DP Mathematics HL Questionbank

• Syllabus

2.2

Sub sections and their related questions

The graph of a function; its equation $y = f\left( x \right)$ .

• 12M.1.hl.TZ1.2a: Draw the graph of y = f (x) on the blank grid below.
• 12M.2.hl.TZ1.11a: Write down the coordinates of the minimum point on the graph of f .
• 12M.2.hl.TZ1.11c: Find the coordinates of the point, on $y = f(x)$ , where the gradient of the graph is 3.
• 12M.1.hl.TZ2.7a: On the axes below, sketch the graph of $y = \frac{1}{{f(x)}}$ , clearly showing the coordinates...
• 12M.1.hl.TZ2.11a: State the range of f and of g .
• 12M.2.hl.TZ2.6a: Sketch the curve...
• 12M.2.hl.TZ2.12b: Sketch the graph of v against t , clearly showing the coordinates of any intercepts, and the...
• 12N.1.hl.TZ0.3a: Using the information shown in the diagram, find the values of a , b and c .
• 08M.2.hl.TZ1.2: (a)     Sketch the curve...
• 08N.2.hl.TZ0.6: (a)     Sketch the curve $y = \left| {\ln x} \right| - \left| {\cos x} \right| - 0.1$ ,...
• 11M.1.hl.TZ2.3a: Sketch the graph of the function. You are not required to find the coordinates of the maximum.
• 11M.1.hl.TZ2.5a: Sketch the graph of $y = \frac{1}{{f(x)}}$.
• 11M.1.hl.TZ2.5b: Sketch the graph of  $y = x{\text{ }}f(x)$ .
• 11M.2.hl.TZ2.5: Sketch the graph of $f(x) = x + \frac{{8x}}{{{x^2} - 9}}$. Clearly mark the coordinates of the...
• 09M.1.hl.TZ2.11: A function is defined as $f(x) = k\sqrt x$, with $k > 0$ and $x \geqslant 0$ . (a)    ...
• SPNone.1.hl.TZ0.9b: Sketch the graph of f , showing clearly the coordinates of the maximum and minimum.
• SPNone.1.hl.TZ0.9c: Hence show that ${{\text{e}}^\pi } > {\pi ^{\text{e}}}$ .
• SPNone.2.hl.TZ0.6: The function f is of the form $f(x) = \frac{{x + a}}{{bx + c}}$, $x \ne - \frac{c}{b}$. Given...
• 13M.1.hl.TZ1.12c: Sketch the graph of $y = f(x)$.
• 13M.2.hl.TZ1.12a: Sketch the graph of ${v_A} = {t^3} - 5{t^2} + 6t$ for $t \geqslant 0$, with ${v_A}$ on the...
• 13M.2.hl.TZ1.12b: Write down the times for which the velocity of the particle is increasing.
• 13M.2.hl.TZ1.12c: Write down the times for which the magnitude of the velocity of the particle is increasing.
• 10M.1.hl.TZ1.5: The graph of $y = \frac{{a + x}}{{b + cx}}$ is drawn below.     (a)     Find the value of...
• 13M.1.hl.TZ2.12e: (i)     On a different diagram, sketch the graph of $y = f(|x|)$ where $x \in D$. (ii)    ...
• 13M.2.hl.TZ2.13c: Sketch the graph of $\theta$, for $0 \leqslant x \leqslant 20$.
• 11N.1.hl.TZ0.4a: sketch the graph of $f$;  
• 11N.2.hl.TZ0.1a: Sketch the graph, clearly labelling the x and y intercepts with their values.
• 11M.1.hl.TZ1.10b: On the axes below, sketch the graph of $y = g(x)$ . On the graph, indicate any asymptotes and...
