1. Translation
1.1. along x-axis f(x-a)
1.1.1. by a units
1.1.2. Graph
1.1.2.1. Asymptote : x=h -> x=h+a
1.1.2.2. point: (x,y) -> (x+a,y)
1.2. along y-axis f(x) + a
1.2.1. by a units
2. Scaling
2.1. parallel to x-axis f(ax)
2.1.1. scale factor, 1/a
2.1.1.1. If a is negative
2.1.1.1.1. reflect in y-axis
2.1.2. Graph
2.1.2.1. Asymptote : x=h -> x=ah
2.1.2.2. point: (x,y) -> (ax,y)
2.1.2.3. No change if point is on y-axis
2.2. parallel to y-axis af(x)
2.2.1. scale factor,a
2.2.1.1. If a is negative
2.2.1.1.1. reflect in x-axis
2.2.2. Graph
2.2.2.1. Asymptote : y=k -> y=ak
2.2.2.2. point: (x,y) -> (x,ay)
2.2.2.3. No change if point is on x-axis
3. Multiple d + cf(bx+a)
3.1. f(bx+a)
3.1.1. 2 ways
3.1.1.1. translate by -a units along x-axis
3.1.1.1.1. scale parallel to x-axis by factor 1/b
3.1.1.2. scale parallel to x-axis by factor 1/b
3.1.1.2.1. translate by -a/b units along x-axis
4. Modulus function
4.1. |f(x)|
4.1.1. reflect those below x-axis in x-axis
4.2. f(|x|)
4.2.1. remove those on left of y-axis
4.2.1.1. reflect those on right of y-axis in y-axis
5. Reciprocal function
5.1. 1 / f(x)
5.1.1. x-intercepts, (a,0)
5.1.1.1. vertical asymptote, x = a
5.1.2. y-intercept (0, b)
5.1.2.1. y-intercept (0, 1/b)
5.1.3. vertical asymptotes, x = a
5.1.3.1. x-intercepts, (a,0)
5.1.4. horizontal asymptote, y = c
5.1.4.1. horizontal asymptote, y = 1/c
5.1.5. oblique asymptote, y =cx + d
5.1.5.1. horizontal asymptote, y = 0
5.1.6. maximum (a,b)
5.1.6.1. minimum (a, 1/b)
5.1.7. minimum (a, b)
5.1.7.1. maximum (a, 1/b)
6. y-squared
6.1. remove f(x) below x-axis
6.1.1. x-intercepts, (a,0)
6.1.1.1. x-intercepts, (a,0)
6.1.2. y-intercept (0, b)
6.1.2.1. y-intercept (0, sqrt(b))
6.1.3. vertical asymptotes, x = a
6.1.3.1. vertical asymptotes, x = a (if curve approaches asymptote)
6.1.4. horizontal asymptote, y = c
6.1.4.1. horizontal asymptote, y = sqrt(c)
6.1.5. maximum (a,b)
6.1.5.1. maximum (a, sqrt(b))
6.1.6. minimum (a, b)
6.1.6.1. minimum (a, sqrt(b))
6.1.7. reflect in x-axis