12. Quadratics NOTES.notebook September 21, 2017

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1 1) Fully factorise 4y 2-5y - 6 Today's Learning: To find the equation of quadratic graphs using substitution of a point. 2) Epand the brackets and simplify: (m + 4)(2m - 3) 3) Calculate 20% of 340 without a calculator. 4) What is 40ml increased by 20%? Quadratic Graphs A quadratic equation involves a squared term e.g = 0 The graph of y = k 2 The simplest quadratic graph is y = 2 happy sad Positive k Negative k y = k 2 graph is stretched by a factor of k 1) Factorise fully: Today's Learning: 2) Without a calculator, find To continue to consider transformations of quadratic graphs. 3) Without a calculator, simplify

2 e.g. Find the equation of the graph of the form y = k 2 1) Without a calculator, find a fifth of 70. 2) Fully factorise: 3g 2-13g ) Multiply out the brackets: (e + 2)(e + 3)(e - 1) y = 2 + q e.g. Find k and q from the graphs of y = k 2 + q: 1) y 2) y (-2, 15) (0,-1) (0,3) (2,-13) positive q negative q 1) Fully factorise: 3m m + 9 2) Simplify the following: Find the equations of these graphs, of the form y = k 2 + q 1) 2) ( 2,15) (0, 1) (0,3) (2, 13) 3) Without a calculator, find 53 31

3 The graph of y = ( + p) 2 e.g. Find p for these graphs of y = ( + p) 2 : a) y b) y positive p negative p (6,0) (-5,0) 1) Find a and b, given: 2a - b = 2 a + b = 7 2) Calculate , giving your answer in scientific notation Sketching Quadratic Graphs We can be asked to label: Turning Point and its nature Roots (where it crosses the -ais) y-intercept Equation of the ais of symmetry 3) Round to 3 sig. fig. 4) Find the area of the sector: 62 o 12cm 77.9cm 2 e.g. 1) Sketch y = -( + 3) 2 and label all of the above. e.g. 2) Sketch the graph of y = -( + 1) Factorise: Factorise: Factorise: State the gradient of the line: 4y + 12 = 2

4 1) Write down the y-intercept of the line 2y = 3-2 a b = 0 2) Without a calculator, find a fifth of 22 What can you say about a and b? 3) Simplify 3e 4 2e -2 4) What is the difference between -4 and 7? e.g. 3) Sketch the graph of y = ( - 2)( + 3) Sketch the graph of y = -( + 2)( - 2) Factorise the following: 1) 3m 2-13m ) 2p 2-18 c) 3gh + 6g 2 Today's Learning: Sketching quadratic graphs.

5 Sketch y = ( + 4)( - 8) a) Write the epression ( + 10)( + 2) in completed square form. b) Hence sketch the graph y = ( + 10)( + 2), marking the coordinates of the turning point and the nature of the turning point. y = ( - 1) 2-25 Roots: 0 = = ( - 4)( + 6) = 4 or -6 Spot the mistake(s)! (-6,0) y (4,0) How do we solve ( + 4)( - 1) = 0 for? y intercept: y = (-1) 2-25 = -26 Equation of ais of symmetry: = -1 (0,-26) (-1,-25) How might we solve = 0 TP occurs at (-1, -25) and is a minimum because 2 >0 Solving Quadratic Equations A quadratic equation can be written as a 2 + b + c = 0 Then, we can solve by factorising. Eamples: 1) = 0 2) = 0 a) Write the epression ( - 5)( + 3) in completed square form. b) Hence sketch the graph y = ( - 5)( + 3), marking the coordinates of the turning point and the nature of the turning point.

6 Eample: Solve = 0 Today's Learning: To write any quadratic equation in the form a 2 + b + c = 0 and to solve equations that don't factorise by using the quadratic formula. The Quadratic Formula If we have an equation a 2 + b + c = 0 that we can't factorise, we can use the Quadratic Formula to find solutions: Eamples: 1) (given in eams) Solve using the Quadratic Formula, giving answers to 2 decimal places: a) b) c) 2) How can we tell how many roots an equation has? The Discriminant For a quadratic equation a 2 + b + c = 0 the discriminant is b 2-4ac. b 2-4ac > 0 means 2 real, distinct roots b 2-4ac = 0 means 2 real, equal roots b 2-4ac < 0 means no real roots e.g. 1) Determine the nature of the roots of 2( + 1) = 2-3

7 e.g. 2) Find the range of values for T such that T = 0 has 2 real, distinct roots. 1) Given f() = 2-4, evaluate f(3) 2) Sketch the graph f() = 2-1. Write the coordinates where this line meets the line f() = 4. 3) Given f() = , find such that f() = The areas of these rectangles are equal. Rationalise the denominator: a) Find the value of. b) Calculate the area of the rectangles. ( + 1) cm ( + 3) cm (2 + 2) cm ( + 4) cm 1) Given the function f() = (5 - ) 2, evaluate: a) f(3) b) f(-1) Simplify: (a 2 ) 3 a -2 a 5 a -5 2) Multiply out the brackets and simplify: (w + 1)(w - 1)(w + 5)

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