Formulas - Evaluation 1

Question 1

Distance formula

Answer 1

d = √(X1-X2)²+(Y1-Y2)²

Question 2

Right angle triangle

Answer 2

a² + b² = c²

Question 3

Isosceles triangle

Answer 3

For triangle ABC AC = BC, 2 equal sides

Question 4

Midpoint formula

Answer 4

M: [ (X1+X2)/2 ] , [ (Y1+Y2)/2 ]

Question 5

Finding intercepts

Answer 5

x-int , set y = 0
y-int, set x = 0

Question 6

Square rule

Answer 6

x² = 36
x = ±6

Question 7

Symmetries

Answer 7

Respect to x-axis, if (x,y) is on the graph (x,-y) is too

Respect to y-axis, if (x,y) is on the graph (-x,y) is too
* Note: y-axis symmetry looks like parabola

Respect to origin, if (x,y) is on the graph (-x,-y) is too

Question 8

Circle equation

Answer 8

Standard form:
(x-h)² + (y-k)² = r²
where centre is (h,k) and r is radius

General form:
x² + y² + ax + by + c = 0
* Note: can be converted to standard form using completing the square method

Question 9

General graphing equations

Answer 9

y = mx + c
y - y1 = m(x - x1)
gradient = rise/run = (y1-y2)/(x1-x2)

Question 10

Function definition

Answer 10

a relation is called a function if any x-value from the domain is associated to only 1 y-value

Question 11

Different quotient

Answer 11

[ f(x+h) - f(x) ] / h

Question 12

Domain of a function

Answer 12

denominator can not be zero, results in undefined answer

Interval notation:
Df: (-∞, 0) U (0, ∞)
* [] -included , () -not included

Set notation:
Df {x/x≠0}

Question 13

Sum, difference, product, quotient of a function

Answer 13

(f+g)x = f(x) + g(x)
(f-g)x = f(x) - g(x)
(f.g)x = f(x) . g(x)
(f/g)x = f(x) / g(x) * where g(x) is not 0

Question 14

Even and odd functions

Answer 14

even:
f(x) = f(-x)
symmetric with respect to y-axis

odd:
f(-x) = -f(x)
symmetric with respect to origin

a function can not be symmetric with respect to the x-axis as then it can not be a function

Question 15

Increasing/decreasing/constant functions

Answer 15

the point up to which the function keeps its nature (inclusive)

Question 16

Local max/min

Answer 16

end points can never be local max/min

empty circle can’t be local max/min as it is not on the graph

notation:
x = _ , f(_) = __

Question 17

Absolute max/min

Answer 17

follows same notation as local max/min

can be end points

if graph goes to infinity (arrow not point at ends) can’t be max/min

Question 18

Average rate of change

Answer 18

gradient of line drawn from point to point

rate of change from a to b, means b-a (b first in formula)

Question 19

Piece wise functions

Answer 19

to find x-int test y = 0 with all conditions (pieces of the graph) whichever value satisfies the equation is the correct x-value

• Be careful while plotting with empty and full circles

Make a point of domain, range, inc/dec/const and symmetry

Question 20

Library functions (parent functions)

Answer 20

f(x) = b (flat line)
f(x) = x (straight line with gradient)
f(x) = x² (parabola)
f(x) = x³ (half upward parabola in ++ quadrant, half downward parabola in – quadrant)
f(x) = √x (half sideway parabola in ++ quadrant)
f(x) = ³√x (half sideway parabola in ++ quadrant, half sideway parabola in – quadrant)
f(x) = 1/x (demand curve in ++ quadrant, reflected in – quadrant)
f(x) = |x| (v graph only in positive y quadrants)

Question 21

Graphing transformations

Answer 21

vertical shift
y = f(x) + k, k > 0 shifts graph up, k < 0 shifts graph down
(x, y) -> (x, y+k)

horizontal shift
y = f(x+h), h > 0 shifts graph left, h < 0 shifts graph right
(x, y) -> (x-h, y)

vertical stretch/compression
y = a . f(x), a > 1 stretches graph, 0 < a < 1 compresses graph
(x, y) -> (x, a.y)

horizontal stretch/compression
y = f(a.x), a > 1 compresses graph, 0 < a < 1 stretches graph
(x, y) -> (1/a .x, y)

reflection about x-axis - multiply f(x) by -1

reflection about y-axis - multiply x by -1

Question 22

Quadratic equation forms

Answer 22

General form:
y= ax² + bx + c
- use completing the square to solve, whatever is added within brackets (half x coefficient squared) should be multiplied by a value and subtracted outside of brackets
- vertex (-b/2a, f(-b/2a) )

Standard form:
y = a(x - h)² + k
- vertex (h, k)
- axis of symmetry: x = h