Section 1
Rectangular Coordinates and Graphs
$$\text { Fill in the blank to correctly complete each sentence.}$$The point $(-1,3)$ lies in quadrant __________ in the rectangular coordinate system.
$$\text { Fill in the blank to correctly complete each sentence.}$$The point $(4,___)$ lies on the graph of the equation $y=3 x-6$.
$$\text { Fill in the blank to correctly complete each sentence.}$$Any point that lies on the $x$ -axis has $y$ -coordinate equal to __________.
$$\text { Fill in the blank to correctly complete each sentence.}$$The $y$ -intercept of the graph of $y=-2 x+6$ is __________.
$$\text { Fill in the blank to correctly complete each sentence.}$$The $x$ -intercept of the graph of $2 x+5 y=10$ is __________.
$$\text { Fill in the blank to correctly complete each sentence.}$$The distance from the origin to the point $(-3,4)$ is __________.
Determine whether each statement is true or false. If false, explain why.The graph of $y=x^{2}+2$ has no $x$ -intercepts.
Determine whether each statement is true or false. If false, explain why.The graph of $y=x^{2}-2$ has two $x$ -intercepts.
Determine whether each statement is true or false. If false, explain why.The midpoint of the segment joining $(0,0)$ and $(4,4)$ is 2.
Determine whether each statement is true or false. If false, explain why.The distance between the points $(0,0)$ and $(4,4)$ is 4.
$$\text {Give three ordered pairs from each table.}$$$$\begin{array}{r|r}x & y \\\hline 2 & -5 \\-1 & 7 \\3 & -9 \\5 & -17 \\6 & -21\end{array}$$
$$\text {Give three ordered pairs from each table.}$$$$\begin{array}{r|r}x & y \\\hline 3 & 3 \\-5 & -21 \\8 & 18 \\4 & 6 \\0 & -6\end{array}$$
$$\text {Give three ordered pairs from each table.}$$(TABLE CANNOT COPY)
For the points $P$ and $Q,$ find ( $a$ ) the distance $d(P, Q)$ and ( $b$ ) the coordinates of the midpoint M of line segment PQ. See Examples 2 and 5(a).$$P(-5,-6), Q(7,-1)$$
For the points $P$ and $Q,$ find ( $a$ ) the distance $d(P, Q)$ and ( $b$ ) the coordinates of the midpoint M of line segment PQ. See Examples 2 and 5(a).$$P(-4,3), Q(2,-5)$$
For the points $P$ and $Q,$ find ( $a$ ) the distance $d(P, Q)$ and ( $b$ ) the coordinates of the midpoint M of line segment PQ. See Examples 2 and 5(a).$$P(8,2), Q(3,5)$$
For the points $P$ and $Q,$ find ( $a$ ) the distance $d(P, Q)$ and ( $b$ ) the coordinates of the midpoint M of line segment PQ. See Examples 2 and 5(a).$$P(-8,4), Q(3,-5)$$
For the points $P$ and $Q,$ find ( $a$ ) the distance $d(P, Q)$ and ( $b$ ) the coordinates of the midpoint M of line segment PQ. See Examples 2 and 5(a).$$P(-6,-5), Q(6,10)$$
For the points $P$ and $Q,$ find ( $a$ ) the distance $d(P, Q)$ and ( $b$ ) the coordinates of the midpoint M of line segment PQ. See Examples 2 and 5(a).$$P(6,-2), Q(4,6)$$
For the points $P$ and $Q,$ find ( $a$ ) the distance $d(P, Q)$ and ( $b$ ) the coordinates of the midpoint M of line segment PQ. See Examples 2 and 5(a).$$P(3 \sqrt{2}, 4 \sqrt{5}), Q(\sqrt{2},-\sqrt{5})$$
For the points $P$ and $Q,$ find ( $a$ ) the distance $d(P, Q)$ and ( $b$ ) the coordinates of the midpoint M of line segment PQ. See Examples 2 and 5(a).$$P(-\sqrt{7}, 8 \sqrt{3}), Q(5 \sqrt{7},-\sqrt{3})$$
Determine whether the three points are the vertices of a right triangle.$$(-6,-4),(0,-2),(-10,8)$$
Determine whether the three points are the vertices of a right triangle.$$(-2,-8),(0,-4),(-4,-7)$$
Determine whether the three points are the vertices of a right triangle.$$(-4,1),(1,4),(-6,-1)$$
Determine whether the three points are the vertices of a right triangle.$$(-2,-5),(1,7),(3,15)$$
Determine whether the three points are the vertices of a right triangle.$$(-4,3),(2,5),(-1,-6)$$
Determine whether the three points are the vertices of a right triangle.$$(-7,4),(6,-2),(0,-15)$$
$$\text { Determine whether the three points are collinear.}$$$$(0,-7),(-3,5),(2,-15)$$
$$\text { Determine whether the three points are collinear.}$$$$(-1,4),(-2,-1),(1,14)$$
$$\text { Determine whether the three points are collinear.}$$$$(0,9),(-3,-7),(2,19)$$
$$\text { Determine whether the three points are collinear.}$$$$(-1,-3),(-5,12),(1,-11)$$
$$\text { Determine whether the three points are collinear.}$$$$(-7,4),(6,-2),(-1,1)$$
$$\text { Determine whether the three points are collinear.}$$$$(-4,3),(2,5),(-1,4)$$
Find the coordinates of the other endpoint of each line segment, given its midpoint and one endpoint. See Example $5(b)$.midpoint $(5,8),$ endpoint $(13,10)$
Find the coordinates of the other endpoint of each line segment, given its midpoint and one endpoint. See Example $5(b)$.midpoint $(-7,6),$ endpoint $(-9,9)$
