GRAPHS AND TRANSFORMATIONS

if p is a root of the function f(x)

the graph y=f(x) crosses the x axis at (p,0)

the graph y=k/x and y=k/x^2, where k is a real constant…

…have asymptotes at x=0 and y=0

the solutions of f(x)=g(x)

the x coordinates where f(x) and g(x) intercept

y=f(x)+a

is y=f(x) translated by the vector (0/a)

translating a function….

…translates any and all asymptotes

y=f(x+a)

is y=f(x) translated by the vector (-a/0)

y=af(x)

is y=f(x) stretched by a scale factor of a in the vertical direction (y axis)

y=f(ax)

is y=f(x) stretched by a scale factor of 1/a in the horizontal direction (x axis)

y=-f(x)

is y=f(x) reflected in the x axis

y=f(-x)

is y=f(x) reflected in the y axis

ax³+bx²+cx+d

where a is positive in the cubic

-ax³+bx²+cx+d

where a is negative in the cubic

ax⁴+bx³+cx²+d^x+e

where a is positive in the quartic

-ax⁴+bx³+cx²+d^x+e

where a is negative in the quartic

y=k/x

k>0, positive, in linear

y=k/x

k<0, negative, in linear

y=k/x²

k>0, positive, in quadratic

y=k/x²

k<0, negative, in quadratic

y=x^1/2

coefficient of x is larger curve gets steeper (5)

y=-x^1/2

reflect curve in x axis

y=x^-1/2

coefficient of x is larger curve gets abit higher (5)

y=-x^-1/2

reflect in x axis

y=x^1/3

y=-x^1/3 reflect in y axis