• 11M.1.hl.TZ1.12d: Now consider the functions $g(x) = \frac{{\ln \left| x \right|}}{x}$ and...
• 14M.2.hl.TZ1.12: Let $f(x) = \left| x \right| - 1$. (a)     The graph of $y = g(x)$ is drawn below.      ...
• 14M.1.hl.TZ2.5a: Sketch the graph of $y = \left| {\cos \left( {\frac{x}{4}} \right)} \right|$ for...
• 14M.2.hl.TZ2.7a: (i)     Sketch the graph of $y = f(x)$, clearly indicating any asymptotes and axes...
• 14M.2.hl.TZ2.14a: Sketch the graph of $y = v(t)$. Indicate clearly the local maximum and write down its coordinates.
• 13N.2.hl.TZ0.3a: Sketch the graph of $y = f(x)$, stating the coordinates of any maximum and minimum points and...
• 14N.3sp.hl.TZ0.1a: Sketch the graph of $y = f(x)$.
• 15M.2.hl.TZ1.11a: Sketch the graph $y = f(x)$.
• 15M.2.hl.TZ2.3a: Sketch the graph of $y = {(x - 5)^2} - 2\left| {x - 5} \right| - 9,{\text{ for }}0 \le x \le 10$.
• 15N.1.hl.TZ0.12e: Sketch the graph of $y = f(x)$ indicating clearly the coordinates of the $x$-intercepts and...
• 16M.2.hl.TZ1.5b.i: Sketch the graph $y = f(x)$.
• 16M.2.hl.TZ1.5b.ii: Write down the range of $f$.
• 16M.1.hl.TZ2.2: The function $f$  is defined as...
• 16M.1.hl.TZ1.7a: Sketch on the same axes the curve $y = \left| {\frac{7}{{x - 4}}} \right|$ and the line...
• 16M.1.hl.TZ1.7b: Find the exact solutions to the equation $x + 2 = \left| {\frac{7}{{x - 4}}} \right|$.
• 16N.2.hl.TZ0.2: Find the acute angle between the planes with equations $x + y + z = 3$ and $2x - z = 2$.
• 16N.2.hl.TZ0.5a: Sketch the graph of $f$ indicating clearly any intercepts with the axes and the coordinates of...
• 17M.1.hl.TZ1.11a.i: Express ${x^2} + 3x + 2$ in the form ${(x + h)^2} + k$.
• 17M.1.hl.TZ1.11a.ii: Factorize ${x^2} + 3x + 2$.
• 17M.1.hl.TZ1.11b: Sketch the graph of $f(x)$, indicating on it the equations of the asymptotes, the coordinates...
• 17M.1.hl.TZ1.11c: Show that $\frac{1}{{x + 1}} - \frac{1}{{x + 2}} = \frac{1}{{{x^2} + 3x + 2}}$.
• 17M.1.hl.TZ1.11e: Sketch the graph of $y = f\left( {\left| x \right|} \right)$.
• 17N.2.hl.TZ0.10b: Sketch the graph of $y = f(x)$ showing clearly the minimum point and any asymptotic behaviour.
• 18M.1.hl.TZ1.9c: Sketch the graph of $y = f\left( x \right)$ showing clearly the position of the points A and B.
• 18M.1.hl.TZ2.2a: Sketch the graphs of $y = \frac{x}{2} + 1$ and $y = \left| {x - 2} \right|$ on the following...
• 18M.1.hl.TZ2.10b.i: Express $g\left( x \right)$ in the form $A + \frac{B}{{x - 2}}$ where A, B are constants.
• 18M.1.hl.TZ2.10b.ii: Sketch the graph of $y = g\left( x \right)$. State the equations of any asymptotes and...
• 18M.2.hl.TZ2.10b: Show that $g\left( t \right) = {\left( {\frac{{1 + t}}{{1 - t}}} \right)^2}$.
• 18M.2.hl.TZ2.10c: Sketch the graph of $y = g\left( t \right)$ for t ≤ 0. Give the coordinates of any intercepts...

