Section 1
The Rectangular Coordinate System and Graphing Utilities
In a rectangular coordinate system, the point where the $x$ - and $y$ -axes meet is called the _____.
The $x$ - and $y$ -axes divide the coordinate plane into four regions called _____.
The distance between two distinct points $\left(x_{1}, y_{1}\right)$ and $\left(x_{2}, y_{2}\right)$ is given by the formula _____.
The midpoint of the line segment with endpoints $\left(x_{1}, y_{1}\right)$ and $\left(x_{2}, y_{2}\right)$ is given by the formula _____.
A _____ to an equation in the variables $x$ and $y$ is an ordered pair $(x, y)$ that makes the equation a true statement.
An $x$ -intercept of a graph has a $y$ -coordinate of _____.
A $y$ -intercept of a graph has an $x$ -coordinate of _____ .
Given an equation in the variables $x$ and $y$, find the $y$ -intercept by substituting _____ for $x$ and solving for _____.
Plot the points on a rectangular coordinate system.$A(-3,-4)$$B\left(\frac{5}{3}, \frac{7}{4}\right)$$C(-1.2,3.8)$$D(\pi,-5)$$E(0,4.5)$$F(\sqrt{5}, 0)$
Plot the points on a rectangular coordinate system.$A(-2,-5)$$B\left(\frac{9}{2}, \frac{7}{3}\right)$$C(-3.6,2.1)$$D(5,-\pi)$$E(3.4,0)$$F(0, \sqrt{3})$
a. Find the exact distance between the points. (See Example 1$)$b. Find the midpoint of the line segment whose endpoints are the given points. (See Example 3)(-2,7) and (-4,11)
a. Find the exact distance between the points. (See Example 1$)$b. Find the midpoint of the line segment whose endpoints are the given points. (See Example 3)(-1,-3) and (3,-7)
a. Find the exact distance between the points. (See Example 1$)$b. Find the midpoint of the line segment whose endpoints are the given points. (See Example 3)(-7,-4) and (2,5)
a. Find the exact distance between the points. (See Example 1$)$b. Find the midpoint of the line segment whose endpoints are the given points. (See Example 3)(3,6) and (-4,-1)
a. Find the exact distance between the points. (See Example 1$)$b. Find the midpoint of the line segment whose endpoints are the given points. (See Example 3)(2.2,-2.4) and (5.2,-6.4)
a. Find the exact distance between the points. (See Example 1$)$b. Find the midpoint of the line segment whose endpoints are the given points. (See Example 3)(37.1,-24.7) and (31.1,-32.7)
a. Find the exact distance between the points. (See Example 1$)$b. Find the midpoint of the line segment whose endpoints are the given points. (See Example 3)$(\sqrt{5},-\sqrt{2})$ and $(4 \sqrt{5},-7 \sqrt{2})$
a. Find the exact distance between the points. (See Example 1$)$b. Find the midpoint of the line segment whose endpoints are the given points. (See Example 3)$(\sqrt{7},-3 \sqrt{5})$ and $(2 \sqrt{7}, \sqrt{5})$
Determine if the given points form the vertices of a right triangle. (See Example 2)$(1,3),(3,1),$ and (0,-2)
Determine if the given points form the vertices of a right triangle. (See Example 2)$(1,2),(3,0),$ and (-3,-2)
Determine if the given points form the vertices of a right triangle. (See Example 2)$(-2,4),(5,0),$ and (-5,1)
Determine if the given points form the vertices of a right triangle. (See Example 2)$(-6,2),(3,1),$ and (1,-2)
Determine if the given points are solutions to the equation.$x^{2}+y=1$a. (-2,-3)b. (4,-17)c. $\left(\frac{1}{2}, \frac{3}{4}\right)$
Determine if the given points are solutions to the equation.$|x-3|-y=4$a. (1,-2)b. (-2,-3)c. $\left(\frac{1}{10},-\frac{11}{10}\right)$
Identify the set of values $x$ for which $y$ will be a real number.$$y=\frac{2}{x-3}$$
Identify the set of values $x$ for which $y$ will be a real number.$$y=\frac{2}{x+7}$$
Identify the set of values $x$ for which $y$ will be a real number.$$y=\sqrt{x-10}$$
Identify the set of values $x$ for which $y$ will be a real number.$$y=\sqrt{x+11}$$
Identify the set of values $x$ for which $y$ will be a real number.$$y=\sqrt{1.5-x}$$
Identify the set of values $x$ for which $y$ will be a real number.$$y=\sqrt{2.2-x}$$
Graph the equations by plotting points.$$y=x$$
Graph the equations by plotting points.$$y=x^{2}$$
Graph the equations by plotting points.$$y=\sqrt{x}$$
Graph the equations by plotting points.$$y=|x|$$
Graph the equations by plotting points.$$y=x^{3}$$
Graph the equations by plotting points.$$y=\frac{1}{x}$$
Graph the equations by plotting points.$$y-|x|=2$$
Graph the equations by plotting points.$$|x|+y=3$$
Graph the equations by plotting points.$$y^{2}-x-2=0$$
Graph the equations by plotting points.$$y^{2}-x+1=0$$
Graph the equations by plotting points.$$x=|y|+1$$
Graph the equations by plotting points.$$x=|y|-3$$
Graph the equations by plotting points.$$y=|x+1|$$
Graph the equations by plotting points.$$y=|x-2|$$
Estimate the $x$ - and $y$ -intercepts from the graph.
