Chapter 1 Main Ideas

Even Symmetry

Symmetry with y-axis

Ex: y = x², y = cos(x), y = |x|

Odd Symmetry

Point symmetry with origin
Ex: y = x, y = x³, y = sin(x)

The Sum of a Function

(f + g)(x) = f(x) + g(x)

The Difference of a Function

(f - g)(x) = f(x) - g(x)

The Product of a Function

(f*g)(x) = f(x)*g(x)

The Quotient of a Function

(f/g)(x) = f(x)/g(x), g(x) ≠ 0

Implicitly Defined Function

a function that is presented as the solution of some equation or system of equations, rather than being given by an explicit formula

ex: x² + y² = 4 → y = +(√4 - x²) & y = -(√4 - x²)

Inverse Relations

When y is replaced with x and vice versa and solved for y; the points of these functions will be symmetric with respect to the line y = x and will reflect across that line

ex: f⁻¹(b) = a if f(a) = b

Horizontal Translation Left by c units

y = f(x + c)

Horizontal Translation Right by c units

y = f(x – c)

Vertical Translation Up by c units

y = f(x) + c

Vertical Translation Down by c units

y = f(x) – c

Reflection across the x-axis

y = –f(x)

ex: y = 2x - 1 → y = -(2x - 1)

Reflection across the y-axis

y = f(–x)

ex: y = 2x - 1 → y = 2(-x) - 1

Horizontal Stretch

y = f(x/c) if c > 1

Horizontal Shrink

y = f(x/c) if c < 1

Vertical Stretch

y = c * f(x) if c > 1

Vertical Shrink

y = c * f(x) if c < 1