91Ó°ÊÓ

MAKE A CONJECTURE Plot the points \( (2, 1) \), \( (-3, 5) \) and \( (7, -3) \) on a rectangular coordinate system. Then change the sign of the \( x \)-coordinate of each point and plot the three new points on the same rectangular coordinate system. Make a conjecture about the location of a point when each of the following occurs. (a) The sign of the \( x \)-coordinate is changed. (b) The sign of the \( y \)-coordinate is changed. (c) The signs of both the \( x \)- and \( y \)-coordinates are changed.

COLLINEAR POINTS Three or more points are collinear if they all lie on the same line. Use the steps below to determine if the set of points {\( A(2, 3) \), \( B(2, 6) \), \( C(6, 3) \)} and the set of points {\( A(8, 3) \), \( B(5, 2) \), \( C(2, 1) \)} are collinear. (a) For each set of points, use the Distance Formula to find the distances from \( A \) to \( B \) from \( B \) to \( C \) and from \( A \) to \( C \). What relationship exists among these distances for each set of points? (b) Plot each set of points in the Cartesian plane. Do all the points of either set appear to lie on the same line? (c) Compare your conclusions from part (a) with the conclusions you made from the graphs in part (b). Make a general statement about how to use the Distance Formula to determine collinearity.

BEAM LOAD The maximum load that can be safely supported by a horizontal beam varies jointly as the width of the beam and the square of its depth, and inversely as the length of the beam. Determine the changes in the maximum safe load under the following conditions. (a) The width and length of the beam are doubled. (b) The width and depth of the beam are doubled. (c) All three of the dimensions are doubled. (d) The depth of the beam is halved.

CAPSTONE Match the equation or equations with the given characteristic. (i) \(y = 3x^3 - 3x\) (ii) \(y = (x+3)^2\) (iii) \(y = 3x - 3\) (iv) \(y = \sqrt[3]{x}\) (v) \(y =3x^2 + 3\) (vi) \(y = \sqrt{x+3}\) (a) Symmetric with respect to the \(y\)-axis (b) Three \(x\)-intercepts (c) Symmetric with respect to the \(x\)-axis (d) \((-2, 1)\) is a point on the graph (e) Symmetric with respect to the origin (f ) Graph passes through the origin

In Exercises 87-92, use the functions given by \(f(x) = \frac{1}{8}x - 3\) and \(g(x) = x^3\) to find the indicated value or function. \((f \circ g)^{-1}\)

In Exercises 87-92, use the functions given by \(f(x) = \frac{1}{8}x - 3\) and \(g(x) = x^3\) to find the indicated value or function. \(g^{-1} \circ f^{-1}\)

In Exercises 93-96, use the functions given by \(f(x) = x + 4\) and \(g(x) = 2x-5\) to find the specified function. \((g \circ f)^{-1}\)

GRAPHICAL ANALYSIS In Exercises 103-106, identify any relationships that exist among the lines, and then use a graphing utility to graph the three equations in the same viewing window. Adjust the viewing window so that the slope appears visually correct\(-\)that is, so that parallel lines appear parallel and perpendicular lines appear to intersect at right angles. (a) \(y = 2x\) (b) \(y = -2x\) (c) \(y = \frac{1}{2}x\)

In Exercises 103-110, find the difference quotient and simplify your answer. \(f(x) = x^3+3x\), \(\frac{f(x+h)-f(x)}{h}\), \(h \neq 0\)

REVENUE The following are the slopes of lines representing daily revenues \(y\) in terms of time \(x\) in days. Use the slopes to interpret any change in daily revenues for a one-day increase in time. (a) The line has a slope of \(m=400\). (b) The line has a slope of \(m=100\). (c) The line has a slope of \(m= 0\).