Chapter 2

\(43-50\) . Solve the equation graphically in the given interval. State each answer rounded to two decimals. $$ 16 x^{3}+16 x^{2}=x+1 ;[-2,2] $$

\(41-48\) Find the \(x\) - and \(y\) -intercepts of the graph of the equation. $$ y=\sqrt{x+1} $$

Show that the triangle with vertices \(A(6,-7), B(11,-3)\) and \(C(2,-2)\) is a right triangle by using the converse of the Pythagorean Theorem. Find the area of the triangle.

Find the slope and y-intercept of the line, and draw its graph. $$ x+3 y=0 $$

Frequency of Vibration The frequency \(f\) of vibration of a violin string is inversely proportional to its length \(L\) . The constant of proportionality \(k\) is positive and depends on the tension and density of the string. (a) Write an equation that represents this variation. (b) What effect does doubling the length of the string have on the frequency of its vibration?

\(43-50\) . Solve the equation graphically in the given interval. State each answer rounded to two decimals. $$ x-\sqrt{x+1}=0 ; \quad[-1,5] $$

\(41-48\) Find the \(x\) - and \(y\) -intercepts of the graph of the equation. $$ x y=5 $$

Show that the points \(A(-2,9), B(4,6), C(1,0),\) and \(D(-5,3)\) are the vertices of a square.

Find the slope and y-intercept of the line, and draw its graph. $$ 2 x-5 y=0 $$

\(43-50\) . Solve the equation graphically in the given interval. State each answer rounded to two decimals. $$ 1+\sqrt{x}=\sqrt{1+x^{2}} ;[-1,5] $$

\(41-48\) Find the \(x\) - and \(y\) -intercepts of the graph of the equation. $$ x^{2}-x y+y=1 $$

Show that the points \(A(-1,3), B(3,11),\) and \(C(5,15)\) are collinear by showing that \(d(A, B)+d(B, C)=d(A, C)\).

Find the slope and y-intercept of the line, and draw its graph. $$ \frac{1}{2} x-1 y+1=0 $$

Is Proportionality Everything? A great many laws of physics and chemistry are expressible as proportionalities. Give at least one example of a function that occurs in the sciences that is not a proportionality.

\(43-50\) . Solve the equation graphically in the given interval. State each answer rounded to two decimals. $$ x^{1 / 3}-x=0 ; \quad[-3,3] $$

\(49-54\) . Find the center and radius of the circle, and sketch its graph. $$ x^{2}+y^{2}=9 $$

Find a point on the \(y\) -axis that is equidistant from the points \((5,-5)\) and \((1,1) .\)

Find the slope and y-intercept of the line, and draw its graph. $$ -3 x-5 y+30=0 $$

\(49-54\) . Find the center and radius of the circle, and sketch its graph. $$ x^{2}+y^{2}=5 $$

Find the lengths of the medians of the triangle with vertices \(A(1,0), B(3,6),\) and \(C(8,2) .\) (A median is a line segment from a vertex to the midpoint of the opposite side.)

Find the slope and y-intercept of the line, and draw its graph. $$ y=4 $$

\(49-54\) . Find the center and radius of the circle, and sketch its graph. $$ (x-3)^{2}+y^{2}=16 $$

Find the point that is one-fourth of the distance from the point \(P(-1,3)\) to the point \(Q(7,5)\) along the segment \(P Q\) .

Find the slope and y-intercept of the line, and draw its graph. $$ x=-5 $$

\(49-54\) . Find the center and radius of the circle, and sketch its graph. $$ x^{2}+(y-2)^{2}=4 $$

Plot the points \(P(-2,1)\) and \(Q(12,-1)\) on a coordinate plane. Which (if either) of the points \(A(5,-7)\) and \(B(6,7)\) lies on the perpendicular bisector of the segment \(P Q ?\)

Find the slope and y-intercept of the line, and draw its graph. $$ 3 x-4 y=12 $$

\(49-54\) . Find the center and radius of the circle, and sketch its graph. $$ (x+3)^{2}+(y-4)^{2}=25 $$

Plot the points \(P(-1,-4), Q(1,1),\) and \(R(4,2)\) on a coordinate plane. Where should the point \(S\) be located so that the figure \(P Q R S\) is a parallelogram?

Find the slope and y-intercept of the line, and draw its graph. $$ 4 y+8=0 $$

\(49-54\) . Find the center and radius of the circle, and sketch its graph. $$ (x+1)^{2}+(y+2)^{2}=36 $$

If \(M(6,8)\) is the midpoint of the line segment \(A B,\) and if \(A\) has coordinates \((2,3),\) find the coordinates of \(B .\)

Find the slope and y-intercept of the line, and draw its graph. $$ 3 x+4 y-1=0 $$

\(55-58\) m Find all real solutions of the equation, rounded to two decimals. $$ x^{3}-2 x^{2}-x-1=0 $$

\(55-62\) . Find an equation of the circle that satisfies the given conditions. Center \((2,-1) ; \quad\) radius 3

(a) Sketch the parallelogram with vertices \(A(-2,-1)\) \(B(4,2), C(7,7),\) and \(D(1,4) .\) (b) Find the midpoints of the diagonals of this parallelogram. (c) From part (b), show that the diagonals bisect each other.

Find the slope and y-intercept of the line, and draw its graph. $$ 4 x+5 y=10 $$

\(55-58\) m Find all real solutions of the equation, rounded to two decimals. $$ x^{4}-8 x^{2}+2=0 $$

\(55-62\) . Find an equation of the circle that satisfies the given conditions. Center \((-1,-4) ; \quad\) radius 8

Use slopes to show that \(A(1,1), B(7,4), C(5,10),\) and \(D(-1,7)\) are vertices of a parallelogram.

\(55-58\) m Find all real solutions of the equation, rounded to two decimals. $$ x(x-1)(x+2)=\frac{1}{6} x $$

A city has streets that run north and south and avenues that run east and west, all equally spaced. Streets and avenues are numbered sequentially, as shown in the figure. The walking distance between points A and B is 7 blocks—that is, 3 blocks east and 4 blocks north. To find the straight-line distances d, we must use the Distance Formula. (a) Find the straight-line distance (in blocks) between A and B. (b) Find the walking distance and the straight-line distance between the corner of 4th St. and 2nd Ave. and the corner of 11th St. and 26th Ave. (c) What must be true about the points P and Q if the walking distance between P and Q equals the straight-line distance between P and Q?

\(55-62\) . Find an equation of the circle that satisfies the given conditions. Center at the origin; passes through \((4,7)\)

Use slopes to show that \(A(-3,-1), B(3,3),\) and \(Q(-9,8)\) are vertices of a right triangle.

\(55-58\) m Find all real solutions of the equation, rounded to two decimals. $$ x^{4}=16-x^{3} $$

\(55-62\) . Find an equation of the circle that satisfies the given conditions. Center \((-1,5) ; \quad\) passes through \((-4,-6)\)

Use slopes to show that \(A(1,1), B(11,3), \quad G(10,8),\) and \(D(0,6)\) are vertices of a rectangle.

\(59-66\) . Find the solutions of the inequality by drawing appropriate graphs. State each answer rounded to two decimals. $$ x^{2} \leq 3 x+10 $$

\(55-62\) . Find an equation of the circle that satisfies the given conditions. Endpoints of a diameter are \(P(-1,1)\) and \(Q(5,9)\)

Use slopes to determine whether the given points are collinear (lie on a line). $$ \begin{array}{l}{\text { (a) }(1,1),(3,9),(6,21)} \\ {\text { (b) }(-1,3),(1,7),(4,15)}\end{array} $$