2.2 Power Functions With Modeling

What is a rational number? Power Functions with Modeling What is a rational number? Any number which can be expressed as a fraction is a rational number. Any number which can be expressed as p/q where q ≠ 0.

Power Functions with Modeling Any function that can be written in the form f(x) = kxn, where k and n are nonzero constants, is a power function. The constant n is the power, and k is the constant of variation, or constant of proportion. Note: We say f(x) varies as the nth power of x, or f(x) is proportional to the nth power of x.

Common Formulas which are power functions Power Functions with Modeling Common Formulas which are power functions Name Formula Power Constant of Variation Circumference C = 2πr 1 2π Area of a circle A = πr2 2 π Force of gravity F = k/d2 -2 k Boyle’s Law V = k/P -1 Circumference of a circle varies directly as its radius. The area enclosed by a circle is directly proportional to the square of its radius. The force of gravity acting on an object is inversely proportional to the squar eo fthe distance from the object to the center of the Earth. The volume of an enclosed gas (at a constant temperature) variables inversely as the applied pressure.

Power Functions with Modeling Consider this… What happens to the circumference of a circle as its radius increases? What happens to the area of a circle as its radius increases? What happens to the force exerted on an object as the square of the distance from the Earth’s center increases? What happens to the volume of gas inside an object as the pressure exerted on the object increases?

Power Functions with Modeling Direct variation Two quantities are vary directly if: As one quantity increases, the other increases, or As one quantity decreases, the other decreases In other words, quantities that are directly related must do the SAME things (both increase or both decrease) Power functions f(x) = kxn that are direct variations have positive powers of n.

Inverse variation Two quantities vary inversely if: As one quantity increases, the other decreases or As one quantity decreases, the other increases. In other words, quantities that are inversely related must do OPPOSITE things (one increases while the other decreases). Power functions f(x) = kxn that are inverse variations have negative powers of n.

Power Functions with Modeling Direct or Inverse? Direct/Inverse Name Formula Power Constant of Variation Circumference C = 2πr 1 2π Area of a circle A = πr2 2 π Force of gravity F = k/d2 -2 k Boyle’s Law V = k/P -1

Inverse Variation Example The volume V of a gas varies inversely as the pressure P on it. If the volume is 240 cm3 under pressure of 30 kg/cm2 , what pressure has to be applied to have a volume of 160 cm3 ?

Writing a power function The area A of an equilateral triangle varies directly as the square of the length s of its side. Express the area as a power function in terms of s.

Which of the twelve basic functions we discussed are power functions Which of the twelve basic functions we discussed are power functions? State the power of each.

State the power and constant of variation for the function, graph it, and analyze it. (domain, range, continuity, increasing/decreasing, symmetry, boundedness, local extrema, asymptotes, and end behavior)

Comparing Graphs of Monomial Functions A monomial function is a single-term polynomial function. Any function that can be written as f(x) = kxa or f(x)=k is a monomial Typical monomial functions: x, x2, x3

Graphs of Monomial Functions F(x) = xn is even if n is even and odd if n is odd

Describe how to obtain the graph of g(x) given the graph of f(x) = xn with the same power of n. g(x) = 1.5x5 g(x) = -2x630

Graphical Behavior of Power functions f(x) = kxn (when x≥0) *All graphs pass through (1,k)*   When k>0, graph is in Quadrant 1 When k<0, graph is in Quadrant 4

In general, for any power function, f(x)=kxn, When x<0 one of three things can happen: f is undefined for x<0 f is an even function (symmetrical about y-axis) f is an odd function (symmetrical about origin)

State the values of k and a for the function f(x) State the values of k and a for the function f(x). Describe the portion of the curve that lies in Quadrant I or IV. Determine whether f is even, odd, or undefined for x < 0. Describe the rest of the curve if any. Graph the function to see whether it matches your description. f(x) = - 4x2/3 f(x) = -x -4

Determining Power Functions from Data Recall r2, the coefficient of determination, can be used to determine the appropriateness of a non-linear model. 0 < r2 < 1 If r2 ≈ 1, then the model is more appropriate and is a good representation for the data If r2 ≈ 0, then the model is less appropriate and is not a good representation for the data.

Light Intensity Data for a 100-W Light Bulb Robert and Preston(yes, from our very own Pre-Calculus Class!) gathered the data shown in the table below from a 100-watt light bulb and a CBL unit with a light-intensity probe. Draw a scatter plot of the data Find a regression model. Is the power close to the a = -2 (the theoretical value)? Sketch the graph of the equation along with the data points. Use the regression model to predict the light intensity at distances of 1.7m and 3.4m. Light Intensity Data for a 100-W Light Bulb Distance (m) Intensity (W/m2) 1.0 1.5 2.0 2.5 3.0 7.95 3.53 2.01 1.27 0.90