2.1 The Rectangular Coordinate Systems and Graphs

Find the distance between two points: $\left(1 , 4\right)$ and $\left(11 , 9\right) .$

Find the midpoint of the line segment with the endpoints $\left(7 , −2\right)$ and $\left(9 , 5\right) .$

Find the midpoint of the line segment with endpoints $\left(−2 , −1\right)$ and $\left(−8 , 6\right) .$

The diameter of a circle has endpoints $\left(−1 , −4\right)$ and $\left(5 , −4\right) .$ Find the center of the circle.

Is it possible for a point plotted in the Cartesian coordinate system to not lie in one of the four quadrants? Explain.

Describe in your own words what the y-intercept of a graph is.

For each of the following exercises, find the x-intercept and the y-intercept without graphing. Write the coordinates of each intercept.

$y = −3 x + 6$

For each of the following exercises, find the x-intercept and the y-intercept without graphing. Write the coordinates of each intercept.

$3 x - 2 y = 6$

For each of the following exercises, find the x-intercept and the y-intercept without graphing. Write the coordinates of each intercept.

$3 x + 8 y = 9$

For each of the following exercises, solve the equation for y in terms of x.

$4 x + 2 y = 8$

For each of the following exercises, solve the equation for y in terms of x.

$2 x = 5 - 3 y$

For each of the following exercises, solve the equation for y in terms of x.

$5 y + 4 = 10 x$

For each of the following exercises, find the distance between the two points. Simplify your answers, and write the exact answer in simplest radical form for irrational answers.

$\left(−4 , 1\right)$ and $\left(3 , −4\right)$

For each of the following exercises, find the distance between the two points. Simplify your answers, and write the exact answer in simplest radical form for irrational answers.

$\left(5 , 0\right)$ and $\left(5 , 6\right)$

For each of the following exercises, find the distance between the two points. Simplify your answers, and write the exact answer in simplest radical form for irrational answers.

Find the distance between the two points given using your calculator, and round your answer to the nearest hundredth.

$\left(19 , 12\right)$ and $\left(41 , 71\right)$

For each of the following exercises, find the coordinates of the midpoint of the line segment that joins the two given points.

$\left(−1 , 1\right)$ and $\left(7 , −4\right)$

For each of the following exercises, find the coordinates of the midpoint of the line segment that joins the two given points.

$\left(0 , 7\right)$ and $\left(4 , −9\right)$

For each of the following exercises, identify the information requested.

What are the coordinates of the origin?

For each of the following exercises, identify the information requested.

If a point is located on the x-axis, what is the y-coordinate?

Find the distance between the two endpoints using the distance formula. Round to three decimal places.

Find the distance that $\left(−3 , 4\right)$ is from the origin.

Which point is closer to the origin?

After graphing it, use the 2^nd CALC button and 1:value button, hit enter. At the lower part of the screen you will see “x=” and a blinking cursor. You may enter any number for x and it will display the y value for any x value you input. Use this and plug in x = 0, thus finding the y-intercept, for each of the following graphs.

$\text{Y}_{1} = \frac{3 x - 8}{4}$

Note: With linear/straight line functions the zero is not really a “guess,” but it is necessary to enter a “guess” so it will search and find the exact x-intercept between your right and left boundaries. With other types of functions (more than one x-intercept), they may be irrational numbers so “guess” is more appropriate to give it the correct limits to find a very close approximation between th…

Note: With linear/straight line functions the zero is not really a “guess,” but it is necessary to enter a “guess” so it will search and find the exact x-intercept between your right and left boundaries. With other types of functions (more than one x-intercept), they may be irrational numbers so “guess” is more appropriate to give it the correct limits to find a very close approximation between th…

If the road was made in the previous exercise, how much shorter would the person’s one-way trip be every day?

After finding the two midpoints in the previous exercise, find the distance between the two midpoints to the nearest thousandth.

In the previous exercise, find the coordinates of the midpoint for each diagonal.

If San Jose’s coordinates are $\left(76 , –12\right)$, where the coordinates represent miles, find the distance between San Jose and San Francisco to the nearest mile.