2016 2014 CAPS Graphs P1 28 Sept 2

Western Cape Education Department Examination Preparation Learning Resource 2016 Functions and Graphs MATHEMATICS Grade 12 Razzia Ebrahim Senior Curriculum Planner for Mathematics E-mail: Razzia.Ebrahim@wced.info/ Razzia.Ebrahim@westerncape.gov.za Website: http://www.wcedcurriculum.westerncape.gov.za/index.php/component/jdownloads/category/1835- grade-12?Itemid=-1 Website: http://wcedeportal.co.za

Mathematics P1 Graphs Index Page 1. 2016 June Paper 1 3–4 2. 2016 Feb-March Paper 1 4–6 3. 2015 November Paper 1 6–8 4. 2015 June Paper 1 9 – 11 5. 2015 Feb-March Paper 1 12 – 13 6. 2014 November Paper 1 14 – 16 7. 2014 Exemplar Paper 1 16 – 18 Copyright reserved 2 Please turn over

Mathematics P1 Graphs DBE/2016 QUESTION 4 The sketch below shows the graphs of f ( x)   x 2  2 x  3 and g ( x)  mx  q . Graph f has x-intercepts at A and B(1 ; 0) and a turning point at C. The straight line g, passing through A and C, cuts the y-axis at E. y g E C f A B x O 4.1 Write down the coordinates of the y-intercept of f. (1) 4.2 Show that the coordinates of C are  1 ; 4  . (3) 4.3 Write down the coordinates of A. (1) 4.4 Calculate the length of CE. (6) 4.5 Determine the value of k if h( x)  2 x  k is a tangent to the graph of f. (5) 4.6 Determine the equation of g 1 , the inverse of g, in the form y = ... (2) 4.7 For which value(s) of x is g  x   g 1  x  ? (3) [21] Copyright reserved 3 Please turn over

Mathematics P1 Graphs DBE/2016 3 f ( x)   q and g ( x)  2 x  r x p  g intersects the vertical asymptote of f at A.  B is the common y-intercept of f and g.  y = 2 is the common horizontal asymptote of f and g. y g B f A 2 f f x O 5.1 Write down the value of r. (1) 5.2 Determine the value of p. (4) 5.3 Determine the coordinates of A. (3) 5.4 For which value(s) of x is f x   g ( x )  0 ? (2) 5.5 If h( x)  f ( x  2) , write down the equation of h. (2) [12] QUESTION 4 DBE/Feb.-Mrt. 2016 Given: J(x) = rx + l 4.1 Determine the coordinates of the y-intercept off (1) 4.2 Sketch the graph of f, clearly indicating ALL intercepts with the axes as well as any asymptotes. (3) 4.3 Calculate the average gradient of f between the points on the graph where x = -2 and X = 1. (3) 4.4 If h(x)=3f(x), write down an equation of the asymptote of h. (1) [SJ Copyright reserved 4 Please turn over

Mathematics P1 Graphs DBE/Feb.-Mar. 2016 QUESTION 5 The graphs of the functions f(x) = a(x + p ) + q and g(x) = _k_ + d are sketched below. 2 x+r Both graphs cut the y-axis at - 4 . One of the points of intersection of the graphs is P(l ; -8), which is also the turning point off The horizontal asymptote of g is y = -2 . y 5.1 Calculate the values of a, p and q. (4) 5.2 Calculate the values of k, r and d. ( 6) 5.3 Determine the value(s)of x in the interval x::51 for which g(x)?:.f(x). (2) 5.4 Determine the value(s)of k for which J(x) = k has two, unequal positive roots. ( 2) 5.5 Write down an equation for the axis of symmetry of g that has a negative gradient. (3) 5.6 The point P is reflected in the line determined in QUESTION 5.5 to give the point Q. Write down the coordinates of Q. (2) [l 9] Copyright reserved 5 Please turn over

Mathematics P1 Graphs DBE/Feb.-Mar. 20I 6 QUESTION 6 I Given: f(x) = -x 2 , x � 0 4 6.1 Determine the equation of 1- 1 in the form 1- 1 (x) = . . . (3) 6.2 On the same system of axes, sketch the graphs of I and 1- 1 • Indicate clearly the intercepts with the axes, as well as another point on the graph of each off and 1- 1 • (3) 6.3 Is 1- 1 a function? Give a reason for your answer. (2) [8] DBE/November 2015 QUESTION 4 Given: f ( x) = 2 x +1 − 8 4.1 Write down the equation of the asymptote of f. (1) 4.2 Sketch the graph of f. Clearly indicate ALL intercepts with the axes as well as the asymptote. (4) 4.3 The graph of g is obtained by reflecting the graph of f in the y-axis. Write down the equation of g. (1) [6] Copyright reserved 6 Please turn over

