PCH6: The Quadratic Polynomial & The Parabola

Question 1

Completing the square

Answer 1

(a + b)^2 = a^2 + 2ab + b^2

(a - b)^2 = a^2 - 2ab + b^2

Question 2

Quadratic Formula

Answer 2

x = - b +/÷ √ b^2 - 4ac
—————————
2a

Question 3

How do you find the roots of quadratic equations

Answer 3

• Solving the quadratic equation (P & S)
• Completing the square
• Quadratic formula

Question 4

What are the equations of a quadratic equation? What determines the concavity of the curve?

Answer 4

1. y = ax^2 + bx + c

if a> 0 = concave up
a 0 then the point is at -p

Question 5

How to solve quadratic inequalities

Answer 5

• Sketch a graph

- Look at points where x/ y values are above specified value given

Question 6

Sum and product of roots

Answer 6

in ax^2 + bx + c where a and b are the roots:

sum of roots (a + b) = - b ÷ a
product of roots (ab) = c ÷ a

Question 7

Axis of symmetry of a parabola

Answer 7

• Straight line midway between the roots of the equation
  1. Axis: x = - b ÷ 2a

2. Greatest or least value for y = 4ac - b^2
   ————-
   4a
   If a > 0 the parabola is concave up (minimum value)
   If a

Question 8

Discriminant

Answer 8

∆ = b^2 - 4ac

If ∆ ≥ 0, roots are real
If ∆ ≤ 0, the roots are not real
If ∆ = 0, the roots are equal
If ∆ is a perfect square the roots are rational

Question 9

Positive definite, negative definite and indefinite expressions

Answer 9

• If ∆ 0, = function is positive definite

- If ∆

Question 10

Equations reducible to quadratics

Answer 10

• Solve using product and sum
• The use of substitution can be used where the power of x is greater than 2
  e.g. x^4 - 13x^2 + 36 = 0
  let x^2 = m
  m^2 - 13m + 36
  (m - 4) (m - 9) = 0
  m = 4 or m = 9
  x^2 = 4 x^2 = 9
  x = +/- 2 x = +/- 3

Question 11

Identities of quadratic expressions

Answer 11

ax^2 + bx + c (three horizontal lines) dx^2 + ex + f

identically equal to

Question 12

What is the equation for a parabola

Answer 12

x^2 = 4ay
y^2 = 4ax

Question 13

Describe the qualities of a parabola

x^2 = 4ay

Answer 13

• Vertex at origin
• Focal length A is the distance from the vertex to the focus
• The axis of symmetry is the y-axis
• The focus has coordinates (0, A)
• The directrix has equation y= -A

Question 14

The parabolas x^2 = - 4ay , y^2 = +/- 4ax

Answer 14

• All have vertex at origin
• Axis of symmetry is the y axis (for x^2 = …) and the x axis (for y^2 = ….)
• The focal length is A units for all
• For x^2 = -4ay The focus is at (0, -A) and the directrix y = A
• For y^2 = 4ax The focus is at (A, 0) and the directrix x = -A
• For y^2= -4ax The focus is at (-A, 0) and the directrix x = A

Question 15

Equation of parabolas where the vertex is not the origin

Answer 15

(x- xo)^2 = 4A(y-yo)

Vertex at (xo, yo)
Axis of symmetry = x= xo
Focus = (xo, yo+ A)
Directrix= y= yo - A

Question 16

How to find the focus and directrix of y= ax^2 + bx + c or x = ay^2 + by + c

Answer 16

• Express the equation in the form of:
  > (x-xo)^2 = 4A(y-yo)
  and then complete the square