2 2 ; x ; 12; y ; . . . are the first four terms of a quadratic sequence. If the second difference is 6, calculate the values of x and y. (5) 3. The quadratic pattern;…………..is given. Determine the value of x. (4) 4 In the quadratic sequence, the first three terms of the first differences are and. Determine the: Value(s) of. (2) Second difference(s). (1) Values of and . (2) 4.4 General term of the quadratic sequence. (4) 5 Given the quadratic sequence 3; 5; 11; 21; … 5.1 Write down the value of the next term, if the pattern continues. (1) 5.2 Determine the value of the 48th term. (5) 5.3 Prove that the terms of this sequence will never be even numbers. (2) 5.4 If all the values of this sequence are increased by 30, determine the general term of the new sequence. (2) 6. A pattern of triangles is formed by increasing the base of the triangle by 2 cm and the perpendicular height by 1cm, in each successive triangle. The first triangle has a base of 2 cm and a height of 2 cm. The pattern continues in this manner.
Shift 3 units UPWARDS Change in the equation Graphical representation Add 3 to the original equation Original equation: New equation: Shift 4 units DOWNWARDS Change in the equation Graphical representation Add -4 to the original equation Original equation: New equation: Consider the functions and R E F L E C T I O N S Reflection in the x-axis Change in the equation Graphical representation Substitute y in the original equation with –y. Then make y the subject of the formula again. Original equation: New equation: Original equation: New equation:
2. The sketch below shows the graphs of for and for . 2.1 Write down the equations of the asymptotes of the graph of . (2) 2.2 The graphs of and intersect at the point . Calculate the value of . (2) 2.3 Determine the equation of in the form ….. (3) 2.4 Write down the equation of if is a reflection of about the -axis. (1)
At the point of inflection the concavity of the graphs changes. Consider the diagram below. Along the curve PA the concavity of the graph is concave down. Along curve AQ the concavity of the curve is concave up. Comparison between graphs of and Function: Derived function: The x-values of the turning points are the x-intercepts of the graph of y’ =…. , which is the parabola on the right. The point of inflection’s x-value is the x-value of the turning point of the graph of y’. These x-intercepts, are the turning point x’s of the cubic graph, because the y- values here are 0. The turning point x-value at i, is the x-value of the point of inflection on the cubic graph. At this x-value, the gradient of the tangent = 0 At this x-intercept, y’ = 0. This x-intercept and the x-value of the parabola on the left are the same x-values.
y A B x –2 O 3 3.1 Determine the gradient of at x = 0. (1) 3.2 For which value of x will the gradient of be the same as the gradient in QUESTION 7.2.1? (1) 3.3 Draw a sketch graph of . Show the x-values of the stationary points and the point of inflection on your sketch. It is not necessary to indicate the intercepts with the axes. (3) 3.1 answer (1) 3.2 x = 1 (1) 3.3 y g x -2 ½ 3 turning point at x = − 2 and at x = 3 point of inflection at x = ½ shape (3) 1. The function defined by is represented by the sketch below. A and B are the turning points of f , while is a tangent to f at A which cuts f at point C.
3.3 For , h is concave up and for , h is concave down explanation (3) 3.4 The graph of h has a point of inflection at OR The graph of h changes from concave up to concave down at OR The graph of h changes concavity at answer (1) answer (1) answer (1) 3.5 Gradient of g is – 12 Gradient of tangent is: factors
answer (2) 2.1 decreasing in depth/verlaging in diepte Derivative Substitution Answer 2.2 or At 9h00 and at minutes later. i.e. at 10h20 Derivative factors answers Answer (4) 3.1 answer (1) 3.2 OR
answer (3) correct formula substitution into the correct formula answer (3) 3.3 h(t) = 0 factors t = 8 –20m/s or 20 m/s downwards (5) 3.4 OR answer (1) answer (1) 3. MINIMA & MAXIMA 1.1 formula substitution (2) 1.2 Therefore the box will have a maximum at derivative derivative = 0 factors x-values choosing (5)
2.1 = 340 ✓ ✓ ✓ answer (3) 2.2 total surface area = = ✓ ✓ subst. of h in the total SA formula (2) 2.3 For the surface area to be as small as possible 4 ✓ ✓ ✓ ✓ derivative. (4)
The sum of a number of terms = SOLUTIONS Activity 1 1.1 (3) 1.2 .1 (2) 1.2.2 (3) 1.3 ………………….(1) ………………...(2) (2) – (1) Subst in eq (1) (4)
1.4 (3) [15] Activity 2 2.1.1 (4) 2.1.2 The sequence consists of arithmetic and geometric sequences, therefore you work out the terms from two separate sequences. (4) 2.1.3 For geometric sequence: For arithmetic sequence: (4) 2.2 Subst in to (6)