Mathematics P1 Graphs DBE/November 2015 QUESTION 5 Given: h( x ) = 2 x − 3 for − 2 ≤ x ≤ 4 . The x-intercept of h is Q. 5.1 Determine the coordinates of Q. (2) 5.2 Write down the domain of h −1 . (3) 5.3 Sketch the graph of h −1 in your ANSWER BOOK, clearly indicating the y-intercept and the end points. (3) 5.4 For which value(s) of x will h( x ) = h −1 ( x ) ? (3) 5.5 P(x ; y) is the point on the graph of h that is closest to the origin. Calculate the distance OP. (5) 5.6 Given: h( x) = f ′( x) where f is a function defined for − 2 ≤ x ≤ 4 . 5.6.1 Explain why f has a local minimum. (2) 5.6.2 Write down the value of the maximum gradient of the tangent to the graph of f. (1) [19] Copyright reserved 7 Please turn over

Mathematcs P1 Graphs DBE/November 2015 QUESTION 6 6.1 The graphs of f ( x) = −2 x 2 + 18 and g ( x) = ax 2 + bx + c are sketched below. Points P and Q are the x-intercepts of f. Points Q and R are the x-intercepts of g. S is the turning point of g. T is the y-intercept of both f and g. y T f g x P 0 Q R S 6.1.1 Write down the coordinates of T. (1) 6.1.2 Determine the coordinates of Q. (3) 6.1.3 Given that x = 4,5 at S, determine the coordinates of R. (2) 6.1.4 Determine the value(s) of x for which g ′′( x) > 0 . (2) a 6.2 The function defined as y = + q has the following properties: x+ p • The domain is x ∈ R, x ≠ −2. • y = x + 6 is an axis of symmetry. • The function is increasing for all x ∈ R, x ≠ −2. Draw a neat sketch graph of this function. Your sketch must include the asymptotes, if any. (4) [12] Copyright reserved 8 Please turn over

9 Mathematics P1/Graphs DBE/2015 QUESTION 4 The diagram below shows the graph of g(x) = _.!!_ _ 2. The point (0; - 5) lies on g. x-1 y g X 0 ! I I I I I I I ------------------------------------------------ ----+------------------------------------------ 4.1 Write down the range of g. (2) 4.2 Determine the value of a. (2) 4.3 If another function h is defined as h(x) = g(x - 3) + 7, determine the coordinates of the point of intersection of the asymptotes of h. (3) [7] Copyright reserved 9 Please turn over

Mathematics P1 Graphs DBE/2015 QUESTIONS Given: l(x) = ( J! 5.1 Determine the value of /(-2). (2) 5.2 Write down the equation of 1- 1 (x) in the form y = ... (2) 5.3 Andrew has no idea how to draw the graph of 1- 1• Explain to Andrew how he may use the graph of I to draw the graph of 1- 1• (1) 5.4 Hence or otherwise, sketch the graph of 1- 1 in your ANSWER BOOK. Clearly indicate ALL intercepts with the axes. (2) 5.5 Write down the domain of 1- 1• (1) 5.6 For which values of x will 1- 1 (x) � -2? (2) 1 5.7 Given: q = log 1 - 4 2 5.7.1 Determine the value of q. (1) 5.7.2 Hence or otherwise, determine the coordinates of the point of intersection of I and 1- 1• (3) (14) Copyright reserved 10 Please turn over

Mathematics P1 Graphs DBE/2015 QUESTI0N6 The sketch below shows the graphs of g(x)=-I2x+I2 and J(x)=-3x2 -9x+30. A and B are the x-intercepts off and C is a point on / D is a point on g such that CD is parallel to the y-axis. H and K are the points of intersection off and g. y C X 6.1 Determine the length of AB. (4) 6.2 Determine the coordinates of K. (5) 6.3 Determine the values of x for which J(x)- g(x) :s; 0. (3) 6.4 Determine the maximum length of CD for - 2 :s; x :s; 3 . (5) [17) Copyright reserved 11 Please turn over

Mathematics P1 Graphs DBE/Feb.-Mar. 2015 QUESTION 4 6 Given: g(x) = -- -1 x+2 4.1 Write down the equations of the asymptotes of g. (2) 4.2 Calculate: 4.2.1 They-intercept of g (1) 4.2.2 The x-intercept of g (2) 4.3 Draw the graph of g, showing clearly the asymptotes and the intercepts with the axes. (3) 4.4 Determine the equation of the line of symmetry that has a negative gradient, in the form y = .... (3) 6 4.5 Determine the value(s) of x for which -- -1 � -x - 3 . x+2 (2) [13] QUESTION 5 The graph of f(x) =ax , a> 1 is shown below. T(2; 9) lies on f y T(2; 9) f 5.1 Calculate the value of a. (2) 5.2 Determine the equation of g(x) if g(x) = j(-x). (1) 5.3 Detem1ine the value(s) of x for which /- 1 (x) � 2. (2) 5.4 Is the inverse of f a function? Explain your answer. (2) [7] Copyright reserved 12 Please turn over

