Algebra Algebra GraphsGraphs

Plotting PointsPlotting Points- To draw straight line graphs we can use a rule to find and plot co-ordinatese.g. Complete the tables below to find co-ordinates in order to plot the following straight lines:a) y = 2x b) y = ½x – 1 c) y = -3x + 2 x y = 2x y = ½x –

1

-2

-1

0

1

2x y = -3x +

2

-2

-1

0

1

2

x-5 -4 -3 -2 -1 1 2 3 4 5

y

-5-4-3-2-1

12345

2 x -1

2 x -2

0

2

-4

4

-2 ½ x -1 – 1

½ x -2 – 1

-1

-½

-2

0

-1 ½

-3 x -1 + 2

-3 x -2 + 2

2

-1

8

-4

5

Gradients of LinesGradients of Lines- The gradient is a number that tells us how steep a line is.- The formula for gradient is:Gradient = rise

run

1st point

2nd point

rise

run

 e.g. Write the gradients of lines A and B

A

B

A =

B =

4

6 8

4

4 = 18 2

6 = 34 2

 e.g. Draw lines with the following gradientsa) 1 b) 3 c) 2 2 5

To draw, write gradients as fractions

= 3 1

When calculating gradients it is best to write as simplest fraction

y = mxy = mx- This is a rule for a straight line, where the gradient (m) is the number directly in front of the x- When drawing graphs of the form y = mx, the line always goes through the origin i.e. (0,0)e.g. Draw the following lines:a) y = 2x b) y = 4x c) y = 3x 5 4

x-5 -4 -3 -2 -1 1 2 3 4 5

y

-5-4-3-2-1

12345

1. Step off the gradient from the origin (0,0) 2. Join the plotted point back to the origin

= 4x 1

To draw, always write gradients as fractions

gradient

Negative Negative GradientsGradients

 e.g. Write the gradients of lines A and B

A =

B =

-3

2

10

-5

-5 = -110 2

-3 2

A

B

When calculating gradients it is best to write as simplest fraction

e.g. Draw the following lines:a) y = -2x b) y = -4x c) y = -3x 5 4

x-5 -4 -3 -2 -1 1 2 3 4 5

y

-5-4-3-2-1

12345

1. Step off the gradient from the origin (0,0) 2. Join the plotted point back to the origin

= -4x 1

To draw, always write gradients as fractions

gradient

InterceptsIntercepts- Is a number telling us where a line crosses the y-axis (vertical axis)i.e. The line y = mx + c has m as the gradient and c as the intercept e.g. Write the intercepts of the lines A, B and C

x-10 -8 -6 -4 -2 2 4 6 8 10

y

-10

-8

-6

-4

-2

2

4

6

8

10

A

B

C

A =

B =

C =

8

4

-3

Drawing Lines: Gradient and Intercept Drawing Lines: Gradient and Intercept MethodMethod- A straight line can be expressed using the rule y = mx + c

e.g. Draw the following lines:a) y = 1x + 2 b) y = -3x – 2 c) y = -4x + 8 2 7

x-10 -8 -6 -4 -2 2 4 6 8 10

y

-10

-8

-6

-4

-2

2

4

6

8

10To draw:1. Mark in intercept2. Step off gradient3. Join up points

= -3x – 2 1

Note: Any rule with no number in front of x has a gradient of 1 1e.g. y = x – 1

Writing Equations of LinesWriting Equations of Lines- A straight line can be expressed using the rule y = mx + c

e.g. Write equations for the following lines

x-10 -8 -6 -4 -2 2 4 6 8 10

y

-10

-8

-6

-4

-2

2

4

6

8

10

A

B

C

A: B: C:m = c = m = c = m = c =

y = 3x – 6 4

y = -2x + 1 3

y = 4x + 4 1

34

-2 3

41 -6 +1 +4

Horizontal and Vertical LinesHorizontal and Vertical Lines- Horizontal lines have a gradient of:

0Rule: y = c (c is the y-axis intercept)- Vertical lines have a

gradient that is:undefined

Rule: x = c (c is the x-axis intercept)e.g. Draw or write equations for the following lines:

a) y = 2 b) c) x = 4 d)

x-5 -4 -3 -2 -1 1 2 3 4 5

y

-5-4-3-2-1

12345

x = -1y = -3

b)

d)

Writing Equations Cont.Writing Equations Cont.When you are given two points and are expected to write an equation:- One method is set up a set of axes and plot the two points.

e.g. Write an equation for the line joining the points A=(1, 3) and B = (3, -1)

x-5 -4 -3 -2 -1 1 2 3 4 5

y

-5-4-3-2-1

12345 m = -2

1c = 5

y = -2x + 5

Sometimes when plotting the points, you may need to extend the axes to find the intercept.

