15) Formula logic and Coordinate geometry

Question 1

Types of questions asked in Formula Logic?

Answer 1

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

Question 2

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?

Answer 2

D2 / D1 = x

D2 = x * D1

Ans 8 —> pull out (1/2)^3 in the denominator

Question 3

If the radius of a circle is decreased by 10 percent, what will be the percentage change in the area of the circle?

Answer 3

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

Question 4

Coordinate pairs (ordered pair)

Answer 4

(x, y)

Every point has a unique set of coordinates consisting of an x and y value

Question 5

Labeling quadrants and what points in each quadrant tell you

Answer 5

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)

Question 6

Creating a line segment

Answer 6

Requires at least two points to create a line

A line segment has a finite length, while a line theoretically extends to infinity

Question 7

When are two points in the same quadrant?

Answer 7

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)

Question 8

Formula for calculating slope of a line

What can the value of slope be?

Answer 8

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)

Question 9

Lines with positive slopes and quadrants

Answer 9

** 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

Question 10

Lines with negative slopes and quadrants

Answer 10

***** 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

Question 11

Lines with 0 slope

Answer 11

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

Question 12

Lines with undefined slope

Answer 12

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

Question 13

Steepness of slope

Answer 13

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)

Question 14

Slope-intercept equation

Answer 14

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

Question 15

What data is SUFFICIENT to fixate a line?

Answer 15

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)

Question 16

Equation for horizontal and vertical lines?

Answer 16

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

Question 17

Checking if a point is on a line

Answer 17

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

Question 18

What is the “Standard form” or “general form” equation of a line

Answer 18

Ax + By = C, where A, B and C are all constants

> best to convert the equation to slope-intercept point

Question 19

Perpendicular lines

Answer 19

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)

Question 20

Parallel lines

Answer 20

Lines with SAME SLOPE but DIFFERENT x and y intercepts

Implication:
> Parallel lines NEVER INTERSECT

Question 21

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

Answer 21

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)

Question 22

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

Answer 22

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)

Question 23

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

Answer 23

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’)

Question 24

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

Answer 24

Reflect EACH of the ENDPOINTS of the line segment

Same thing applies to polygon like a triangle —> reflect each VERTEX then CONNECT the dots

Question 25

Reflections of a point, a figure, a line, or a line segment OVER a line or a point:

Reflection over y = x

Answer 25

y = x —> line with slope 1 and y int = 0

(x, y) —> (y, x)

Flip x and y

Question 26

Reflections of a point, a figure, a line, or a line segment OVER a line or a point:

Reflection over y = -x

Answer 26

y = -x —> line with slope -1 and y int = 0

(x, y) —> (-y, -x)

Flip x and y
Also change sign of each

Question 27

Reflections of a point, a figure, a line, or a line segment OVER a line or a point:

Reflection over y = b

Answer 27

y = b —> horizontal line

(x, y) —-> (x, 2b - y)

Same x, y changes by 2b - y

e.g., point (5, -2) reflected over y = 1 becomes (5, 4)

OR VISUAL APPROACH –> equal distance from the reflection line

PAY ATTENTION TO x and y values

Question 28

Reflections of a point, a figure, a line, or a line segment OVER a line or a point:

Reflection over x = a

Answer 28

x = a —> vertical line

(x, y) —> (2a - x, y)

Same y, x changes by 2a - x

e.g., point (5, -2) reflected over x = 2 becomes (-1, -2)

OR VISUAL APPROACH –> equal distance from the reflection line

PAY ATTENTION TO x and y values

Question 29

Distance between two points

Answer 29

Distance = sqrt[ change x^2 + change y^2]
= sqrt[ (x2-x1)^2 + (y2-y1)^2 ]

Derived from the pythagorean theorem for right triangles
> pay attention to x2 vs x1 and y2 vs y1 (order matters)

”+” (not like difference of squares)

HOWEVER, if two points share an “x” or “y” coordinate –> short cut
> ABSOLUTE difference between different coordinate

e.g., share x coordinate —> look at change in y coordinates
e.g., share y coordinate –> look at change in x coordinates

Question 30

A square is inscribed in a circle that has a diameter with endpoints (3, 4) and (3, -2). What is the area of the square?

