17/01/2019 Straight lines

17/01/2019 y = mx + c All straight line graphs follow the general equation y = mx + c For example: y = 5x + 1 y = 3x - 2

PLOTTING STRAIGHT LINE GRAPHS 17/01/2019 PLOTTING STRAIGHT LINE GRAPHS

17/01/2019 x 1 2 3 4 5 y Example: Draw the graph of y = x + 2 Step 1: We start by drawing a table of values for the graph we are trying to draw... x 1 2 3 4 5 y

17/01/2019 Co-ordinates (2, 3) Along the corridor Up the stairs

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17/01/2019 Example: Draw the graph of y = x + 2 Step 2: We use our table of coordinates to plot each point on a set of axes... Step 3: Join these points to form a straight line x 1 2 3 4 5 y 6 7 1 2 3 4 5 6

17/01/2019 Task – Draw the graphs of the following equations on a +10 to -10 axis. Start by copying and completing these tables x 1 2 3 4 5 y 1) y = x + 4 x 1 2 3 4 5 y y = x – 1 x 1 2 3 4 5 y 3) y = 2x+3 x 1 2 3 4 5 y 4) y = 3x-1

17/01/2019 Other special lines

17/01/2019 On a spare grid can you Draw the following lines? X = 2 X=-3 X=1 X = -5

17/01/2019 On a second one can you Draw the following lines? y = 4 Y =-5 Y =2 y = -3

17/01/2019 𝒚=𝒎𝒙+𝒄

17/01/2019 𝑦 = m𝑥 + c All straight lines have equations which can be written in the form m is the GRADIENT (for each 1 unit right, how many units up?) c is the 𝒚-INTERCEPT (where does the graph cut the 𝒚-axis?)

‘For every one to the right I go up m’ 17/01/2019 Gradient Gradient means steepness of the slope ‘For every one to the right I go up m’ The gradient, m, is always the coefficient of 𝒙 (the number attached to the 𝑥 in the equation) 𝑦 = 3𝑥 + 2 gradient is 3 So for every 1 across to the right, I go UP 3 𝑦 = -4𝑥 - 1 gradient is -4 So for every 1 across to the right, I go DOWN 4 𝑦 = 2 + 7𝑥 gradient is 7 So for every 1 across to the right, I go UP 7

‘Where does the graph meet or cross the y-axis?’ 17/01/2019 𝑦-intercept Intercept sounds like intersect, which means meet or cross ‘Where does the graph meet or cross the y-axis?’ The 𝑦-intercept is the number with no letter attached to it 𝑦 = 3𝑥 + 2 intercept is at 2 So the graph crosses y axis at 2 𝑦 = -4𝑥 - 1 intercept is at -1 So the graph crosses y axis at 4 𝑦 = 2 + 7𝑥 intercept is at 2 So the graph crosses y axis at 2

On your mini whiteboards… Quick Questions! 17/01/2019 𝒚 = m𝒙 + c On your mini whiteboards… Quick Questions!

17/01/2019 What is the GRADIENT of 𝑦 = 2𝑥 + 3

17/01/2019 What is the GRADIENT of 𝑦 = 5𝑥 – 3

17/01/2019 What is the GRADIENT of 𝑦 = -5𝑥 – 3

17/01/2019 What is the GRADIENT of 𝑦 = 100𝑥 – 3

17/01/2019 Which is STEEPER? 𝑦 = 3𝑥 + 5 or 𝑦 = 8𝑥 + 6

17/01/2019 Which is STEEPER? 𝑦 = -3𝑥 + 5 or 𝑦 = -8𝑥 + 6

17/01/2019 Sketching graphs

𝑦 = 3𝑥 + 2 gradient 3 3 𝑦-intercept (0,2) 17/01/2019 Example (0,2)

𝑦 = -2𝑥 + 1 gradient -2 𝑦-intercept (0,1) -2 17/01/2019 Example (0,1)

17/01/2019 Task – Draw the following graphs on your grids using the gradient/intercept method 1) y = 2x + 1 2) y = 3x – 2 3) y = 2x – 1 y = x + 4 5) y = x – 3 6) y = 3x + 1 y = - 2x + 3 8) y = 3 – x 9) y = -3x – 1 y = 1 2 x + 3 11) y = 1 3 x – 2 12) y = 1 4 x y = 2 3 x + 2 14) y = 2 – 1 2 x 15) y = -x

17/01/2019 Solutions - Green

17/01/2019 Solutions - Orange

On your mini whiteboards… Quick Questions! 17/01/2019 𝒚 = m𝒙 + c On your mini whiteboards… Quick Questions!

