Algebra Graphs

Plotting Points - To draw straight line graphs we can use a rule to find and plot co-ordinates e.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 = 2xy = ½x – x y = -3x x -1 2 x ½ x -1 – 1 ½ x -2 – 1 -½ ½ -3 x x

Gradients of Lines - The gradient is a number that tells us how steep a line is. - The formula for gradient is:Gradient = rise run e.g. Write the gradients of lines A and B A =B = = = e.g. Draw lines with the following gradients a) 1 b) 3 c) To draw, write gradients as fractions = 3 1 When calculating gradients it is best to write as simplest fraction

y = 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 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 Gradients e.g. Write the gradients of lines A and B 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 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

Intercepts - 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 A = B = C =

Drawing Lines: Gradient and Intercept Method - 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 To draw: 1. Mark in intercept 2. Step off gradient 3. Join up points = -3x – 2 1 Note: Any rule with no number in front of x has a gradient of 1 1 e.g. y = x – 1

Writing Equations of Lines - A straight line can be expressed using the rule y = mx + c e.g. Write equations for the following lines A:B:C:m =c =m =c =m =c = y = 3x – 6 4 y = -2x y = 4x

Horizontal and Vertical Lines - Horizontal lines have a gradient of:0 Rule: 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 = -1y = -3 b) d)

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) m = -2 1 c =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 y = -2x + 5

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 = 4b) 4x – 3y =12 1. Cover up ‘y’ term to find x intercept - x = 4 ÷ -1 x = Cover up ‘x’ term to find y intercept 2y = 4 ÷ 2 y = 2 3. Join up intercepts with a straight line 4x = 12 ÷ 4 x = 3 -3y = 12 ÷ -3 y = -4 It is also possible to rearrange equations into the form y = mx + c e.g. Rearrange 2x – y = 6 -2x - y = 6 – 2x ÷ -1 y = x y = 2x – 6

Applications e.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 Draw a graph of the equation 2. What are the costs if they sell 8 pizzas? $130 3a. What is the cost per pizza? $10 3b. How is this shown by the graph? The gradient of the line 4a. What are the costs if they sell no pizzas? $50 4b. How is this shown by the graph? Where the line crosses the y-axis

The Basic Parabola - The parabola is a quadratic graph linking y and x 2 - The basic parabola is y = x 2 e.g. Complete the table below by using the rule y = x 2 to find and plot co-ordinates to draw the basic parabola. xy = x (-1) 2 (-2) 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 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 = x 2 – 3 b) y = x c) y = (x + 1) 2 d) y = (x – 1) 2 x y = x 2 – 3y = x x y = (x + 1) 2 y = (x – 1) (-1) 2 – 3 (-2) 2 – (-1) (-2) (-1 + 1) 2 (-2 + 1) (-1 – 1) 2 (-2 – 1)

Transformations of the Basic Parabola 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 = x 2 b) y = x c) y = x 2 – 5 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 = x 2 b) y = (x + 3) 2 c) y = (x – 2) 2 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) c) y = (x – 7) d) y = (x + 6) 2 – 5 To draw combined transformations, first find the position of the vertex Then draw in basic parabola shape

Changing the Shape of the Basic Parabola 1. When x 2 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 = 2x 2 and y = ¼x 2 xy = 2x 2 y = ¼x × (-1) 2 2 × (-2) ¼ × (-2) 2 ¼ × (-1) 2 1 ¼ 0 ¼ 1 Use the grid to determine the x- values to put into your table

1. When x 2 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 = -x 2 y = -(x + 2) 2 y = -(x – 1) First find placement of the vertex When plotting points move down instead of up.

Factorised Parabolas Method 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 xy = (x – 3)(x + 1) (-3 – 3)(-3 + 1)12 (-2 – 3)(-2 + 1)

Method 2: Calculating and plotting specific features e.g. Draw the parabola y = (x – 3)(x + 1) 1. x-axis intercepts (where y = 0)solving quadratics: 0 = (x – 3)(x + 1) x = 3 and y-axis intercept (where x = 0)y = (0 – 3)(0 + 1) y = 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 = Join the points with a smooth curve Vertex = (1, -4)

e.g. Draw the parabola y = x(x – 4) 1. x-axis intercepts0 = x(x – 4)x = 0 and 4 2. y-axis intercepty = 0(0 – 4)y = 0 3. Position of vertexy = 2(2 – 4)y = -4Vertex = (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 = 4Vertex = (3, 4) Note: -x indicates parabola will be upside down

Expanded Form Parabolas - 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 = x 2 – 2x – 8 and y = x 2 + 2x Factorised Expressiony = 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)

Writing 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 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) y = (x + 1)(x + 5) or Vertex = (-3, 4) y = (x + 3) Add in a negative sign if parabola upside down - -

Writing Harder Equations - 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 a)b) 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× = k y = 0.5(x – 1)(x – 5) y = k(x – 20) = k(10 – 20) = k× = k×100 y = -0.4(x – 20) = k Coefficients can be written as decimals or fractions (3, -2) (10, 0)