- Or, substitute the gradient and a point into y = mx + c to find ‘c’, the intercept

m = -2 1

using point

(1, 3)

y = mx + c 3 = -2 x 1 + c 3 = -2 + c 5 = c

+2 +2

y = -2x + 5

Equations in the Form ‘ax + by = c’Equations in the Form ‘ax + by = c’- Can use the cover up rule to find the two intercepts:

e.g. Draw the following lines:a) 2y – x = 4 b) 4x – 3y =12

x-5 -4 -3 -2 -1 1 2 3 4 5

y

-5

-4

-3

-2

-1

1

2

3

4

51. Cover up ‘y’ term to find x intercept

- x = 4÷ -1 ÷ -1

x = -4

2. Cover up ‘x’ term to find y intercept

2y = 4÷ 2 ÷ 2

y = 2

3. Join up intercepts with a straight line

4x = 12÷ 4 ÷ 4

x = 3

-3y = 12÷ -3 ÷ -3

y = -4

It is also possible to rearrange equations into the form y = mx + ce.g. Rearrange 2x – y = 6

-2x -2x- y = 6 – 2x÷ -1 ÷ -1

y = -6 + 2xy = 2x – 6

x1 2 3 4 5 6 7 8 9 10

y

102030405060708090

100110120130140

ApplicationsApplicationse.g. A pizzeria specializes in selling large size pizzas. The relationship between x, number of pizzas sold daily, and y, the daily costs is given by the equation, y = 10x + 50

1. Draw a graph of the equation

2. What are the costs if they sell 8 pizzas?$1303a. What is the cost per pizza?$103b. How is this shown by the graph?

The gradient of the line4a. What are the costs

if they sell no pizzas?

$504b. How is this shown by the graph?

Where the line crosses the y-axis

The Basic The Basic ParabolaParabola

- The parabola is a quadratic graph linking y and x2- The basic parabola is y = x2

e.g. Complete the table below by using the rule y = x2 to find and plot co-ordinates to draw the basic parabola.

x-5 -4 -3 -2 -1 1 2 3 4 5

y

-2-1

123456789

10

x y = x2

-2

-1

0

1

2

(-1)2

(-2)2

0

1

4

4

1

Note: the points of a basic parabola are easily drawn from the vertex by stepping out one and up one, then out two and up four, then out three and up nine etc...

VERTEX

Plotting PointsPlotting Points- As with straight line graphs we can use a rule to find and plot co-ordinates in order to draw any parabola.

e.g. Complete the tables below to find co-ordinates in order to plot the following parabolas:a) y = x2 – 3 b) y = x2 + 2 c) y = (x + 1)2 d) y = (x – 1)2 x y = x2 – 3 y = x2 + 2

-2

-1

0

1

2

x y = (x + 1)2

y = (x – 1)2

-2

-1

0

1

2

x-5 -4 -3 -2 -1 1 2 3 4 5

y

-4-3-2-1

12345678

(-1)2 – 3

(-2)2 – 3

-3

-2

1

1

-2 (-1)2 + 2

(-2)2 + 2

2

3

6

6

3

(-1 + 1)2

(-2 + 1)2

1

4

1

9

0 (-1 – 1)2

(-2 – 1)2

1

0

9

1

4

Transformations of the Basic Transformations of the Basic ParabolaParabola

1. Up or Down Movement- When a number is added or subtracted at the end, the basic parabola moves vertically

e.g. Draw the following parabolas:a) y = x2 b) y = x2 + 1 c) y = x2 – 5

x-5 -4 -3 -2 -1 1 2 3 4 5

y

-5-4-3-2-1

1234567

To draw vertical transformations, first find the position of the vertex Then draw in basic parabola shape

2. Left or Right Movement

- When a number is added or subtracted in the brackets, the basic parabola moves horizontally but opposite in direction

e.g. Draw the following parabolas:a) y = x2 b) y = (x + 3)2 c) y = (x – 2)2

x-5 -4 -3 -2 -1 1 2 3 4 5

y

-5-4-3-2-1

1234567

To draw horizontal transformations, first find the position of the vertex Then draw in basic parabola shape

3. Combined Movements

e.g. Draw the following parabolas:a) y = (x – 4)2 – 8 b) y = (x + 3)2 + 3 c) y = (x – 7)2 + 4 d) y = (x + 6)2 – 5

x-10 -8 -6 -4 -2 2 4 6 8 10

y

-10

-8

-6

-4

-2

2

4

6

8

10

To draw combined transformations, first find the position of the vertex Then draw in basic parabola shape

Changing the Shape of the Basic Changing the Shape of the Basic ParabolaParabola

1. When x2 is multiplied by a positive number other than 1- the parabola becomes wider or narrower- Set up a table and use the rule to find and plot co-ordinates

e.g. Complete the tables and draw y = 2x2 and y = ¼x2

x y = 2x2 y = ¼x2

-2

-1

0

1

2

2 × (-1)2

2 × (-2)2

0

2

8

8

2

¼ × (-2)2

¼ × (-1)2

1

¼

0

¼ 1

x-5 -4 -3 -2 -1 1 2 3 4 5

y

-2-1

123456789

10

Use the grid to determine the x-values to put into your table

1. When x2 is multiplied by a negative number- it produces an upside down parabola- all transformations are the same as for a regular parabola

e.g. Draw the following parabolas: y = -x2

y = -(x + 2)2

y = -(x – 1)2 + 2 First find placement of the vertex

x-5 -4 -3 -2 -1 1 2 3 4 5

y

-5-4-3-2-1

12345

When plotting points move down instead of up.

