 1/x^k for k = 0, 1, ..., 10, we would expect that where the graph of x^k is big, the graph of 1/x^k should be small and vice versa.  That is:

This is visible in the following animation:

As before, numerically, they all go through the points (1, 1) and (-1, ±1), depending on whether the exponent is even or odd.  However, just like the trigonometric functions, when they are undefined (i.e., at x = 0, due to division by 0), these graphs have a vertical asymptote.  Moreover, similar to exponential functions, they have a horizontal asymptote at y = 0.

To summarize our observations on power functions with integral exponents:

With this understanding of the important points on the graphs and the overall shape of the graphs, we can use our transformational graphing technique to graph any function built from a power function, such as f(x) = -4(x - 1)^-3 + 2 = -4/(x - 1)³ + 2.  This decomposes as:

• subtract 1 from the input,
• take this to the -3 power,
• multiply the result by 4,
• negate this, and
• add 2.

This leads to the following analysis:

Drawing in the asymptotes, plotting the points, and "connecting-the-dots" gives:

You should notice how quickly the graph approaches the horizontal asymptote, due to the higher exponent; that is, the graph of g(x) = -4/(x - 1) + 2 would look about the same, but approach the horizontal asymptote more gradually:

As usual, you can see how this graph is "built" out of the graph of x^-3 by selecting the Examples in the following applet:

Make sure that you have learned these general features of power graphs by completing the following exercises.

Proportions

There are many situations in which the functional relation between two variables behaves very nicely with respect to multiplication and proportions.  For example, we can see that in the following function:

• when the input doubles, so does the output:
  • when x doubles from 2 to 4, the output doubles from -3 to -6;
  • when x doubles from 6 to 12, the output doubles from -9 to -18;
• when the input triples, so does the output:
  • when x triples from 2 to 6 the output triples from -3 to -9;
  • when x triples from 4 to 12 the output triples from -6 to -18.

This means that the proportion of the inputs is the same as the proportion of the corresponding outputs (e.g., 2/12 = 1/6 = -3/(-18), etc.).  In such a case, we say that x and y are directly proportional.  The term "direct" connotes the fact that as the input gets larger so does the output.  We can see that f is given as a linear function passing through the origin, f(x) = (-3/2)x.

The following function displays a related, but opposite, pattern:

• when the input doubles, the output halves:
  • when x doubles from 2 to 4, the output drops from -3 to -3/2;
  • when x doubles from 6 to 12, the output drops from -1 to -1/2;
• when the input triples, the output drops to a third:
  • when x triples from 2 to 6 the output drops from -3 to -1;
  • when x triples from 4 to 12 the output drops from -3/2 to -1/2.

This means that the proportion of the inputs is the reciprocal of the proportion of the corresponding outputs (e.g., 2/6 = 1/3 = 1/(-3/(-1)), etc.).  In such a case, we say that x and y are inversely proportional.  The term "inverse" connotes the fact that as the input gets larger the output gets smaller, and vice versa.  In this case, g can be given as g(x) = -6/x.

The terminology of direct and inverse proportions is often used in the sciences to describe functional relationships between two or more variables.  As the two previous examples suggest, the two phrases "is directly proportional to" and "is inversely proportional to" translate directly into mathematical equations:

Many physical relationships can be described using the language of proportions and power functions:

• Hooke's Law says that the amount a spring is stretched, s, is "directly proportional" to the force, F, applied to stretch it.  That is, s = kF, for some constant, k.
• Galileo showed that the distance, d, that an object has fallen is "directly proportional" to the square of the time, t, is has been falling.  That is, d = kt², for some constant, k. 
• Bolye's Law says that the pressure, P, of a fixed amount of gas kept at a constant temperature is "inversely proportional" to the volume, V, in which it is contained.  That is, P = k/V, for some constant, k. 
• It is known that the perceived intensity of a sound, I, is "inversely proportional" to the square of one's distance, r, from it.  That is, I = k/r², for some constant, k. 

Notice that we can even describe an inverse proportion as a direct proportion with a reciprocal (i.e., negative power).  For example, we can rewrite Boyle's Law as P = kV^-1.  Note: In this way, we could eliminate the use of the term "inverse proportion" entirely; however, since it is used so commonly in the sciences, we will continue to use it.

Often such descriptions are combined to give one variable as a function of several others.  For example, the Ideal Gas Law says that the pressure, P (in atmospheres) of a gas is inversely proportional to its volume, V (in liters), but directly proportional to both its amount, n (in moles), and temperature, T (in ºKelvin).  This means that P is a function of three variables, P = f(V, n, T) = knT/V,  where k is the universal gas constant (normally called, R, which approximately equals 0.0821 L atm/(mol ºK)).  Similarly, Newton's Law of Universal Gravitation says that the force of gravity, F (in Newtons) between two objects is inversely proportional to the distance, r (between their centers, in meters), but directly proportional to the mass of each object, m₁ and m₂ (in kilograms).  This translates into the functional equation F = g(r, m₁, m₂) = km₁m₂/r²,  where k is the universal gravitational constant (usually called, G, which approximately equals 6.67 x 10^-11 N m²/kg²).  

Because so many such laws abound in Physics, Chemistry, and Biology, power functions show up frequently in experimental data.  For example, consider very carefully measuring the weight, w, of 1 kg. block of metal as you rise steadily in a hot air balloon.  Since the weight is just the force of gravity, and since the mass of the block and the mass of the earth are constant in this experiment, from Newton's Law of Gravitation, you could expect the data to fit a power function, w = s(h) = kh^-2, h is your height (measured in meters from the center of the earth, i.e., you would need to add the radius of the earth to your height above the ground).  Note: You could even use our data-fitting technique with log-log graphs to verify that the exponent is -2.

Practice using the language of proportions by completing the following Exercises.  

Go to Polynomials and Roots

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Send questions or comments to jwicks@northpark.edu

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