Sketch the curve with equation
y=x2(2x−a) y = x^2(2x - a) y=x2(2x−a)where a>0a > 0a>0.
The polynomial p(x)p(x)p(x) is given by
p(x)=x2(2x−a)−64 p(x) = x^2(2x - a) - 64 p(x)=x2(2x−a)−64It is given that x−8x - 8x−8 is a factor of p(x)p(x)p(x). Use the factor theorem to show that a=15a = 15a=15.
State the transformation which maps the curve with equation y=x2(2x−15)y = x^2(2x - 15)y=x2(2x−15) onto the curve with equation y=x2(2x−15)−64y = x^2(2x - 15) - 64y=x2(2x−15)−64.
The polynomial x2(2x−15)−64x^2(2x - 15) - 64x2(2x−15)−64 can be written as (x−8)(2x2+bx+c)(x - 8)(2x^2 + bx + c)(x−8)(2x2+bx+c). Without finding the values of bbb and ccc, use your knowledge of the transformation in part (b)(ii) and the sketch in part (a) to explain why
b2<8c b^2 < 8c b2<8c532 exam-style questions on AQA A Level Maths 1.5 B: Algebra and functions, covering 1.5.1 Laws of indices, 1.5.2 Surds, 1.5.3 Quadratic functions, 1.5.4 Simultaneous equations, 1.5.5 Linear and quadratic inequalities, 1.5.6 Polynomials and rational expressions, 1.5.7 Graphs of functions and proportion, 1.5.8 Composite and inverse functions (A-level only), 1.5.9 Transformations of graphs, 1.5.10 Partial fractions (A-level only), and 1.5.11 Functions in modelling (A-level only). Each one has a worked solution and a mark scheme showing where the marks go.