Solve Quadratics Using the Formula

Transcription

1 Clip 6 Solve Quadratics Using the Formula a + b + c = 0, = b± b 4 ac a ) Solve the equation = 0 Give our answers correct to decimal places. ) Solve the equation = 0 ) Solve the equation = 0 4) Solve the equation 7 + = 0 5) Solve the equation + 6 = 0 6) Solve the equation 0 = 0 7) Solve the equation = 0 8) Solve the equation = 0 9) + 0 = 00 Find the positive value of. Give our answer correct to significant figures. 0) ( + )( ) = a) Show that 7 = 0 b) Solve the equation 7 = 0 Page 5

2 Clip 6 Completing the Square ) Show that if = + 8 then 9 for all values of. ) Show that if = then 5 for all values of. ) The epression can be written in the form ( + p) + q for all values of. Find the values of p and q. 4) Given that = ( p) + q for all values of, find the value of p and the value of q. 5) For all values of, + 6 = ( + p) + q a) Find the values of p and q. b) Find the minimum value of ) For all values of, 8 5 = ( p) + q a) Find the value of p and the value of q. b) On the aes, sketch the graph of = 8 5. O c) Find the coordinate of the minimum point on the graph of = ) The epression 0 can be written in the form p ( q) for all values of. a) Find the values of p and q. b) The epression 0 has a maimum value. (i) Find the maimum value of 0. (ii) State the value of for which this maimum value occurs. Page 54

3 Clip 6 Algebraic Fractions ) Simplif full a) 9 c) 8 ab ab e) ab 4 ab 6ab b) 0 5 d) f) ) Simplif full a) c) + b) d) ) a) Factorise b) Simplif ) Write as single fractions in their simplest form a) + c) b) 5 4 d) ) a) Factorise b) Write as a single fraction in its simplest form ) Solve a) + = c) = e) = 4 b) = d) = f) + + = Page 55

4 Clip 64 Rearranging Difficult Fomulae ) Make c the subject of the formula. v = a + b + c ) Make t the subject of the formula. A = π t + 5t ) Make s the subject of the formula. R = s + π s + t l 4) k = m l a) Make l the subject of the formula. b) Make m the subject of the formula. 5) A k ( = + 5) Make the subject of the formula. 6) R = u+ v u+ v Make u the subject of the formula. 7) + = 5 0+ Make the subject of the formula. 8) a = 5 4b Rearrange this formula to give a in terms of b. 9) S = πd h + d Rearrange this formula to make h the subject. Page 56

5 Clip 65 Simultaneous Equations With a Quadratic ) Solve these simultaneous equations. = = 6 ) Solve these simultaneous equations. = 4 = ) Solve these simultaneous equations. = = + 4) Solve these simultaneous equations. = 5 = 5 5) Solve these simultaneous equations. + = = 6) Sarah said that the line = 7 cuts the curve + = 5 at two points. a) B eliminating show that Sarah is not correct. b) B eliminating, find the solutions to the simultaneous equations + = 5 = 9 Page 57

6 Clip 66 Gradients of Lines ) Diagram NOT accuratel drawn B (0, 7) A (0, ) 0 A is the point (0, ) B is the point (0, 7) The equation of the straight line through A and B is = + a) Write down the equation of another straight line that is parallel to = + b) Write down the equation of another straight line that passes through the point (0, ). c) Find the equation of the line perpendicular to AB passing through B. ) A straight line has equation = 5 The point P lies on the straight line. The coordinate of P is -6 a) Find the coordinate of P. A straight line L is parallel to = 5 and passes through the point (, ). b) Find the equation of line L. c) Find the equation of the line that is perpendicular to line L and passes through point (, ). ) In the diagram A is the point (0, -) B is the point (-4, ) C is the point (0, ) B C a) Find the equation of the line that passes through C and is parallel to AB. b) Find the equation of the line that passes through C and is perpendicular to AB O - - A - Page 58

7 Clip 67 Transformations of Functions ) This is a sketch of the curve with equation = f(). It passes through the origin O. The onl verte of the curve is at A (, -) a) Write down the coordinates of the verte of the curve with equation (i) = f( ) (ii) = f() 5 (iii) = f() (iv) = f() A (, -) b) The curve = has been translated to give the curve = f(). Find f() in terms of. ) The graph of = f() is shown on the grids. a) On this grid, sketch the graph of = f( ) b) On this grid, sketch the graph of = f() ) Sketch the graph of = ( ) + State the coordinates of the verte. 4) Sketch the graph of = + 4 State the coordinates of the verte and the points at which the curve crosses the - ais. Page 59

8 Clip 68 Graphs of Trigonometric Functions - of ) On the aes below below, draw a sketch-graph to show = sin Given that sin 0 = 0.5, write down the value of: (i) sin 50 (ii) sin 0 ) On the aes below, draw a sketch-graph to show = cos Given that cos 60 = 0.5, write down the value of: (i) cos 0 (ii) cos 40 Page 60

9 Clip 68 Graphs of Trigonometric Functions - of ) On the aes below, draw a sketch-graph to show = tan ) The diagram below shows the graph of = cos a + b, for values of between 0 and 00. ) Work Here out is the the graph values of of the a curve and b. = cos for 0 < < a) Use the graph to solve cos = 0.75 for 0 < < 60 b) Use the graph to solve cos = for < < Page 6

10 Clip 69 Transformation of Trig. Functions ) The diagram below shows the graph of = sin, for values of between 0 and 60. B A The curve cuts the ais at the point A. The graph has a maimum at the point B. a) (i) Write down the coordinates of A. (ii) Write down the coordinates of B. b) On the same diagram, sketch the graph of = sin + for values of between 0 and 60. ) The diagram below shows the graph of = cos a + b, for values of between 0 and 00. Work out the values of a and b Page 6

11 Clip 70 Graphs of Eponential Functions ) (4, 75) (, ) The sketch-graph shows a curve with equation = pq. The curve passes through the points (, ) and (4, 75). Calculate the value of p and the value of q. ) The graph shows the number of bacteria living in a petri dish. The number N of bacteria at time t is given b the relation: N = a b t The curve passes through the point (0, 400). a) Use this information to show that a = 400. The curve also passes through (, 900). b) Use this information to find the value of b. Number of bacteria N t c) Work out the number of bacteria in the dish at time t =. Time (hours) Page 6