15) Formula logic and Coordinate geometry Flashcards
Types of questions asked in Formula Logic?
Formula Logic Qs ask us to understand the impact of a HYPOTHETICAL CHANGE to A FORMULA
> similar to proportions qs
(1) By what factor has the formula’s result changed? (FACTOR CHANGE)
> keep track of the ORIGINAL formula’s value in order to understand the “what factor” impact
> Factor = New value of formula / Old value of formula
(2) What will the percent change be if “x” occurs? (PERCENT CHANGE)
> percent change in the new outcome of a formula compared to the original outcome of a formula
Can either solve algebraically or using smart numbers
> algebraically –> rewrite new value in terms of old variables
Ex: On a certain planet, the density D can be expressed by the equation: D = 1/(3k)^3, where k is the radius of the planet. If k is halved, the density D would change by what factor?
D2 / D1 = x
D2 = x * D1
Ans 8 —> pull out (1/2)^3 in the denominator
If the radius of a circle is decreased by 10 percent, what will be the percentage change in the area of the circle?
Area of circle: pi * r^2
R2 = 0.9*r
Percent change formula: (area 2 - area1)/area 1 * 100
= (0.81r^2 - r^2)/r^2 * 100
= -19%
Ans: 19 percent
Coordinate pairs (ordered pair)
(x, y)
Every point has a unique set of coordinates consisting of an x and y value
Labeling quadrants and what points in each quadrant tell you
Top right is quad 1
Count going counterclockwise:
Top left: Quad 2
Bottom left: Quad 3
Bottom right: Quad 4
Few things to note:
> Quadrants are indicative of SIGNS of the ordered pair
> Quad 1 and Quad 3: x and y have the SAME SIGN (xy > 0 or x/y > 0)
> Quad 2 and Quad 4: x and y have OPPOSITE signs (xy < 0 or x/y < 0)
Creating a line segment
Requires at least two points to create a line
A line segment has a finite length, while a line theoretically extends to infinity
When are two points in the same quadrant?
x coordinates have the same sign AND y coordinates have the same sign
product of x coordinates > 0
AND
product of y coordinates > 0
(but don’t need x and y to necessarily have the same sign)
Formula for calculating slope of a line
What can the value of slope be?
slope m = Rise / Run = (Y2 - Y1) / (X2 - X1)
Positive slope = upward sloping
Negative slope = downward sloping
Value of slope of a line can be:
> 0 (horizontal) -> y = #
> positive
> negative
> undefined (Vertical line has undefined slope)
Lines with positive slopes and quadrants
** Rule 1: All positively sloped lines MUST cross through Quadrants I and III (though it may or may not intersect II and IV)
Rule 2: If the x intercept of a positively sloped line is negative, then it’s y-intercept WILL BE POSITIVE AND the line will intersect Quadrant II
Rule 3: If the x-intercept of a positively sloped line is 0, then it’s y-intercept is also 0 AND the line only passes through Quadrants I and III
> x-intercept and y-intercept have OPPOSITE signs
Rule 4: If the x-intercept of a positively sloped line is positive, then its y-intercept WILL BE NEGATIVE AND the line will intersect Quadrant IV
> x-intercept and y-intercept have OPPOSITE signs
How to check these rules?
