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The quadratic polynomial and the parabola (9.1 – 9.5)

9.1 The quadratic polynomial ax² + bx + c. Graph of a quadratic function. Roots of a quadratic equation. Quadratic inequalities.

9.2 General theory of quadratic equations, relation between roots and coefficients. The discriminant.

9.3 Classification of quadratic expressions; identity of two quadratic expressions.

9.4 Equations reducible to quadratics.

9.5 The parabola defined as a locus. The equation x² = 4ay. Use of change of origin when vertex is not at (0, 0)

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GRAPHING A PARABOLA

You can recognise the equation of a parapola because the highest power of x is 2 and the power of y is 1 or the highest power of y is 2 and the power of x is 1. There are no negative indices.

The simplest parabolas are y = x² and x = y².

Note that the square of a number is always positive so for y = x², y is always positive, and for x = y², x is always positive. When drawing up a number table, assign values to the pronumeral that is squared and calculate the values of the other pronumeral.

Q.1. Complete the following table for the parabola y = x² and then draw the graph.

y = x²

x	0	1	-1	2	-2	3	-3	4	-4
y

Q.2. Draw a similar table for the parabola x = y² and graph the results.

Remember that since y is squared you assign values to y and then calculate the corresponding x-values.

The graph of y = x² is called a function because for every x value there is only one value of y.

The graph of x = y² is called a relation because for all x values other than zero, there is more than one y-value.

For the rest of this topic we will be dealing with parabolic functions, i.e. parabolas that are symmetrical about the y-axis or a line parallel to the y-axis.

The point where the parabola changes direction is called the vertex.

The vertex occurs at the maximum or minimum value of the parabola.

A parabola with a minimum value is concave up;t

A parabola with a maximum value is concave down;u

A line through the vertex that divides the parabola into two halves where each half is a reflection of the other is called the axis of symmetry.

The x-values where the graph cuts the x-axis are called the roots of the equation. They are the values where y = 0.

The general equation of a parabola is: y = ax² +bx +c

Exercise: Draw graphs of the following parabolas. For each parabola, label the vertex with its coordinates, draw the axis of symmetry and write its equation and write down the roots of the equation.

(a) y = 4x²

(b) y = x² + x –2

(c) 2x² + 2x – 12

(d) 2x² – x - 10

FACTORISATION AND THE PARABOLA

1. Factorise the following expressions:

(i) x² – 9 (ii) x² – 25 (iii) x² + 4x + 4 (iv) x² – 2x + 1

(v) x² + x – 6 (vi) x² – 2x – 8 (vii) x² + 2x – 8 (viii) x² + 2x – 3

(ix) x² -3x – 4 (x) x² -6x + 5

2. Draw the following parabolas:

(i) y = x² – 9 (ii) y = x² – 25 (iii) y = x² + 4x + 4

(iv) y = x² – 2x + 1 (v) y = x² + x – 6 (vi) y = x² – 2x – 8

(vii) y = x² + 2x – 8 (viii) y = x² + 2x – 3 (ix) y = x² -3x – 4

(x) y = x² -6x + 5

3. Use your graph of the parabolas in Q.2. to solve the following equations:

(i) x² – 9 = 0 (ii) x² – 25 = 0 (iii) x² + 4x + 4 = 0

(iv) x² – 2x + 1= 0 (v) x² + x – 6 = 0 (vi) x² – 2x – 8 = 0

(vii) x² + 2x – 8 = 0 (viii) x² + 2x – 3 = 0 (ix) x² -3x – 4 = 0

(x) x² -6x + 5= 0

Answers:

1. (i) (x + 3)(x – 3) (ii) (x + 5)(x – 5) (iii) (x + 2)(x + 2)

(iv) (x - 1)(x – 1) (v) (x + 3)(x – 2) (vi) (x + 2)(x – 4)

(vii) (x + 4)(x – 2) (viii) (x + 2)(x – 1) (ix) (x + 1)(x – 4)

(x) (x - 1)(x – 5)

2.

3. (i) x = 3 (ii) x = 5 (iii) x = -2 (iv) x = 1 (v) x = 2 or -3

(vi) x = 4 or -2 (vii) x = 2 or -4 (viii) x = 1 or -2 (ix) x = 4 or -1

(x) x = 1 or 5

PARABOLA 2

The point where the parabola changes direction is called the vertex.

The vertex occurs at the maximum or minimum value of the parabola.

A parabola with a minimum value is concave up;t

A parabola with a maximum value is concave down;u

A line through the vertex that divides the parabola into two halves where each half is a reflection of the other is called the axis of symmetry.

The x-values where the graph cuts the x-axis are called the roots of the equation. They are the values where y = 0.

The general equation of a parabola is: y = ax² +bx +c

Exercise: Draw graphs of the following parabolas. For each parabola, label the vertex with its coordinates, draw the axis of symmetry and write its equation and write down the roots of the equation.

(e) y = 8x²

(f) y = -8x²

(g) y = x² + 2x – 4

(h) y =2x² + x – 12

(i) y = -2x² + x + 10

(j) y > 2x²

(k) y < 6x²

(l) y > 2x² – 4x +3

SIMULTANEOUS EQUATION PROBLEMS

1. Three pencils and an eraser cost $1-10. Four pencils and two erasers cost $1-80. What is the cost of each pencil and each eraser?

2. Jill is four years older than Sue. Six years ago Jill was twice Sue’e age. How old are they now?

3. Amber is six years older than Sam. The sum of their ages is 22. How old is Amber and how old is Sam?

4. Emma scored 5 more marks in her Maths test than she did in her Science test. If the sum of her marks for Maths and Science was 155, how many marks did she score in each test?

5. Two numbers have a sum of 17 and a product of 60. What are the numbers?

6. A pie and a sandwich weigh 250 grams. Three pies and two sandwiches weigh 650 grams. What is the weight of two pies and a sandwich?

1. pencil = 20c, eraser = 50c. 2. Jill = 14, Sue = 10

3. Sam = 8, Amber = 14. 4. Maths = 80, Science = 75.

5. 12 & 5. 6. 400 grams.