Determine the $x$ - and $y$ -intercepts of the graph whose points are defined in the spreadsheet.$$\begin{array}{|c|c|c|c|c|c|c|c|c||}\hline & \mathrm{A} & \mathrm{B} & \mathrm{C} & \mathrm{D} & \mathrm{E} & \mathrm{F} & \mathrm{G} & \mathrm{H} \\\hline 1 & \mathrm{x} & -1 & 0 & 1 & 2 & 3 & 4 & 5 \\\hline 2 & \mathrm{y} & -4 & -2 & 0 & 2 & 4 & 6 & 8 \\\hline\end{array}$$
Determine the $x$ - and $y$ -intercepts of the graph whose points are defined in the spreadsheet.$$\begin{array}{|c|c|c|c|c|c|c|c|c||}\hline \hline & \mathrm{A} & \mathrm{B} & \mathrm{C} & \mathrm{D} & \mathrm{E} & \mathrm{F} & \mathrm{G} & \mathrm{H} \\\hline 1 & \mathrm{x} & -10 & -5 & 0 & 5 & 10 & 15 & 20 \\\hline 2 & \mathrm{y} & -3 & 0 & 4 & 9 & 15 & 22 & 30 \\\hline\end{array}$$
Find the $x$ - and $y$ -intercepts.$$-2 x+4 y=12$$
Find the $x$ - and $y$ -intercepts.$$-3 x-5 y=60$$
Find the $x$ - and $y$ -intercepts.$$x^{2}+y=9$$
Find the $x$ - and $y$ -intercepts.$$x^{2}=-y+16$$
Find the $x$ - and $y$ -intercepts.$$y=|x-5|-2$$
Find the $x$ - and $y$ -intercepts.$$y=|x+4|-3$$
Find the $x$ - and $y$ -intercepts.$$x=y^{2}-1$$
Find the $x$ - and $y$ -intercepts.$$x=y^{2}-4$$
Find the $x$ - and $y$ -intercepts.$$|x|=|y|$$
Find the $x$ - and $y$ -intercepts.$$x=|5 y|$$
Find the $x$ - and $y$ -intercepts.$$\frac{(x-3)^{2}}{4}+\frac{(y-4)^{2}}{9}=1$$
Find the $x$ - and $y$ -intercepts.$$\frac{(x+6)^{2}}{16}+\frac{(y+3)^{2}}{4}=1$$
A map of a wilderness area is drawn with the origin placed at the parking area. Two fire observation platforms are located at points $A$ and $B$. If a fire is located at point $C,$ which observation tower is closer to the fire?
A map of a state park is drawn so that the origin is placed at the visitor center. The distance between grid lines is $1 \mathrm{mi}$. Suppose that two hikers are located at points $A$ and $B$.a. Determine the distance between the hikers.b. If the hikers want to meet for lunch, determine the location of the midpoint between the hikers.
Assume that the units shown in the grid are in feet.a. Determine the exact length and width of the rectangle shown.b. Determine the perimeter and area.
The endpoints of a diameter of a circle are shown. Find the center and radius of the circle.
An isosceles triangle is shown (an isosceles triangle has two sides of equal length). Find the area of the triangle. Assume that the units shown in the grid are in meters. (Hint: Find the midpoint of the base of the triangle. Then use the distance formula to find the base and height.)
Read the scenario given and select a graph (a-f) that best models the situation.Average daily temperature ( $y$ ) for Albuquerque, New Mexico, versus the number of days $(x)$ after January 1 . Assume that the $x$ -axis represents a period of 1 yr.
