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Chapter 3: Problem 13
Find the coordinates of the vertex for the parabola defined by the given quadratic function. $$f(x)=2 x^{2}-8 x+3$$
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Determine whether each statement makes sense or does not make sense, and explain your reasoning. My graph of \(y=\frac{x-1}{(x-1)(x-2)}\) has vertical asymptotes at \(x=1\) and \(x=2\)
Use the four-step procedure for solving variation problems given on page 424 to solve. \(C\) varies jointly as \(A\) and \(T . C=175\) when \(A=2100\) and \(T=4 .\) Find \(C\) when \(A=2400\) and \(T=6\).
Write the equation of a rational function \(f(x)=\frac{p(x)}{q(x)}\) having the indicated properties, in which the degreeof \(p\) and \(q\) are as small as possible. More than one correct finction may be possible. Graph your function using a graphing utility to verify that it has the required propertics I has a vertical asymptote given by \(x=1,\) a slant asymptote whose equation is \(y=x, y\) -intercept at \(2,\) and \(x\) -intercepts at \(-1\) and 2
In this exercise, we lead you through the steps involved in the proof of the Rational Zero Theorem. Consider the polynomial equation $$a_{n} x^{n}+a_{n-1} x^{n-1}+a_{n-2} x^{n-2}+\cdots+a_{1} x+a_{0}=0$$ and let \(\frac{P}{q}\) be a rational root reduced to lowest terms. a. Substitute \(\frac{p}{q}\) for \(x\) in the equation and show that the equation can be written as $$a_{n} p^{n}+a_{n-1} p^{n-1} q+a_{n-2} p^{n-2} q^{2}+\cdots+a_{1} p q^{n-1}=-a_{0} q^{n}$$ b. Why is \(p\) a factor of the left side of the equation? c. Because \(p\) divides the left side, it must also divide the right side. However, because \(\frac{P}{q}\) is reduced to lowest terms, \(p\) and \(q\) have no common factors other than \(-1\) and 1 Because \(p\) does divide the right side and has no factors in common with \(q^{n},\) what can you conclude? d. Rewrite the equation from part (a) with all terms containing \(q\) on the left and the term that does not have a factor of \(q\) on the right. Use an argument that parallels parts (b) and (c) to conclude that \(q\) is a factor of \(a_{n}\).
The illumination from a light source varies inversely as the square of the distance from the light source. If you raise a lamp from 15 inches to 30 inches over your desk, what happens to the illumination?
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