Coordinate Geometry Flashcards
Coordinate plane graphic
See screenshot
what is a segment?
a section of a line (that extends to infinity) between two points
what is the slope of a line?
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
what are the possible values for the slope of a line and in what situations do those occur?
see screenshot
things to note about positively sloped lines:
see screenshot
things to note about negatively sloped lines:
see screenshot
things to note about horizontal (no slope) lines
see screenshot
things to note about vertical lines (with undefined slope)
see screenshot
the larger the absolute value of the slope the STEEPER or LESS STEEP the line?
steeper
*see screenshot
what do all the components of y=mx+b mean?
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
for the y=mx+b formula, what is the x coordinate of every possible value for b (y intercept)?
it will always be (0,y)
though not explicitly stated, is it possible to determine the x-intercept of a line in the form y=mx+b?
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
how to graph a line given a slope intercept equation?
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
can a linear equation with y and x be converted to slope intercept form?
yes, just rearrange it
equations for horizontal and vertical lines
see screenshot
how to tell if a point is on a line?
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
what is the standard form of the equation of a line?
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)
does knowing that a line passes through a given point in the xy plane tell us much about that line?
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
if you know one point on a line, what two things could you be given that would allow you to define the line?
- the slope of the line, or the slope of a line that is perpendicular or parallel to the line
- a second point on the line (could be the y-intercept, x-intercept, or any other point on the line)
what is true about parallel lines?
same slope, different y intercepts -> they will never intersect
what is unique about the product of slopes of perpendicular lines?
they are negative reciprocals, so their slopes multiply to -1
reflections over the x-axis, y-axis, and origin
see screenshot
how do you reflect a line segment across axes or the origin?
same as a single point, you just do it for both the endpoints
*see screenshot
line segment reflection across origin example
see screenshot
reflections over the line y=x
a point (a,b) reflected over line y=x is (b,a)
*see screenshot
reflections over the line y=-x
a point (a,b) reflected over line y=-x is (-b,-a)
*see screenshot
reflections over the line y=b
a point (x,y) reflected over line y=b is (x,2b-y)
*see screenshot
reflections over line x=a
a point (x,y) reflected over line x=a is (2a-x,y)
*see screenshot
additional reflections summary
see screenshot
(euclidean) distance between two points?
d = sqrt((x2-x1)^2+(y2-y1)^2)
if two points share some x or y coordinates we don’t need the distance formula
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
what is Pythagorean theorem?
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)
how do you find the midpoint of a line segment?
see screenshot formula
what is the equation for a circle centered at (a,b) drawn in the coordinate plane?
(x-a)^2+(y-b)^2 = r^2
T/F: all the points on a circle are at equal distance, called radius, from a point in the plane, called the center
T
what is the equation for a circle centered at the origin?
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
when graphing inequalities always ensure equation is in slope intercept form and y is on left hand side of inequality
see screenshot
when graphing greater than/less than inequalities, what form does the line take to indicate it is “greater than” or “less than”
dash, and the region of solutions is shaded
what is the only difference when graphing greater than or equal to vs greater than?
solid line used on graph, and solutions on the line satisfy the inequality
what is the only way for two points to be in the same quadrant?
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
how do you reflect a whole line vs a point or set of points?
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.
what is a stationary point in reflections?
a point that is the same after being reflected that it was before
