5d Graphing Techniques

Question 1

Graph Sketching

Basis

Answer 1

S: Shape + Stationary Points
A: Asymptotes
I: Intercepts
L: Label graphs

Acronym: SAIL

Question 2

Rational Functions, y = ax + b/ cx + d

Sketching

where c ≠ 0, numerator and denominator have no common factor

Answer 2

1. Express y in p + q/ (cx + d) through long division
   a. Vertical asymptote: x = -d/ c
   b. Horizontal asymptote: y = p

Question 3

Rational Functions, y = ax² + bx + c/ dx + e

Sketching

where a ≠ 0, d ≠ 0, numerator and denominator have no common factor

Answer 3

1. Express y in the form (px + q) + r/ (dx + e)
   a. Oblique asymptote: y = px + q
   b. Vertical asymptote: y = -e/ d

Question 4

Conics

Parabola

General Formula

Answer 4

(x - h)² = a(y - k)
OR
(y - k)² = a(x - h)

Vertex (h, k)

Question 5

Conics

Circle

General Formula

Answer 5

(x - h)² + (y - k)² = r²

Radius = r

Centre (h, k)

Question 6

Conics

Ellipse

General Formula

Answer 6

(x - h)²/ a² + (y - k)²/ b² = 1

Centre (h, k)
Horizontal Radius = a
Vertical Radius = b

Question 7

Conics

Sketching Graph

Ellipse

Answer 7

1. Divide equation so that RHS = 1
2. Find a
3. Find b
4. Find centre

Question 8

Conics

Hyperbola

General Formula

Answer 8

(x - h)²/ a² - (y - k)²/ b² = 1
OR
(y - k)²/ a² - (x - h)²/ b² = 1

Point of Intersection of Asymptotes (h, k)

Horizontal distance from Asymptote = a for 1st equation
Vertical distance from Asymptote = b for 2nd equation