Graph Notes

Equation If a > 0 If a < 0 𝑦 = 𝑎 𝑥 Asymptote: x = 0 y = 0 𝑦 = 𝑎 𝑥2 Asymptote: x = 0 y = 0 𝑦 = 𝑎𝑥 1 2 𝑦 = 𝑎𝑥 1 3

𝑦2 = 𝑎𝑥 𝑦 = 𝑎𝑥3 + 𝑑 y intercept = (0, d) x intercept = ( √𝑑 𝑎 3 , 0) 𝑦 = 𝑎(𝑥 − 𝑝)(𝑥 − 𝑞)2 Touches x axis at x = q Crosses x axis at x = p Roots: (p, 0) and (q, 0) where p < q (p, 0) then (q, 0) Touches x axis at x = q Crosses x axis at x = p Roots: (p, 0) and (q, 0) where q < p (q, 0) then (p, 0) 𝑦 = 𝑎(𝑥 − 𝑝)(𝑥 − 𝑞)(𝑥 − 𝑟) 3 real and distinct roots

Logarithm Graphs Graph of y = ax, where a > 1. • Domain: (−∞, ∞) • Range: (0, ∞) • As value of x increases, value of y increases. • y intercept: NIL (Graph does not touch x axis, asymptote: y = 0) • x intercept: (0, 1) Graph y = ax, where a < 1. • Domain: (−∞, ∞) • Range: (0, ∞) • As value of x increases, value of y decreases. • y intercept: NIL • x intercept (0,1)

Graphical Transformation Equation Transformation Values 𝑦 = 𝑓(𝑥) + 𝑎 Graph translates parallel to y axis by a units. (x, y + a) → y values change 𝑦 = 𝑓(𝑥 + 𝑎) Graph translates parallel to x axis by –a units. (x + a, y) → x values change 𝑦 = 𝑓(𝑎𝑥) Graph is stretched parallel to the x axis by scale factor 1/a. (x/a, ay) → x values change 𝑦 = 𝑎𝑓(𝑥) Graph is stretched parallel to the y axis by scale factor a. (x, ay) → y values change 𝑦 = −𝑓(𝑥) Graph is reflected in the x axis. x, -y → y values change 𝑦 = 𝑓(−𝑥) Graph is reflected in the y axis. -x, y → x values change For transformations that affect y values, f(x) remains intact. For transformations that affect x values, change is within bracket.