> shifting a positively sloped line
Lines with negative slopes and quadrants
***** Rule 1: All negatively sloped lines MUST cross through Quadrants II and IV (though it may or may not intersect I and III)
Rule 2: If the x intercept of a negatively sloped line is negative, then it’s y-intercept WILL ALSO be negative AND the line will intersect Quadrant III
Rule 3: If the x-intercept of a negatively sloped line is 0, then it’s y-intercept is also 0 AND the line only passes through Quadrants II and IV
> All negatively sloped lines have an x-intercept and y-intercept WITH THE SAME SIGN
Rule 4: If the x-intercept of a negatively sloped line is positive, then its y-intercept WILL ALSO be positive AND the line will intersect Quadrant I
> All negatively sloped lines have an x-intercept and y-intercept WITH THE SAME SIGN
Lines with 0 slope
Horizontal lines - not mandatory quadrants it has to intersect
Rule 1: If y intercept of a horizontal line is POSITIVE, then the line intersects Quadrants I and II
Rule 2: If y intercept of a horizontal line is zero, then the line is the x-axis and does not pass through any of the quadrants
> only time the horizontal line intersects with the x axis
Rule 3: If y intercept of a horizontal line is negative, then the line intersects quadrants III and IV
Horizontal lines also have points that all have the SAME Y COORDINATE
> helpful for DS when given two points with unknown coordinates
(a, b) and (c, d) —> b=d
Lines with undefined slope
Undefined lines (zero nominator) - not mandatory quadrants it has to intersect
Rule 1: If x intercept of a vertical line is POSITIVE, then the line intersects Quadrants I and IV
Rule 2: If x intercept of a vertical line is zero, then the line is the y-axis and does not pass through any of the quadrants
> only time the vertical line intersects with the y axis
Rule 3: If x intercept of a vertical line is negative, then the line intersects quadrants II and III
Vertical lines also have points that all have the SAME X COORDINATE
Steepness of slope
Compare ABSOLUTE Value of slope
> The larger the absolute value of the slope of a line = the STEEPER the line
e.g. m = -3 —> read as “one over, 3 down” (NOT 3 left, one up)
Slope-intercept equation
y=mx+b
y = y coordinate for a point on the line
m = slope
x = corresponding x coordinate for a point on the line
b = y intercept of the line –> when x = 0
x intercept –> (x, 0)
Set y equal to zero to calculate x intercept
> just need to know (1) Slope of the line (2) y intercept
Working with the slope-intercept equation:
> need to REARRANGE and ISOLATE variable y in order to DETERMINE slope of the line and y intercept (if equation is presented in general form)
e.g., 3x + 5y = 8
What data is SUFFICIENT to fixate a line?
Aka to create an equation for a line
(1) two points
> can calc slope
> can calc y and x intercept
(2) Slope and one point
> incl. slope of a PARALLEL or perpendicular line
If you just have one point, NS (can rotate line about that point)
Equation for horizontal and vertical lines?
Equation for horizontal line is y = b —-> y intercept is b
> recognize special horizontal line is the x axis, y = 0
Equation for vertical line is x = a —-> x intercept is a
> recognize special vertical line is y axis, x = 0
Checking if a point is on a line
Sub in the value of x and y of the point INTO the equation of the line —> if it WORKS, then Yes, the point is on the line
Therefore, if some point A is on some line L, the x and y coordinates of point A MUST KEEP THE EQUATION OF LINE L in equality
for DS –> just need to see if you can plug in point into a full y=mx+b equation
What is the “Standard form” or “general form” equation of a line
Ax + By = C, where A, B and C are all constants
> best to convert the equation to slope-intercept point
Perpendicular lines
Product of slopes = -1
Implication:
> Slopes are NEGATIVE RECIPROCALS
e.g., 1 and -1
e.g., 2 and -1/2
e.g., 1/4 and -4
> Lines cross at one point and meet at a RIGHT ANGLE (creates 4 right angles)
Parallel lines
Lines with SAME SLOPE but DIFFERENT x and y intercepts
Implication:
> Parallel lines NEVER INTERSECT
Reflections of a point, a figure, a line, or a line segment OVER a line or a point:
Reflection of a POINT (x, y) over x axis
Same x coordinate, flip sign of y coordinate
(x, y) —> (x, -y)
e.g., (2, 3) —> (2, -3)
Implication:
> point and its reflection point are the SAME DISTANCE from the object being reflected over
> aka, the MIDPOINT between (x,y) and its reflection is on the x axis, y = 0
Notation: point A and A prime (A’)
ALSO: broader formula (y=b)
(x, y) —> (x, 2b - y)
Reflections of a point, a figure, a line, or a line segment OVER a line or a point:
Reflection of a POINT (x, y) over y axis
Same y coordinate, flip sign of x coordinate
(x, y) —> (-x, y)
e.g., (2, 3) –> (-2, 3)
Implication:
> point and its reflection point are the SAME DISTANCE from the object being reflected over
> aka, the MIDPOINT between (x,y) and its reflection is on the y axis, x = 0
Notation: point A and A prime (A’)
ALSO: broader formula (x=a)
(x, y) —> (2a - x, y)
Reflections of a point, a figure, a line, or a line segment OVER a line or a point:
Reflection of a POINT (x, y) over origin
Origin is a POINT –> change signs of BOTH x and y coordinate
(x, y) —> (-x, -y)
e.g., (2, 3) –> (-2, -3)
Implication:
> point and its reflection point are the SAME DISTANCE from the object being reflected over (origin is the midpoint)
Notation: point A and A prime (A’)
Reflections of a point, a figure, a line, or a line segment OVER a line or a point:
Reflection of a LINE SEGMENT AB over x axis
Reflect EACH of the ENDPOINTS of the line segment
Same thing applies to polygon like a triangle —> reflect each VERTEX then CONNECT the dots
