Chapter 1

Write an equation of: (a) a vertical line passing through the given point; (b) a horizontal line passing through the given point. $$(5,8)$$

Determine any \(x\) - or \(y\) -intercepts for the graph of the equation. Note: You're not asked to draw the graph. (a) \(y=6 x^{3}+9 x^{2}+x\) (b) \(y=9 x^{3}+6 x^{2}+x\)

The given points Pand Q are the endpoints of a diameter of a circle. Find (a) the center of the circle; (b) the radius of the circle. 20\. \(P(1,-3)\) and \(Q(-5,-5)\)

Solve each equation. $$\frac{3}{2 x+1}-\frac{4}{x+1}=\frac{2}{2 x^{2}+3 x+1}$$

Evaluate each expression, given that \(a=-2\) \(b=3,\) and \(c=-4\). $$|b+c|-|b|-|c|$$

Is the graph of the line \(x=0\) the \(x\) -axis or the \(y\) -axis?

Solve each equation. $$\frac{5}{x-4}-\frac{3}{2 x^{2}-5 x-12}=\frac{1}{2 x+3}$$

Evaluate each expression, given that \(a=-2\) \(b=3,\) and \(c=-4\). $$|a+b|^{2}-|b+c|^{2}$$

Is the graph of the line \(y=0\) the \(x\) -axis or the \(y\) -axis?

Graph the equation after determining the \(x\) - and \(y\) -intercepts and whether the graph possesses any of the three types of symmetry described on page 58 $$y=2 x^{2}$$

(a) \(\frac{2}{3 x}=\frac{3}{x}\) (b) \(\frac{2}{3 x}=\frac{3}{x+1}\) (c) \(\frac{2}{3 x}=\frac{3}{x}+1\)

Evaluate each expression, given that \(a=-2\) \(b=3,\) and \(c=-4\). $$\frac{|a|+|b|+|c|}{|a+b+c|}$$

Find an equation of the line with the given slope and \(y\) -intercept. (a) slope \(-4 ; y\) -intercept 7 (b) slope \(2 ; y\) -intercept \(3 / 2\)

(a) \(\frac{3}{x-2}=\frac{5}{9 x}\) (b) \(\frac{3}{x-2}=\frac{5}{9 x-2}\) (c) \(\frac{3}{x-2}=\frac{5}{\frac{5}{3} x-2}\)

Evaluate each expression, given that \(a=-2\) \(b=3,\) and \(c=-4\). $$\frac{a+b+|a-b|}{2}$$

Find an equation of the line with the given slope and \(y\) -intercept. (a) slope \(0 ; y\) -intercept 14 (b) slope \(14 ; y\) -intercept 0

Have you or a friend ever run in a \(10 \mathrm{K}(10,000 \text { meter })\) race? When the author polled his precalculus class at UCLA in Fall \(1997,\) he found that there were five students in the class (of 160 ) who said they had run a \(10 \mathrm{K}\) in under 50 minutes. Of those five, two (one male, one female) said they had run a \(10 \mathrm{K}\) in under 40 minutes. The world record for this event is well under 30 minutes. In this exercise you'll look at some of the world records in this event over the past decade. (a) The table that follows lists the world records in the (men's) 10,000 meter race as of the end of the years \(1993,1995,\) and \(1997 .\) After converting the times into seconds, plot the three points corresponding to these records in a coordinate system similar to the one shown. $$\begin{array}{lll}\text { Year } & \text { Time } & \text { Runner } \\\\\hline 1993 & 26: 58.38 & \text { Yobes Ondieki (Kenya) } \\\1995 & 26: 43.53 & \text { Haile Gebrselassie (Kenya) } \\\1997 & 26: 27.85 & \text { Paul Tergat (Kenya) } \\\\\hline\end{array}$$ (GRAPH CAN'T COPY) (b) Use the midpoint formula and the data for 1993 and 1995 to compute an estimate for what the world record might have been by the end of \(1994 .\) Then compute the percentage error (rounded to two decimal places), given that the record at the end of 1994 was 26: 52.23 (set by William Seigei of Kenya). Was your estimate too high or too low? (c) Use the midpoint formula and the data for 1995 and 1997 to compute an estimate for what the world record might have been by the end of \(1996 .\) Then compute the percentage error given that the record at the end of 1996 was 26: 38.08 (set by Salah Hissou of Morocco). Was your estimate too high or too low? Is the percentage error more or less than that obtained in part (b)? (d) Using a coordinate system similar to the one shown in part (a), or using a photocopy, plot the points corresponding to the (actual, not estimated) world records for the years \(1993,1994,1995,1996,1997,\) and 1998 Except for \(1998,\) all the records have been given above. The world record at the end of 1998 was 26: 22.75 (set by Haile Gebrselassie of Kenya). Use the picture you obtain to say whether or not the record times seem to be leveling off.