Read the scenario given and select a graph (a-f) that best models the situation.Temperature of a cake $(y)$ versus the number of minutes $(x)$ after the cake has come out of the oven.
Read the scenario given and select a graph (a-f) that best models the situation.Average number of doctors visits per year $(y)$ versus the age of patient $(x)$.
Read the scenario given and select a graph (a-f) that best models the situation.Speed of a car $(y)$ versus time $(x)$. Assume that the experiment begins at a time at which the car begins merging onto a highway until the time when the driver merges off the highway.
Read the scenario given and select a graph (a-f) that best models the situation.Speed of a shopping cart $(y)$ versus time $(x)$. Assume that the shopping cart starts at rest, and then is blown across a parking lot by the wind until it finally comes to rest after hitting a dumpster.
Read the scenario given and select a graph (a-f) that best models the situation.The weight of an individual ( $y$ ) versus the age $(x)$ of the individual over the individual's lifetime.
The graph shows average life expectancy for males in the United States based on an individual's birth year. The average life expectancy for a man born in 1975 is 68.8 yr. The average life expectancy for a man born in 1995 is 72.5 yr. Assuming a linear trend, estimate the life expectancy for a man born in 1985. (Source: U.S. National Center for Health Statistics, www.cdc.gov/nchs).
The city of Guayaquil in Ecuador is located on the coast and has an altitude of $10 \mathrm{~m}$ above sea level. The city of Quito, located in the mountains of Ecuador, has an altitude of $2810 \mathrm{~m}$ above sea level. The high temperature for each city is given for a day in November. Assuming a linear relationship between altitude and temperature, estimate the November high temperature for a city in Ecuador that is $1410 \mathrm{~m}$.
Determine if points $A, B,$ and $C$ are collinear. Three points are collinear if they all fall on the same line. There are several ways that we can determine if three points, $A, B,$ and $C$ are collinear. One method is to determine if the sum of the lengths of the line segments $\overline{A B}$ and $\overline{B C}$ equals the length of $\overline{A C}$.$$(2,2),(4,3), \text { and }(8,5)$$
Determine if points $A, B,$ and $C$ are collinear. Three points are collinear if they all fall on the same line. There are several ways that we can determine if three points, $A, B,$ and $C$ are collinear. One method is to determine if the sum of the lengths of the line segments $\overline{A B}$ and $\overline{B C}$ equals the length of $\overline{A C}$.$$(2,1.5),(4,2), \text { and }(8,3)$$
Determine if points $A, B,$ and $C$ are collinear. Three points are collinear if they all fall on the same line. There are several ways that we can determine if three points, $A, B,$ and $C$ are collinear. One method is to determine if the sum of the lengths of the line segments $\overline{A B}$ and $\overline{B C}$ equals the length of $\overline{A C}$.$$(-2,8),(1,2), \text { and }(4,-3)$$
Determine if points $A, B,$ and $C$ are collinear. Three points are collinear if they all fall on the same line. There are several ways that we can determine if three points, $A, B,$ and $C$ are collinear. One method is to determine if the sum of the lengths of the line segments $\overline{A B}$ and $\overline{B C}$ equals the length of $\overline{A C}$.$$(-1,5),(0,3), \text { and }(5,-13)$$
Suppose that $d$ represents the distance between two points $\left(x_{1}, y_{1}\right)$ and $\left(x_{2}, y_{2}\right) .$ Explain how the distance formula is developed from the Pythagorean theorem.
Explain how you might remember the midpoint formula to find the midpoint of the line segment between $\left(x_{1}, y_{1}\right)$ and $\left(x_{2}, y_{2}\right)$.
Explain how to find the $x$ - and $y$ -intercepts from an equation in the variables $x$ and $y$.
Given an equation in the variables $x$ and $y$, what does the graph of the equation represent?