Factorised Factorised ParabolasParabolasMethod 1: Set up a table, calculate and plot points

e.g. Draw the parabola y = (x – 3)(x + 1)Use the grid to determine the x-values to put into your table

x-5 -4 -3 -2 -1 1 2 3 4 5

y

-5-4-3-2-1

12345

x y = (x – 3)(x + 1)

-3

-2

-1

0

1

2

3

(-3 – 3)(-3 + 1)12

(-2 – 3)(-2 + 1)5

0-3

-4

-3

0

Method 2: Calculating and plotting specific featurese.g. Draw the parabola y = (x – 3)(x + 1)

x-5 -4 -3 -2 -1 1 2 3 4 5

y

-5-4-3-2-1

12345

1. x-axis intercepts (where y = 0) solving quadratics: 0 = (x – 3)(x + 1)x = 3 and -1

2. y-axis intercept (where x = 0) y = (0 – 3)(0 + 1)y = -3

3. The position of the vertex- is halfway between x-axis intercepts- substitute x co-ordinate into equation to find y co-ordinate

y = (1 – 3)(1 + 1)y = -4

4. Join the points with a smooth curve

Vertex = (1, -4)

e.g. Draw the parabola y = x(x – 4)

x-8-7-6-5-4-3-2-1 1 2 3 4 5 6 7 8

y

-9-8-7-6-5-4-3-2-1

12345678

1. x-axis intercepts 0 = x(x – 4) x = 0 and 4 2. y-axis intercept y = 0(0 – 4) y = 0 3. Position of vertex y = 2(2 – 4) y = -4 Vertex = (2, -4)

e.g. Draw the parabola y = (1 – x)(x – 5)1. x-axis intercepts 2. y-axis intercept 3. Position of vertex

0 = (1 – x)(x – 5) x = 1 and 5 y = (1 – 0)(0 – 5) y = -5 y = (1 – 3)(3 – 5) y = 4 Vertex = (3, 4)

Note: -x indicates parabola will be upside down

Expanded Form Expanded Form ParabolasParabolas

- Remember you can always set up a table and calculate co-ordinates to plot.- Or simply factorise the expression and plot specific points as shown earlier

e.g. Draw the parabolas y = x2 – 2x – 8 and y = x2 + 2x

Factorised Expression y = x(x + 2)y = (x – 4)(x + 2)

1. x-axis intercepts 2. y-axis intercept 3. Position of vertex

x = -2 and 4 y = -8 Vertex = (1, -9)

x = 0 and -2 y = 0 Vertex = (-1, -1)

x-8-7-6-5-4-3-2-1 1 2 3 4 5 6 7 8

-10-9-8-7-6-5-4-3-2-1

12345678

Writing EquationsWriting Equations- If the parabola intercepts x-axis, you can substitute into y = (x – a)(x – b) - Or, you can substitute the vertex co-ordinates into y = (x – a)2 + b

e.g. Write equations for the following parabolas

x-5 -4 -3 -2 -1 1 2 3 4 5

y

-5-4-3-2-1

12345

(a)

(b)

(c)

a)

c)

b)

Always substitute in the opposite sign x-value

y = (x – 2)(x – 4)or

Vertex = (3, -1)y = (x – 3)2 – 1

Vertex = (-2, 1)y = (x + 2)2 + 1

y = (x + 1)(x + 5)or

Vertex = (-3, 4)y = (x + 3)2 + 4

Add in a negative sign if parabola upside down

-

-

Writing Harder Writing Harder EquationsEquations

- Used when the co-efficient is not equal to 1. - Use either of the equations y = k(x – a)(x – b) or y = k(x + c)2 + d

e.g. Write equations for the following parabolas

x-4 -2 2 4 6 8 10

y

-4-2

2468

1012a) b)

x5 10 15 20 25 30 35

y

10

20

30

40

50

y = k(x – 1)(x – 5)

Substitute in the values of a specific point to find the coefficient k

-2 = k(3 – 1)(3 – 5)-2 = k×-4

0.5 = k

y = 0.5(x – 1)(x – 5)

y = k(x – 20)2 + 40

0 = k(10 – 20)2 + 40 0 = k×100 + 40

-40 = k×100

y = -0.4(x – 20)2 + 40-0.4 = k

Coefficients can be written as decimals or fractions

(3, -2) (10, 0)