Solve each equation by factoring. $$x^{2}-5 x-6=0$$

Evaluate each expression, given that \(a=-2\) \(b=3,\) and \(c=-4\). $$\frac{a+b-|a-b|}{2}$$

(a) Use a graphing utility to graph the equation. (b) Use a graphing utility, as in Example \(5,\) to estimate to one decimal place the \(x\) -intercepts. (c) Use algebra to determine the exact values for the \(x\) -intercepts. Then use a calculator to check that the answers are consistent with the estimates obtained in part (b). $$y=x^{2}-2 x-2$$

Find an equation for the line that is described, and sketch the graph. Write the final answer in the form \(y=m x+b ;\) (a) Passes through (-3,-1) and has slope 4 (b) Passes through \((5 / 2,0)\) and has slope \(1 / 2\) (c) Has \(x\) -intercept 6 and \(y\) -intercept 5 (d) Has \(x\) -intercept -2 and slope \(3 / 4\) (e) Passes through (1,2) and (2,6)

(a) Sketch the parallelogram with vertices \(A(-7,-1)\) \(B(4,3), C(7,8),\) and \(D(-4,4)\) (b) Compute the midpoints of the diagonals \(\overline{A C}\) and \(\overline{B D}\) (c) What conclusion can you draw from part (b)?

Solve each equation by factoring. $$x^{2}-5 x=-6$$

Rewrite each expression without using absolute value notation. $$|\sqrt{2}-1|-1$$

(a) Use a graphing utility to graph the equation. (b) Use a graphing utility, as in Example \(5,\) to estimate to one decimal place the \(x\) -intercepts. (c) Use algebra to determine the exact values for the \(x\) -intercepts. Then use a calculator to check that the answers are consistent with the estimates obtained in part (b). $$y=2 x^{2}+x-5$$

Find an equation for the line that is described, and sketch the graph. Write the final answer in the form \(y=m x+b ;\) (a) Passes through (-7,-2) and (0,0) (b) Passes through (6,-3) and has \(y\) -intercept 8 (c) Passes through (0,-1) and has the same slope as the line \(3 x+4 y=12\) (d) Passes through (6,2) and has the same \(x\) -intercept as the line \(-2 x+y=1\) (e) Has \(x\) -intercept -6 and \(y\) -intercept \(\sqrt{2}\)

The vertices of \(\triangle A B C\) are \(A(1,1), B(9,3),\) and \(C(3,5)\) (a) Find the perimeter of \(\triangle A B C\) (b) Find the perimeter of the triangle that is formed by joining the midpoints of the three sides of \(\triangle A B C\) (c) Compute the ratio of the perimeter in part (a) to the perimeter in part (b). (d) What theorem from geometry provides the answer for part (c) without using the results in (a) and (b)?

Solve each equation by factoring. $$10 z^{2}-13 z-3=0$$

(a) Use a graphing utility to graph the equation. (b) Use a graphing utility, as in Example \(5,\) to estimate to one decimal place the \(x\) -intercepts. (c) Use algebra to determine the exact values for the \(x\) -intercepts. Then use a calculator to check that the answers are consistent with the estimates obtained in part (b). $$y=2 x^{3}-5 x$$

Find an equation for the line that is described, and sketch the graph. Write the answer in the form \(A x+B y+C=0\). Passes through (-3,4) and is parallel to the \(x\) -axis.

