Chapter 2: Quadratic Functions

Question 1

State the features of a quadratic graph

Answer 1

1) Parabola
- Can open upwards or downwards
2) Vertex (h,k)
- Turning point
- If parabola opens upwards: vertex is minimum point
If parabola opens downwards: vertex is maximum point
3) Axis of symmetry (x=h)
4) Axial intercepts
- x-intercept (y=0)
- y-intercept (x=0)

Question 2

What are the forms of a quadratic equation?

Answer 2

1) Factorised/x-intercept form
y=a(x-p)(x-q)
- If a>0, parabola opens upwards. If a<0, parabola opens downwards
- x-intercepts: p and q
- Axis of symmetry: x=p+q/2

2) Completed square/vertex form
y=a(x-h)² OR y=a(x-h)²+k
- If a>0, parabola opens upwards. If a<0, parabola opens downwards
- Vertex: (h,0) or (h,k)
- Axis of symmetry: x=h

3) General form
y=ax²+bx+c
- If a>0, parabola opens upwards. If a<0, parabola opens downwards
- y-intercept: c
- Axis of symmetry: x=-b/2a
- Find discriminant to determine number of solutions

Question 3

What are the conditions for positive/negative definite quadratics?

Answer 3

Positive-definite:
- a>0
- Δ<0

Negative-definite:
- a<0
- Δ<0

Question 4

What are the 3 ways graphs can intersect?

Answer 4

1) Cutting (2 points of intersection)
2) Touching/tangential (1 point of intersection)
3) Missing (0 points of intersection)

Solve by:
- If all terms of equations known:
Equate both equations and solve for x (no. of solutions vary)
- If 1 term of equations unknown:
Equate equations and find expression for Δ and equate to (>0, =0, <0), depending on no. of intersections

Question 5

Know how to draw a sign diagram*

Answer 5

Sign changes when graph cuts the x-axis

Question 6

Use sign diagrams to solve a quadratic inequality (ax²+bx+c≥0) or (ax²+bx+c≤0)

Answer 6

1) Shift all terms to LHS
2) Factorise LHS
3) Equate equation to 0 to find roots
4) Draw sign diagram
5) Determine range