Which line(s) has a y intercept of 4? 17/01/2019 Which line(s) has a y intercept of 4? 3 2 -4 4 -1 1 -3 -2

Which line(s) has a gradient of 2? 17/01/2019 Which line(s) has a gradient of 2? 3 2 -4 4 -1 1 -3 -2

What are the equations of these lines: 17/01/2019 Summary of skills What are the equations of these lines: Line 1 Line 2 Line 3

PLOTTING QUADRATICS (Double brackets) 17/01/2019 PLOTTING QUADRATICS (Double brackets)

17/01/2019 Quadratic graphs Quadratic equations… 𝑦=𝑎 𝑥 2 +𝑏𝑥+𝑐 …have graphs of a certain shape which we call a parabola If 𝑎 is positive, then the shape will be a “u” If 𝑎 is negative, then the shape will be an “n” To plot Substitute values of 𝑥 into the equation to find the corresponding 𝑦 values Plot points and join with a smooth curve

17/01/2019 Examples Here are examples of three quadratic functions: y = x2 y = x2 – 3x y = –3x2

17/01/2019 Example Plot the graph of y = x2 – 3 for values of x between –3 and 3. We can use a table of values to generate coordinates that lie on the graph as follows: x y = x2 – 3 –3 –2 –1 1 2 3 6 1 –2 –3 –2 1 6 (–3, 6) (–2, 1) (–1, –2) (0, –3) (1, –2) (2, 1) (3, 6)

17/01/2019 Example x y = x2 – 3 –3 –2 –1 1 2 3 6 1 –2 –3 –2 1 6 1 2 3 6 1 –2 –3 –2 1 6 The points given in the table are plotted … x –2 –1 –3 1 2 3 4 5 y … and the points are then joined together with a smooth curve. The shape of this graph is called a parabola. It is characteristic of a quadratic function.

17/01/2019 Task – For each equation, make a table of values, draw the axes and plot the graph 1) y = x2 { - 7 < x < 7 } Ans 6) y = x2- 20 { - 5 < x < 7 } Ans 2) y = 2x2 { - 4 < x < 4 } Ans 7) y = x2 + 2x {- 5 < x < 5 } Ans 3) y = 3x2 { - 4 < x < 4 } Ans 8) y = x2 - 3x {- 5 < x < 5 } Ans 4) y = ½x2 { - 6 < x < 6 } Ans 9) y = x2 + 3x + 10 {- 5 < x < 5 } Ans 5) y = x2 + 5 { - 6 < x < 6 } Ans 10) y = x2 - 4x - 10 {- 5 < x < 7 } Ans

x - 7 - 6 - 5 - 4 - 3 - 2 - 1 1 2 3 4 5 6 7 y 9 16 25 36 49 49 36 25 16 9 4 1 y 50 x x y = x2 40 x x 30 x x 20 x x 10 x x x x x x x x -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 Back to questions

x - 5 - 4 - 3 - 2 - 1 1 2 3 4 5 y 50 32 18 8 y 50 x x y = 2x2 40 x x 30 20 x x 10 x x x x x x -5 -4 -3 -2 -1 1 2 3 4 5 Back to questions

x - 4 - 3 - 2 - 1 1 2 3 4 y 48 27 12 y 50 x x 40 y = 3x2 30 x x 20 x x 10 x x x x -5 -4 -3 -2 -1 1 2 3 4 5 Back to questions

x - 6 - 5 - 4 - 3 - 2 - 1 1 2 3 4 5 6 y 18 12.5 8 4.5 0.5 y x 18 16 14 12 10 8 6 4 2 -2 x y = ½x2 x x x x x x x x x x x x -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 Back to questions

x - 6 - 5 - 4 - 3 - 2 - 1 1 2 3 4 5 6 y 41 30 21 14 9 y 50 x 40 x y = x2 + 5 x 30 x x 20 x x x 10 x x x x x x -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 Back to questions

x - 5 - 4 - 3 - 2 - 1 1 2 3 4 5 6 7 y - 11 - 16 - 19 - 20 16 29 y 30 25 20 15 10 5 -5 -10 -15 -20 x y = x2 - 20 x x x x -5 -4 -3 -2 -1 1 2 3 4 5 6 7 x x x x x x x x x Back to questions

x - 5 - 4 - 3 - 2 - 1 1 2 3 4 5 y 15 8 24 35 y 50 40 x 30 y = x2 + 2x x 20 x x 10 x x x x x x x x -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 Back to questions

x - 5 - 4 - 3 - 2 - 1 1 2 3 4 5 y 40 28 18 10 y 50 40 x 30 y = x2 – 3x x 20 x x 10 x x x x x x x x -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 Back to questions

x - 5 - 4 - 3 - 2 - 1 1 2 3 4 5 y 20 14 10 8 28 38 50 y 50 x 40 x y = x2 + 3x + 10 30 x 20 x x x x 10 x x x x x -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 Back to questions

x - 5 - 4 - 3 - 2 - 1 1 2 3 4 5 6 7 y 35 22 11 - 10 - 13 - 14 y x 35 30 25 20 15 10 5 y = x2 - 4x - 10 x x x x x x -5 -4 -3 -2 -1 1 2 3 4 5 6 7 x -5 -10 -15 -20 x x x x x x Back to questions