Core Mathematics 1 Graphs of Functions

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1 Regent College Maths Department Core Mathematics 1 Graphs of Functions

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3 Graphs of Functions September 2011 C1 Note Graphs of functions; sketching curves defined by simple equations. Here are some curves you should know. y x y x 2 1 y x x 2 1 y x x 1 y 1 x 2 x 2 Copyright - For AS, A2 notes and IGCSE / GCSE worksheets 19

4 September 2011 C1 Note Geometrical interpretation of algebraic solution of equations. Use of intersection points of graphs of functions to solve equations. Functions to include simple cubic functions and the reciprocal function y = k/x with x 0. Graphs of cubics y ax bx cx d If a 0 then curve goes from If a 0 then curve goes from the bottom left to the top right: the top left to the bottom right: Three distinct linear factors x 2 is a factor 1 x is a factor x 4 is a factor So we see that y k x 2 x 1 x 4. The curve crosses the y axis at (0, 8). So k k and so k 1. Copyright - For AS, A2 notes and IGCSE / GCSE worksheets 20

5 September 2011 C1 Note One linear factor repeated x 2 is a repeated factor, as it turns at this point. x 3 is a factor So we see that y k x 2 2 x 3. 2 The curve crosses the y axis at (0, -24). So k k 24 and so k 2. Knowledge of the term asymptote is expected. 1 An asymptote is a line to which a given curve gets closer and closer. y (as drawn on the x previous page) has a vertical asymptote at x = 0 and a horizontal asymptote at y = 0. Copyright - For AS, A2 notes and IGCSE / GCSE worksheets 21

6 1. On separate diagrams, sketch the graphs of (a) y = (x + 3) 2, (b) y = (x + 3) 2 + k, where k is a positive constant. Show on each sketch the coordinates of each point at which the graph meets the axes. (2) May 2006, Q3 2. Given that f(x) = (x 2 6x)(x 2) + 3x, (a) express f(x) in the form x(ax 2 + bx + c), where a, b and c are constants. (b) Hence factorise f(x) completely. (c) Sketch the graph of y = f(x), showing the coordinates of each point at which the graph meets the axes. May 2006, Q9 3. (a) On the same axes sketch the graphs of the curves with equations (i) y = x 2 (x 2), (ii) y = x(6 x), and indicate on your sketches the coordinates of all the points where the curves cross the x-axis. (2) (b) Use algebra to find the coordinates of the points where the graphs intersect. 4. The curve C has equation y = x 3 and the line l has equation y = 2x + 5. (7) Jan 2007, Q10 (a) Sketch the graphs of C and l, indicating clearly the coordinates of any intersections with the axes. (b) Find the coordinates of the points of intersection of C and l. (6) June 2008, Q6

7 5. The point P(1, a) lies on the curve with equation y = (x + 1) 2 (2 x). (a) Find the value of a. (1) (b) Sketch the curves with the following equations: (i) y = (x + 1) 2 (2 x), (ii) y = x 2. On your diagram show clearly the coordinates of any points at which the curves meet the axes. (5) (c) With reference to your diagram in part (b), state the number of real solutions to the equation (x + 1) 2 (2 x) = x 2. (1) Jan 2009,Q8 6. (a) Factorise completely x 3 6x 2 + 9x (b) Sketch the curve with equation y = x 3 6x 2 + 9x showing the coordinates of the points at which the curve meets the x-axis. (4) Using your answer to part (b), or otherwise, (c) sketch, on a separate diagram, the curve with equation y = (x 2) 3 6(x 2) 2 + 9(x 2) showing the coordinates of the points at which the curve meets the x-axis. (2) June 2009, Q10

8 7. (a) Factorise completely x 3 4x. (b) Sketch the curve C with equation y = x 3 4x, showing the coordinates of the points at which the curve meets the axis. The point A with x-coordinate 1 and the point B with x-coordinate 3 lie on the curve C. (c) Find an equation of the line which passes through A and B, giving your answer in the form y = mx + c, where m and c are constants. (5) (d) Show that the length of AB is k 10, where k is a constant to be found. 8. (a) On the axes below sketch the graphs of (i) y = x (4 x), (ii) y = x 2 (7 x), (2) Jan 2010, Q9 showing clearly the coordinates of the points where the curves cross the coordinate axes. (b) Show that the x-coordinates of the points of intersection of (5) y = x (4 x) and y = x 2 (7 x) are given by the solutions to the equation x(x 2 8x + 4) = 0. The point A lies on both of the curves and the x and y coordinates of A are both positive. (c) Find the exact coordinates of A, leaving your answer in the form (p + q 3, r + s 3), where p, q, r and s are integers. (7) May 2010, Q10

9 9. (a) Sketch the graphs of (i) y = x(x + 2)(3 x), (ii) y = x 2. showing clearly the coordinates of all the points where the curves cross the coordinate axes. (6) (b) Using your sketch state, giving a reason, the number of real solutions to the equation x(x + 2)(3 x) + x 2 = 0. (2) May 2010, Q The curve C has equation y = x(5 x) and the line L has equation 2y = 5x + 4. (a) Use algebra to show that C and L do not intersect. (b) Sketch C and L on the same diagram, showing the coordinates of the points at which C and L meet the axes. (4) Jan 2012, Q5 (4)

10 11. Figure 1 Figure 1 shows a sketch of the curve C with equation y = x 1 + 1, x 0. The curve C crosses the x-axis at the point A. (a) State the x-coordinate of the point A. The curve D has equation y = x 2 (x 2), for all real values of x. (1) (b) On a copy of Figure 1, sketch a graph of curve D. Show the coordinates of each point where the curve D crosses the coordinate axes. (c) Using your sketch, state, giving a reason, the number of real solutions to the equation x 2 (x 2) = x (1) May 2014, Q14