solving for y (0,y)

solving for x (x,0)

symetry towards the:

x-axsis- relace y with -y

y-axsis- replace x with -x

origin- replace x and y with -x,-y

*if an equivilant equation results it is symetric to that axsis 

f(-x)= -f(x)

if and only if the graph is symetrical with respect to the origin

f(-x)= f(x)

if and only if the graph is symetrical with respect to the y-axsis

f(x)= x²-3 (left)

f(x)= x²+3 (right)

f(x)= /x/+2 up

f(x)= √(x+3) left

f(x)= (x+3)³ left

If a > 0 then the vertex (h; f(h)) is the minimum value for f(x) on the graph

If a < 0 then the vertex (h; f(h)) is the maximum value for f(x) on the graph

The values for which f(x)  0 or f(x) < 0 fall beneath the x-axis

The value for which f(x)  0 or f(x) > 0 rest above the x-axis

the highest power of x present amongst all terms

within the polynomial

degree n is of the form f(x) = axn where a 6= 0 is a real number

and n > 0 is an integer

The graph of odd powered functions are s-shaped in nature, becoming ever more angular as the power increases

The graph of even powered functions are parabolic in nature, becoming ever more angular as the power increases

For an even function f(x) = axn, if a > 0, then the parabola opens upwards,

whereas if a < 0, then the parabola opens down.

For an odd function, if a > 0, then the s-shaped graph goes "uphill" in reading

direction, whereas if < 0, then the graph tends "downhill" in reading direction

For m odd, the graph cuts through the x-axis at x = r

For m even, the graph touches the x-axis at x = r

i. If m = n; then the horizontal asymptote of R(x) is the line y = an

bm

ii. If m > n; then the horizontal asymptote of R(x) is the line y = 0

ii. If m < n; there is no horizontal asymptote

If a vertical asymptote has even multiplicity, then the graph approaches the

asymptote in the same direction from both sides

If a vertical asymptote has odd multiplicity, then the graph approaches the

asymptote in opposite directions from both sides

We can think of a 1-1 function in the sense that for each y-value, there is a unique corresponding x-value, no repeating y values

use the horixontal line test 

An exponential function is of the form f(x) = ax, where a > 0 and a 6= 1

If 0 < a < 1, then the graph of f(x) = ax decreases in reading direction

If a > 1, then the graph of f(x) = ax increases in reading direction

i The points (a1; a^-a); (a; 1); and (1; a) are on the graph of f(x)

ii There are no x-intercepts

iii the Horizontal Asymptote is the line y = 0 i.e. the x-axis

iv For 0 < a < 1, the graph is always decreasing and 1 - 1 (i.e. invertible)

For a > 1, the graph is always increasing and 1 - 1 (i.e. invertible)

v The domain is all real numbers; (∞1;1)

The range is all positive numbers; (0;1)

i loga 1 = 0 because a0 = 1 for all a > 0

ii loga a = 1 because a1 = a

iii aloga x = x because loga x is the exponent to which a must be raised to get x

iv loga ax = x because ax = ax                                                                   

i loga(MN) = logaM + loga N

ii loga(M

(x - h)² = 4a(y - k) (h; k) (h; k + a) y = k - a opens up

(x - h)² = -4a(y - k) (h; k) (h; k - a) y = k + a opens down

(y - k)² = 4a(x - h) (h; k) (h + a; k) x = h - a opens right

(y - k)²= -4a(x - h) (h; k) (h - a; k) x = h + a opens left

a function whose domain is a subset of the positive integers,

denoted {a_n}