Find the coordinates of the other endpoint of each line segment, given its midpoint and one endpoint. See Example $5(b)$.midpoint $(12,6),$ endpoint $(19,16)$
Find the coordinates of the other endpoint of each line segment, given its midpoint and one endpoint. See Example $5(b)$.midpoint $(-9,8),$ endpoint $(-16,9)$
Find the coordinates of the other endpoint of each line segment, given its midpoint and one endpoint. See Example $5(b)$.midpoint $(a, b),$ endpoint $(p, q)$
Find the coordinates of the other endpoint of each line segment, given its midpoint and one endpoint. See Example $5(b)$.midpoint $(6 a, 6 b),$ endpoint $(3 a, 5 b)$
Solve each problem.The graph shows a straight line that approximates the percentage of Americans 25 years and older who had earned bachelor's degrees or higher for the years $1990-2012 .$ Use the midpoint formula and the two given points to estimate the percent in 2001 . Compare the answer with the actual percent of 26.2.(GRAPH CANNOT COPY)
Solve each problem.The graph shows a straight line that approximates national advertising revenue, in millions of dollars, for newspapers in the United States for the years $2006-2012$. Use the midpoint formula and the two given points to estimate revenue in $2009 .$ Compare the answer with the actual figure of 4424 million dollars.(GRAPH CANNOT COPY)
Solve each problem.The table lists how poverty level income cutoffs (in dollars) for a family of four have changed over time. Use the midpoint formula to approximate the poverty level cutoff in 2012 to the nearest dollar.(TABLE CANNOT COPY)
Solve each problem.Enrollments in public colleges for recent years are shown in the table. Assuming a linear relationship, estimate the enrollments for(a) 2003 and (b) $2009 .$ Give answers to the nearest tenth of thousands if applicable.(TABLE CANNOT COPY)
Show that if $M$ is the midpoint of the line segment with endpoints $P\left(x_{1}, y_{1}\right)$ and $Q\left(x_{2}, y_{2}\right),$ then$$d(P, M)+d(M, Q)=d(P, Q) \quad \text { and } \quad d(P, M)=d(M, Q)$$
Write the distance formula $d=\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}}$ using a rational exponent.
For each equation, (a) give a table with at least three ordered pairs that are solutions, and (b) graph the equation. See Examples 7 and 8.$$y=\frac{1}{2} x-2$$
For each equation, (a) give a table with at least three ordered pairs that are solutions, and (b) graph the equation.$$y=-\frac{1}{2} x+2$$
For each equation, (a) give a table with at least three ordered pairs that are solutions, and (b) graph the equation.$$2 x+3 y=5$$
For each equation, (a) give a table with at least three ordered pairs that are solutions, and (b) graph the equation.$$3 x-2 y=6$$
For each equation, (a) give a table with at least three ordered pairs that are solutions, and (b) graph the equation.$$y=x^{2}$$
For each equation, (a) give a table with at least three ordered pairs that are solutions, and (b) graph the equation.$$y=x^{2}+2$$
For each equation, (a) give a table with at least three ordered pairs that are solutions, and (b) graph the equation.$$y=\sqrt{x-3}$$
For each equation, (a) give a table with at least three ordered pairs that are solutions, and (b) graph the equation.$$y=\sqrt{x}-3$$
For each equation, (a) give a table with at least three ordered pairs that are solutions, and (b) graph the equation.$$y=|x-2|$$
For each equation, (a) give a table with at least three ordered pairs that are solutions, and (b) graph the equation.$$y=-|x+4|$$
For each equation, (a) give a table with at least three ordered pairs that are solutions, and (b) graph the equation.$$y=x^{3}$$
For each equation, (a) give a table with at least three ordered pairs that are solutions, and (b) graph the equation.$$y=-x^{3}$$
$$\text { Answer the following.}$$If a vertical line is drawn through the point $(4,3),$ at what point will it intersect the $x$ -axis?
$$\text { Answer the following.}$$If a horizontal line is drawn through the point $(4,3),$ at what point will it intersect the $y$ -axis?
$$\text { Answer the following.}$$If the point $(a, b)$ is in the second quadrant, then in what quadrant is $(a,-b) ?$ $(-a, b) ? \quad(-a,-b) ? \quad(b, a) ?$
$$\text { Answer the following.}$$Show that the points $(-2,2),(13,10),(21,-5),$ and $(6,-13)$ are the vertices of a rhombus (all sides equal in length).
$$\text { Answer the following.}$$Are the points $A(1,1), B(5,2), C(3,4),$ and $D(-1,3)$ the vertices of a parallelogram (opposite sides equal in length)? of a rhombus (all sides equal in length)?
$$\text { Answer the following.}$$Find the coordinates of the points that divide the line segment joining $(4,5)$ and $(10,14)$ into three equal parts.