Coordinate Geometry

Question 1

Coordinate plane graphic

Answer 1

See screenshot

Question 2

what is a segment?

Answer 2

a section of a line (that extends to infinity) between two points

Question 3

what is the slope of a line?

Answer 3

its a measure of the steepness of the line and is calculated as
m = (rise)/(run) = (y2-y1)/(x2-x1), where:

y2 = second y coordinate
y1 = first y coordinate
x2 = second x coordinate
x1 = first x coordinate

Question 4

what are the possible values for the slope of a line and in what situations do those occur?

Answer 4

see screenshot

Question 5

things to note about positively sloped lines:

Answer 5

see screenshot

Question 6

things to note about negatively sloped lines:

Answer 6

see screenshot

Question 7

things to note about horizontal (no slope) lines

Answer 7

see screenshot

Question 8

things to note about vertical lines (with undefined slope)

Answer 8

see screenshot

Question 9

the larger the absolute value of the slope the STEEPER or LESS STEEP the line?

Answer 9

steeper

*see screenshot

Question 10

what do all the components of y=mx+b mean?

Answer 10

y = y coordinate for a point on the line
x = the corresponding x coordinate for the point on the line
m = slope of the line
b = y intercept of the line

Question 11

for the y=mx+b formula, what is the x coordinate of every possible value for b (y intercept)?

Answer 11

it will always be (0,y)

Question 12

though not explicitly stated, is it possible to determine the x-intercept of a line in the form y=mx+b?

Answer 12

yes, this occurs at the point (x,0). Specifically, isolate x = (y-b)/m, then note that y=0 at the x intercept, so x = -b/m

Question 13

how to graph a line given a slope intercept equation?

Answer 13

ex: y=2x+4
choose a couple of values for x and find their corresponding y values algebraically. this creates a couple of ordered pairs from which you can graph the line

Question 14

can a linear equation with y and x be converted to slope intercept form?

Answer 14

yes, just rearrange it

Question 15

equations for horizontal and vertical lines

Answer 15

see screenshot

Question 16

how to tell if a point is on a line?

Answer 16

ex. is the point (2,3) on the line y = (1/2)x+2?

if the equality holds when substituting then yes it holds
3 = 2/2 + 2 -> 3=3

Question 17

what is the standard form of the equation of a line?

Answer 17

Ax+By = C, where a, b, and c are all constants

typically best to convert to slope intercept form:
y=-(A/B)*x+(C/B)

Question 18

does knowing that a line passes through a given point in the xy plane tell us much about that line?

Answer 18

no, since there are an infinite number of lines that pass through that point.

if we know one other point on that line though, we know everything about the line, since any two points define a line

Question 19

if you know one point on a line, what two things could you be given that would allow you to define the line?

Answer 19

1. the slope of the line, or the slope of a line that is perpendicular or parallel to the line
2. a second point on the line (could be the y-intercept, x-intercept, or any other point on the line)

Question 20

what is true about parallel lines?

Answer 20

same slope, different y intercepts -> they will never intersect

Question 21

what is unique about the product of slopes of perpendicular lines?

Answer 21

they are negative reciprocals, so their slopes multiply to -1

Question 22

reflections over the x-axis, y-axis, and origin

Answer 22

see screenshot

Question 23

how do you reflect a line segment across axes or the origin?

Answer 23

same as a single point, you just do it for both the endpoints

*see screenshot

Question 24

line segment reflection across origin example

Answer 24

see screenshot

Question 25

reflections over the line y=x

Answer 25

a point (a,b) reflected over line y=x is (b,a) *see screenshot

Question 26

reflections over the line y=-x

Answer 26

a point (a,b) reflected over line y=-x is (-b,-a) *see screenshot

Question 27

reflections over the line y=b

Answer 27

a point (x,y) reflected over line y=b is (x,2b-y) *see screenshot

Question 28

reflections over line x=a

Answer 28

a point (x,y) reflected over line x=a is (2a-x,y) *see screenshot

Question 29

additional reflections summary

Answer 29

see screenshot

Question 30

(euclidean) distance between two points?

Answer 30

d = sqrt((x2-x1)^2+(y2-y1)^2)

Question 31

if two points share some x or y coordinates we don't need the distance formula

Answer 31

if they share the same x coordinates, then distance is the abs val of difference between y coordinates if they share same y coordinates, then distance is abs val of difference in x coordinates

Question 32

what is Pythagorean theorem?

Answer 32

relates the length of the legs of a right triangle, where a and b are the two shorter legs and c is the longest leg (across from the 90 degree angle) c^2 = a^2 +b^2 -> c= sqrt(a^2 + b^2)

Question 33

how do you find the midpoint of a line segment?

Answer 33

see screenshot formula

Question 34

what is the equation for a circle centered at (a,b) drawn in the coordinate plane?

Answer 34

(x-a)^2+(y-b)^2 = r^2

Question 35

T/F: all the points on a circle are at equal distance, called radius, from a point in the plane, called the center

Answer 35

T

Question 36

what is the equation for a circle centered at the origin?

Answer 36

x^2+y^2 = r^2 (since x and y are 0) the full formula (x-a)^2+(y-b)^2 = r^2 basically just adds adjustments so its centered at origin

Question 37

when graphing inequalities always ensure equation is in slope intercept form and y is on left hand side of inequality

Answer 37

see screenshot

Question 38

when graphing greater than/less than inequalities, what form does the line take to indicate it is "greater than" or "less than"

Answer 38

dash, and the region of solutions is shaded

Question 39

what is the only difference when graphing greater than or equal to vs greater than?

Answer 39

solid line used on graph, and solutions on the line satisfy the inequality

Question 40

what is the only way for two points to be in the same quadrant?

Answer 40

two points (x1, y1) and (x2,y2) to be in the same quadrant are for both x1 and x2 to be the same sign and for y1 and y2 to be the same sign

Question 41

how do you reflect a whole line vs a point or set of points?

Answer 41

say we are trying to reflect a line across y=x. we know that to reflect a point (x,y) across y=x, we simply flip the coordinates, so its reflection is (y,x). we can do this for a line by simply switching the x and y variables. So the reflection of y =2x+3 across y=x is x=2y+3.

Question 42

what is a stationary point in reflections?

Answer 42

a point that is the same after being reflected that it was before