Investigation of key features of graphs, such as maximum and minimum values, intercepts, horizontal and vertical asymptotes and symmetry, and consideration of domain and range.

• 12M.1.hl.TZ1.2a: Draw the graph of y = f (x) on the blank grid below.
• 12M.2.hl.TZ1.11a: Write down the coordinates of the minimum point on the graph of f .
• 12M.1.hl.TZ2.7a: On the axes below, sketch the graph of $y = \frac{1}{{f(x)}}$ , clearly showing the coordinates...
• 12M.1.hl.TZ2.11a: State the range of f and of g .
• 12M.2.hl.TZ2.6a: Sketch the curve...
• 12M.2.hl.TZ2.12b: Sketch the graph of v against t , clearly showing the coordinates of any intercepts, and the...
• 12N.1.hl.TZ0.3a: Using the information shown in the diagram, find the values of a , b and c .
• 08N.2.hl.TZ0.6: (a)     Sketch the curve $y = \left| {\ln x} \right| - \left| {\cos x} \right| - 0.1$ ,...
• 11M.1.hl.TZ2.3a: Sketch the graph of the function. You are not required to find the coordinates of the maximum.
• 11M.2.hl.TZ2.5: Sketch the graph of $f(x) = x + \frac{{8x}}{{{x^2} - 9}}$. Clearly mark the coordinates of the...
• 10M.1.hl.TZ1.5: The graph of $y = \frac{{a + x}}{{b + cx}}$ is drawn below.     (a)     Find the value of...
• 10N.2.hl.TZ0.8: The diagram shows the graphs of a linear function f and a quadratic function g.     On the...
• 11M.1.hl.TZ1.10b: On the axes below, sketch the graph of $y = g(x)$ . On the graph, indicate any asymptotes and...
• 14M.2.hl.TZ1.10a: Find the equations of the horizontal and vertical asymptotes of the curve $y = f(x)$.
• 14M.2.hl.TZ1.12: Let $f(x) = \left| x \right| - 1$. (a)     The graph of $y = g(x)$ is drawn below.      ...
• 14M.2.hl.TZ2.7a: (i)     Sketch the graph of $y = f(x)$, clearly indicating any asymptotes and axes...
• 14M.2.hl.TZ2.14a: Sketch the graph of $y = v(t)$. Indicate clearly the local maximum and write down its coordinates.
• 13N.1.hl.TZ0.3b: State the range of ${f^{ - 1}}$.
• 13N.2.hl.TZ0.3a: Sketch the graph of $y = f(x)$, stating the coordinates of any maximum and minimum points and...
• 14N.1.hl.TZ0.1b: State the equations of the asymptotes of the graph of $g$.
• 15M.1.hl.TZ1.11c: Find the coordinates of any local maximum and minimum points on the graph of $y(x)$. Justify...
• 15M.1.hl.TZ1.11d: Find the coordinates of any points of inflexion on the graph of $y(x)$. Justify whether any...
• 15M.1.hl.TZ2.10a: Sketch the graph of $y = f(x)$, indicating clearly any asymptotes and points of intersection...
• 15M.2.hl.TZ2.12b: Sketch a displacement/time graph for the particle, $0 \le t \le 5$, showing clearly where the...
• 15N.1.hl.TZ0.12e: Sketch the graph of $y = f(x)$ indicating clearly the coordinates of the $x$-intercepts and...
• 16M.2.hl.TZ1.5b.i: Sketch the graph $y = f(x)$.
• 16M.2.hl.TZ1.5b.ii: Write down the range of $f$.
• 16M.1.hl.TZ2.2: The function $f$  is defined as...
• 16M.1.hl.TZ1.7a: Sketch on the same axes the curve $y = \left| {\frac{7}{{x - 4}}} \right|$ and the line...
• 16M.1.hl.TZ1.7b: Find the exact solutions to the equation $x + 2 = \left| {\frac{7}{{x - 4}}} \right|$.
• 16N.2.hl.TZ0.2: Find the acute angle between the planes with equations $x + y + z = 3$ and $2x - z = 2$.
• 16N.2.hl.TZ0.5a: Sketch the graph of $f$ indicating clearly any intercepts with the axes and the coordinates of...
• 17M.1.hl.TZ1.11a.i: Express ${x^2} + 3x + 2$ in the form ${(x + h)^2} + k$.
• 17M.1.hl.TZ1.11a.ii: Factorize ${x^2} + 3x + 2$.
• 17M.1.hl.TZ1.11b: Sketch the graph of $f(x)$, indicating on it the equations of the asymptotes, the coordinates...
• 17M.1.hl.TZ1.11c: Show that $\frac{1}{{x + 1}} - \frac{1}{{x + 2}} = \frac{1}{{{x^2} + 3x + 2}}$.
• 17M.1.hl.TZ1.11e: Sketch the graph of $y = f\left( {\left| x \right|} \right)$.
• 17N.1.hl.TZ0.6a: Sketch the graph of $y = \frac{{1 - 3x}}{{x - 2}}$, showing clearly any asymptotes and stating...
• 17N.2.hl.TZ0.10b: Sketch the graph of $y = f(x)$ showing clearly the minimum point and any asymptotic behaviour.
• 18M.1.hl.TZ1.9c: Sketch the graph of $y = f\left( x \right)$ showing clearly the position of the points A and B.
• 18M.1.hl.TZ2.2a: Sketch the graphs of $y = \frac{x}{2} + 1$ and $y = \left| {x - 2} \right|$ on the following...
• 18M.1.hl.TZ2.10b.i: Express $g\left( x \right)$ in the form $A + \frac{B}{{x - 2}}$ where A, B are constants.
• 18M.1.hl.TZ2.10b.ii: Sketch the graph of $y = g\left( x \right)$. State the equations of any asymptotes and...
• 18M.2.hl.TZ2.10b: Show that $g\left( t \right) = {\left( {\frac{{1 + t}}{{1 - t}}} \right)^2}$.
• 18M.2.hl.TZ2.10c: Sketch the graph of $y = g\left( t \right)$ for t ≤ 0. Give the coordinates of any intercepts...