A point in three-dimensional space can be represented in a three-dimensional coordinate system. In such a case, a $z$ -axis is taken perpendicular to both the $x$ - and $y$ -axes. A point $A$ is assigned an ordered triple $A(x, y, z)$ relative to a fixed origin where the three axes meet. For Exercises $91-94$, determine the distance between the two given points in space. Use the distance formula$d=\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}+\left(z_{2}-z_{1}\right)^{2}}$.(5,-3,2) and (4,6,-1)
A point in three-dimensional space can be represented in a three-dimensional coordinate system. In such a case, a $z$ -axis is taken perpendicular to both the $x$ - and $y$ -axes. A point $A$ is assigned an ordered triple $A(x, y, z)$ relative to a fixed origin where the three axes meet. For Exercises $91-94$, determine the distance between the two given points in space. Use the distance formula$d=\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}+\left(z_{2}-z_{1}\right)^{2}}$.(6,-4,-1) and (2,3,1)
A point in three-dimensional space can be represented in a three-dimensional coordinate system. In such a case, a $z$ -axis is taken perpendicular to both the $x$ - and $y$ -axes. A point $A$ is assigned an ordered triple $A(x, y, z)$ relative to a fixed origin where the three axes meet. For Exercises $91-94$, determine the distance between the two given points in space. Use the distance formula$d=\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}+\left(z_{2}-z_{1}\right)^{2}}$.(3,7,-2) and (0,-5,1)
A point in three-dimensional space can be represented in a three-dimensional coordinate system. In such a case, a $z$ -axis is taken perpendicular to both the $x$ - and $y$ -axes. A point $A$ is assigned an ordered triple $A(x, y, z)$ relative to a fixed origin where the three axes meet. For Exercises $91-94$, determine the distance between the two given points in space. Use the distance formula$d=\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}+\left(z_{2}-z_{1}\right)^{2}}$.(9,-5,-3) and (2,0,1)
In advanced courses, a complex number $a+b i$ is represented by an ordered pair $(a, b)$. In such a case, we can graph a complex number in a plane in which the horizontal axis represents the real part and the vertical axis represents the imaginary part. This is called the complex plane. Graph the complex number in the complex plane.a. $3+4 i$b. $-2-i$c. 5d. $4 i$
In advanced courses, a complex number $a+b i$ is represented by an ordered pair $(a, b)$. In such a case, we can graph a complex number in a plane in which the horizontal axis represents the real part and the vertical axis represents the imaginary part. This is called the complex plane. Graph the complex number in the complex plane.a. $2-3 i$b. $-4-i$c. 3d. $2 i$
The absolute value of a complex number $z=a+b i$ is defined as $|z|=\sqrt{a^{2}+b^{2}}$. Geometrically, this is the distance from the point $(a, b)$ in the complex plane to the origin. For Find the absolute value of the complex number.$$3-4 i$$
The absolute value of a complex number $z=a+b i$ is defined as $|z|=\sqrt{a^{2}+b^{2}}$. Geometrically, this is the distance from the point $(a, b)$ in the complex plane to the origin. For Find the absolute value of the complex number.$$-6+8 i$$
The absolute value of a complex number $z=a+b i$ is defined as $|z|=\sqrt{a^{2}+b^{2}}$. Geometrically, this is the distance from the point $(a, b)$ in the complex plane to the origin. For Find the absolute value of the complex number.$$2-7 i$$
The absolute value of a complex number $z=a+b i$ is defined as $|z|=\sqrt{a^{2}+b^{2}}$. Geometrically, this is the distance from the point $(a, b)$ in the complex plane to the origin. For Find the absolute value of the complex number.$$1+9 i$$
What is meant by a viewing window on a graphing device?
Which of the viewing windows would show both the $x$ - and $y$ -intercepts of the graph of $780 x-42 y=5460 ?$a. [-20,20,2] by [-40,40,10]b. [-10,10,1] by [-10,10,1]c. [-10,10,1] by [-10,150,10]d. [-10,10,1] by [-150,10,10]
Graph the equation with a graphing utility on the given viewing window.$$y=2 x-5 \text { on }[-10,10,1] \text { by }[-10,10,1]$$
Graph the equation with a graphing utility on the given viewing window.$$y=-4 x+1 \text { on }[-10,10,1] \text { by }[-10,10,1]$$
Graph the equation with a graphing utility on the given viewing window.$y=1400 x^{2}-1200 x$ on [-5,5,1] by [-1000,2000,500]
Graph the equation with a graphing utility on the given viewing window.$y=-800 x^{2}+600 x$ on [-5,5,1] by [-1000,500,200]
Graph the equations on the standard viewing window.a. $y=x^{3}$b. $y=|x|-9$
Graph the equations on the standard viewing window.a. $y=\sqrt{x+4}$b. $y=|x-2|$
Use a graphing device to create a table of values for the given values of $x$. Then identify the $x$ - and $y$ -intercepts shown in the table.$y=x^{3}-3 x^{2}-x+3$ for $x=-2,-1,0,1,2,3,4$
Use a graphing device to create a table of values for the given values of $x$. Then identify the $x$ - and $y$ -intercepts shown in the table.$y=x^{3}-x^{2}-4 x+4$ for $x=-3,-2,-1,0,1,2,3$