Solve each equation by factoring. $$3 t^{2}-t-4=0$$

Rewrite each expression without using absolute value notation. $$|x-3| \text { given that } x \geq 3$$

(a) Use a graphing utility to graph the equation. (b) Use a graphing utility, as in Example \(5,\) to estimate to one decimal place the \(x\) -intercepts. (c) Use algebra to determine the exact values for the \(x\) -intercepts. Then use a calculator to check that the answers are consistent with the estimates obtained in part (b). $$y=3 x^{3}+5 x^{2}+x$$

Find an equation for the line that is described, and sketch the graph. Write the answer in the form \(A x+B y+C=0\). Passes through (-3,4) and is parallel to the \(y\) -axis.

(A numerologist's delight) Using the Pythagorean theorem and your calculator, compute the area of a right triangle in which the lengths of the hypotenuse and one leg are 2045 and \(693,\) respectively.

Solve each equation by factoring. $$(x+1)^{2}-4=0$$

Rewrite each expression without using absolute value notation. \(|x-3|\) given that \(x<3\)

Find the \(x\) - and \(y\) -intercepts of the line, and find the area and the perimeter of the triangle formed by the line and the axes. (a) \(3 x+5 y=15\) (b) \(3 x-5 y=15\)

Solve each equation by factoring. $$x^{2}+3 x-40=0$$

Rewrite each expression without using absolute value notation. $$\left|t^{2}+1\right|$$

Find the \(x\) - and \(y\) -intercepts of the line, and find the area and the perimeter of the triangle formed by the line and the axes. (a) \(5 x+4 y=40\) (b) \(2 x+4 y=\sqrt{2}\)

Solve each equation by factoring. $$x(2 x-13)=-6$$

Rewrite each expression without using absolute value notation. $$\left|x^{4}+1\right|$$

Use a graphing utility to graph the equations and to approximate the \(x\) -intercepts. In approximating the \(x\) -intercepts, use a "solve" key or a sufficiently magnified view to ensure that the values you give are correct in the first three decimal places. Remark: None of the \(x\) -intercepts for these four equations can be obtained using factoring techniques.) $$y=x^{3}-3 x+1$$

Determine whether each pair of lines is parallel, perpendicular, or neither. (a) \(3 x-4 y=12 ; 4 x-3 y=12\) (b) \(y=5 x-16 ; y=5 x+2\) (c) \(5 x-6 y=25 ; 6 x+5 y=0\) (d) \(y=-\frac{2}{3} x-1 ; y=\frac{3}{2} x-1\)

Suppose that the coordinates of points \(P, Q,\) and \(M\) are $$\begin{array}{c}P\left(x_{1}, y_{1}\right) \quad Q\left(x_{2}, y_{2}\right) \\\M\left(\frac{x_{1}+x_{2}}{2}, \frac{y_{1}+y_{2}}{2}\right)\end{array}$$ Follow steps (a) and (b) to prove that \(M\) is the midpoint of the line segment from \(P\) to \(Q\) (a) By computing both of the distances \(P M\) and \(M Q\) show that \(P M=M Q\). (This shows that \(M\) lies on the perpendicular bisector of line segment \(\overline{P Q}\), but it does not show that \(M\) actually lies on \(P Q .\) ) (b) Show that \(P M+M Q=P Q\). (This shows that \(M\) does lie on \(\overline{P Q} .)\)

Solve each equation by factoring. $$x(3 x-23)=8$$

Rewrite each expression without using absolute value notation. $$|-\sqrt{3}-4|$$

Say whether the statement is TRUE or FALSE. (In Exercises \(37-40\), do not use a calculator or table; use instead the approximations \(\sqrt{2} \approx 1.4 \text { and } \pi \approx 3.1 .)\) $$-5<-50$$

Use a graphing utility to graph the equations and to approximate the \(x\) -intercepts. In approximating the \(x\) -intercepts, use a "solve" key or a sufficiently magnified view to ensure that the values you give are correct in the first three decimal places. Remark: None of the \(x\) -intercepts for these four equations can be obtained using factoring techniques.) $$y=8 x^{3}-6 x-1$$