Transformation

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Transformation by

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