Mathematics P1 Graphs DBE/Feb.-Mar. 2015 QUESTION 6 The graphs of f(x) = ax 2 +bx+ c ; a* 0 and g(x) = mx + k are drawn below. D( 1 ; - 8) is a common point on f and g. • f intersects the x-axis at (- 3 ; 0) and (2; 0). • g is the tangent to fat D. y 6.1 For which value(s) of x is /(x) � 0? (2) 6.2 Determine the values of a, b and c. (5) 6.3 Determine the coordinates of the turning point off (3) 6.4 Write down the equation of the axis of symmetry of h if h(x) = f(x - 7) + 2. (2) 6.5 Calculate the gradient of g. (3) [15) Copyright reserved 13 Please turn over

Mathematics P1 Graphs DBE/November 2014 QUESTION 4 2 The diagram below shows the hyperbola g defined by g ( x)   q with asymptotes x p y = 1 and x = – 1. The graph of g intersects the x-axis at T and the y-axis at (0 ; 3). The line y = x intersects the hyperbola in the first quadrant at S. y 3 y=x g S 1 T x 1 O g 4.1 Write down the values of p and q. (2) 4.2 Calculate the x-coordinate of T. (2) 4.3 Write down the equation of the vertical asymptote of the graph of h, if h( x)  g ( x  5) (1) 4.4 Calculate the length of OS. (5) 4.5 For which values of k will the equation g(x) = x + k have two real roots that are of opposite signs? (1) [11] Copyright reserved 14 Please turn over

Mathematics P1 Graphs DBE/November 2014 f QUESTION 5 Given: f ( x)  log a x where a  0 . S ;1 is a point on the graph of f. 1 3  y f x O . S  1 ; 1 3  5.1 Prove that a  3 . (2) 5.2 Write down the equation of h, the inverse of f, in the form y = ... (2) 5.3 If g ( x)   f ( x) , determine the equation of g. (1) 5.4 Write down the domain of g. (1) 5.5 Determine the values of x for which f x   3. (3) [9] Copyright reserved 15 Please turn over

Mathematics P1 Graphs DBE/November 2014 QUESTION 6 Given: g ( x)  4 x 2  6 and f ( x)  2 x . The graphs of g and f are sketched below. S is an x-intercept of g and K is a point between O and S. The straight line QKT with Q on the graph of f and T on the graph of g, is parallel to the y-axis. y  f g Q x O K S T 6.1 Determine the x-coordinate of S, correct to TWO decimal places. (2) 6.2 Write down the coordinates of the turning point of g. (2) 6.3 6.3.1 Write down the length of QKT in terms of x, where x is the x-coordinate of K. (3) 6.3.2 Calculate the maximum length of QT. (6) [13] NSC – Grade 12 Exemplar/ DBE/2014 QUESTION 4 2 4.1 Given: f ( x) = −3 x +1 4.1.1 Calculate the coordinates of the y-intercept of f. (2) 4.1.2 Calculate the coordinates of the x-intercept of f. (2) 4.1.3 Sketch the graph of f in your ANSWER BOOK, showing clearly the asymptotes and the intercepts with the axes. (3) 4.1.4 One of the axes of symmetry of f is a decreasing function. Write down the equation of this axis of symmetry. (2) Copyright reserved 16 Please turn over

Mathematics P1 Graphs DBE/2014 4.2 The graph of an increasing exponential function with equation f ( x ) = a.b x + q has the following properties: • Range: y > −3 • The points (0 ; – 2) and (1; – 1) lie on the graph of f. 4.2.1 Determine the equation that defines f . (4) 4.2.2 Describe the transformation from f ( x ) to h( x ) = 2.2 x + 1 (2) [15] QUESTION 5 The sketch below shows the graphs of f ( x) = −2 x 2 − 5 x + 3 and g ( x) = ax + q . The angle of inclination of graph g is 135° in the direction of the positive x-axis. P is the point of intersection of f and g such that g is a tangent to the graph of f at P. y P g 135º x 0 f 5.1 Calculate the coordinates of the turning point of the graph of f. (3) 5.2 Calculate the coordinates of P, the point of contact between f and g. (4) 5.3 Hence or otherwise, determine the equation of g. (2) 5.4 Determine the values of d for which the line k ( x) = − x + d will not intersect the graph of f. (1) [10] Copyright reserved 17 Please turn over

Mathematics P1 DBE/2014 QUESTION 6 The graph of g is defined by the equation g ( x) = ax . The point (8 ; 4) lies on g. 6.1 Calculate the value of a. (2) 6.2 If g ( x) > 0 , for what values of x will g be defined? (1) 6.3 Determine the range of g. (1) 6.4 Write down the equation of g −1 , the inverse of g, in the form y = ... (2) 6.5 If h( x) = x − 4 is drawn, determine ALGEBRAICALLY the point(s) of intersection of h and g. (4) 6.6 Hence, or otherwise, determine the values of x for which g ( x ) > h( x ) . (2) [12] 18