The graphs of the functions $y = \left| {f\left( x \right)} \right|$ and $y = f\left( {\left| x \right|} \right)$ .

• 12M.1.hl.TZ1.2a: Draw the graph of y = f (x) on the blank grid below.
• 16M.2.hl.TZ1.5b.i: Sketch the graph $y = f(x)$.
• 16M.2.hl.TZ1.5b.ii: Write down the range of $f$.
• 16M.1.hl.TZ1.7a: Sketch on the same axes the curve $y = \left| {\frac{7}{{x - 4}}} \right|$ and the line...
• 16M.1.hl.TZ1.7b: Find the exact solutions to the equation $x + 2 = \left| {\frac{7}{{x - 4}}} \right|$.
• 16N.1.hl.TZ0.3a: state the value of $a$ and the value of $c$;
• 16N.1.hl.TZ0.3b: find the value of $b$.
• 17M.1.hl.TZ1.11e: Sketch the graph of $y = f\left( {\left| x \right|} \right)$.
• 18M.1.hl.TZ2.2a: Sketch the graphs of $y = \frac{x}{2} + 1$ and $y = \left| {x - 2} \right|$ on the following...
• 18M.2.hl.TZ2.10b: Show that $g\left( t \right) = {\left( {\frac{{1 + t}}{{1 - t}}} \right)^2}$.
• 18M.2.hl.TZ2.10c: Sketch the graph of $y = g\left( t \right)$ for t ≤ 0. Give the coordinates of any intercepts...

The graph of $\frac{1}{{f\left( x \right)}}$ given the graph of $y = f(x)$ .

• 08M.2.hl.TZ2.8: The graph of $y = f(x){\text{ for }} - 2 \leqslant x \leqslant 8$ is shown.     On the set...
• 16M.2.hl.TZ1.5b.i: Sketch the graph $y = f(x)$.
• 16M.2.hl.TZ1.5b.ii: Write down the range of $f$.
• 16M.1.hl.TZ1.7b: Find the exact solutions to the equation $x + 2 = \left| {\frac{7}{{x - 4}}} \right|$.
• 16N.1.hl.TZ0.3a: state the value of $a$ and the value of $c$;
• 16N.1.hl.TZ0.3b: find the value of $b$.
• 18M.1.hl.TZ2.2a: Sketch the graphs of $y = \frac{x}{2} + 1$ and $y = \left| {x - 2} \right|$ on the following...
• 18M.2.hl.TZ2.10b: Show that $g\left( t \right) = {\left( {\frac{{1 + t}}{{1 - t}}} \right)^2}$.
• 18M.2.hl.TZ2.10c: Sketch the graph of $y = g\left( t \right)$ for t ≤ 0. Give the coordinates of any intercepts...