Formulas - Evaluation 1 Flashcards
Distance formula
d = √(X1-X2)²+(Y1-Y2)²
Right angle triangle
a² + b² = c²
Isosceles triangle
For triangle ABC AC = BC, 2 equal sides
Midpoint formula
M: [ (X1+X2)/2 ] , [ (Y1+Y2)/2 ]
Finding intercepts
x-int , set y = 0
y-int, set x = 0
Square rule
x² = 36
x = ±6
Symmetries
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
Circle equation
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
General graphing equations
y = mx + c
y - y1 = m(x - x1)
gradient = rise/run = (y1-y2)/(x1-x2)
Function definition
a relation is called a function if any x-value from the domain is associated to only 1 y-value
Different quotient
[ f(x+h) - f(x) ] / h
Domain of a function
denominator can not be zero, results in undefined answer
Interval notation:
Df: (-∞, 0) U (0, ∞)
* [] -included , () -not included
Set notation:
Df {x/x≠0}
Sum, difference, product, quotient of a function
(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
Even and odd functions
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
Increasing/decreasing/constant functions
the point up to which the function keeps its nature (inclusive)
Local max/min
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(_) = __
Absolute max/min
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
Average rate of change
gradient of line drawn from point to point
rate of change from a to b, means b-a (b first in formula)
Piece wise functions
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
Library functions (parent functions)
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)
Graphing transformations
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
Quadratic equation forms
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