Answer 30

Geometry rules:

Square can be comprised of 2 isosceles triangles with sides following this ratio:
1x: 1x: x*sqrt(2)

So once you know x*sqrt(2) = distance, then you can solve for x^2 = area of square

Since the two points share x coordinate, the distance between the two points is | 4 - -2 | = 6

therefore x^2 = 18

Question 31

Calculating area of right triangles in xy plane

Answer 31

Plot the three vertexes to identify which two points create the hypotenuse

Question 32

Midpoint of a line segment formula

Answer 32

Trying to find (x, y) of MIDPOINT: find the average of each coordinate

x coordinate = (x1 + x2)/2
y coordinate = (y1 + y2)/2

Question 33

Line segment k, with a slope of zero, has endpoints (a, b) and (c, d). If the midpoint of line segment k is (10, 2), what is the value of a+b?

(1) d = 2
(2) a = c-2

Answer 33

Coordinate geometry
> always draw a picture so you can see slopes and hidden info
> in this case, line segment k has 0 slope so it is HORIZONTAL —-> endpoints have the SAME Y value

b = d = 2 based on the midpoint

(a+c)/2 = 10
a = 20 - c

Therefore, a + b = a + 2
= 20 - c + 2 = 22 - c

So we need value of a or value of c

(1) NS –> we already know this

(2) a = c - 2 = 20 - c —-> 1 var can solve for c
2c = 22
c = 11

so a = 11

Sufficient

B

Question 34

Equation for a circle on the coordinate plane

Center at point (a, b) and radius r

Answer 34

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

Might not need to expand equation PS

Derived from DISTANCE formula between center point (a, b) and any point on the circumference of the circle (x, y) = radius

A circle with a center at the ORIGIN (0, 0) will have equation

x^2 + y^2 = r^2

Question 35

Graphing inequalities and their solution set

e.g., y > 2x + 4
e.g., 2x + 4 < y

Answer 35

Important notes:
> all inequalities HAVE TO BE in slope-intercept form (must have y isolated on the LEFT side of the inequality)
e.g., 2x + 4 < y —-> y > 2x + 4 —-> looking for region above 2x+4

Graphing inequalities:
(1) make sure you express as slope-intercept form

(2) graph normal linear equation first
e.g., y = 2x+4

(3) Turn line into inequality using DASHED LINES (for < >) or SOLID LINES (for <= >=) and SHADE region of applicable points
> there’s an infinite number of points that satisfy this inequality

What does this mean?
y > 2x +4 tells us that for ANY value of x, the value of y in our solution set will always be greater than the value of 2x+4

Similarly, region y < 2x + 4 tells us that for any value of x, the value of y in our solution set will always be less than the value of 2x + 4

THEREFORE GOAL IS TO SATISFY THE INEQUALITY

Question 36

In the xy-plane, region R is defined by the inequality: y > 6x + 8. Which of the following points could be a solution to the inequality?

(2, 0)
(4, 12)
(30, 40)
(10, 50)
(4, 33)

Answer 36

Concept:
> for any value of x, our solution set must be greater than 6x + 8

Plug in values of x into 6x + 8 and pick answer that has y value GREATER than “manufactured” 6x+8 value

Ans (4, 33)

6x+8 when x=4 equals 32. Manufactured y (32) is LOWER than 33 provided

In other words:

33 > 6*4 + 8 (Yes)

Question 37

Dealing with multiple inequalities in coordinate plane

Answer 37

Remember where the solution set of an inequality lies:

“Less than” inequality –> solution set lies only below the line
“Less than or equal to” inequality –> solution set lies ON the line AND below the line
“Greater than” inequality –> solution set lies only above the line
“Greater than or equal to” inequality –> solution set lies ON the line AND ABOVE the line

Question 38

Graphing absolute value functions

e.g., | x | + | y | = 5

Answer 38

Easiest thing to do is to PLOT THE X and Y INTERCEPTS first

e.g., x = 0, y = +/-5
y = 0, x = +/-5