379 questions · 83 question types identified
Questions that require sketching y = |f(x)| and then solving an inequality of the form |f(x)| > k or |f(x)| < k for a specific constant k.
Questions requiring sketching two polynomial curves and then finding intersection points algebraically by solving the resulting equation.
Questions asking to find the area of a shaded region bounded by a curve and a straight line, typically requiring integration.
Questions involving sketching a rational function (reciprocal or reciprocal squared) with another curve (polynomial or linear) to determine number of intersections or solve equations.
Questions asking to find asymptotes and stationary points for a given rational function, typically with straightforward sketching or verification tasks.
Questions where students must describe or apply multiple distinct transformations (e.g., translation, reflection, or stretch) and sketch the results on separate diagrams.
Questions providing f'(x) and additional information (like a point on the curve or range), asking to find f(x) or sketch the curve.
Questions asking to sketch a polynomial curve given explicitly in factored form like y = (x-a)(x-b)(x-c) or y = (x-a)²(x-b), showing intercepts and shape, without additional transformations or follow-up parts.
Questions asking to find parameter values given that a curve has specific properties like tangency, number of stationary points, or asymptotes, requiring algebraic or calculus-based analysis.
Questions involving sketching a polynomial curve and a straight line, then finding their intersection points or analyzing their relationship.
Questions where a factored or expanded polynomial is sketched, then a horizontal translation is applied by replacing x with (x - a), requiring students to find new roots or describe the transformation.
Rational functions with quadratic numerator and linear or quadratic denominator, requiring polynomial division or algebraic manipulation to find oblique or horizontal asymptotes (e.g., y = x²/(2x+1), y = x²/(x-2), y = (x²-3x+6)/(1-x)).
Questions where a cubic f(x) is given and a vertical translation g(x) = f(x) + k is applied, requiring students to expand, verify a root, and fully factorise the new cubic g(x).
Questions where the curve behaviour (asymptotes, turning points, sketch) is analysed for different ranges or signs of the parameter, or where conditions on the parameter are derived (e.g., show no turning points if λ < 0, find values of p for two distinct turning points).
Questions where variables satisfy y = Ax^p or similar, with a graph of ln(y) against ln(x) given, requiring finding constants A and p from the linear graph.
Questions where a given curve is sketched and a specific straight line must be drawn to solve an equation graphically by finding intersection points.
Questions where the real-world scenario is modelled by a linear relationship (y = mx + c or direct/inverse proportionality), requiring interpretation of parameters or finding specific values.
Questions asking to sketch a rational function by applying a transformation to a standard rational function (e.g., translating 1/x).
Questions asking to sketch a factored polynomial curve and then use the sketch to solve inequalities or identify solution regions.
Questions asking for the equation after applying one specific transformation (translation, reflection, or stretch) to a given curve.
Questions asking to sketch both y = af(x) and y = f(bx) starting from the same given function f(x), requiring direct application of vertical and horizontal stretch transformations.
Questions requiring differentiation to find coordinates of stationary points on polynomial curves (cubic, quartic etc.) with justification of nature using second derivative or sign change, where the function is a standard polynomial expression.
Questions asking to sketch y = -f(x), y = f(-x), involving reflections in the x-axis or y-axis.
Questions where the real-world scenario is modelled by a quadratic equation (parabola), typically involving projectile motion, arches, or optimization, requiring analysis of turning points, intercepts, or specific values.
Questions asking to determine the number of solutions or range of parameter values by analyzing a given graph, using horizontal line intersection counting.
Questions asking to sketch y = af(x) where a is a constant multiplier, involving vertical stretches or compressions of the given curve.
Questions where the function is given in factored form and students must sketch first (possibly using the factored form directly), then expand to polynomial form.
Questions where the function is given in expanded polynomial form and students must first factorise it before sketching the curve.
Questions where the function has a linear or factored denominator (with vertical asymptotes), requiring proof that y cannot take values in an open interval, typically using discriminant of a quadratic formed by y=k substitution.
Questions involving curves defined by y² = f(x) or similar implicit/parametric forms, requiring sketching with attention to symmetry and domain restrictions.
Questions involving discrete optimization scenarios such as worker-task assignment, route planning, or resource allocation using operations research techniques rather than curve analysis.
Questions requiring sketching rational function(s) and solving inequalities by comparing f(x) with another function g(x) (linear or otherwise), typically finding intersection points.
Questions that sketch a rational function and then require sketching both y = |f(x)| and y² = f(x) or other multiple transformations.
Questions that provide two explicit curve equations and ask to describe the single transformation mapping one to the other, where both equations are fully given.
Questions asking students to sketch one or two simple transformations (e.g., f(x+a), af(x), f(ax)) separately, typically with a smooth curve and turning points to track.
Questions that provide a sketch of y = f(x) and ask students to sketch a single horizontal or vertical translation (e.g., y = f(x+a) or y = f(x)+a), requiring identification of how key features transform.
Questions that ask to sketch the curve and then find the derivative, gradient at a point, or equation of a tangent line.
Rational functions with linear numerator and linear denominator, typically having horizontal and vertical asymptotes found directly without division (e.g., y = (3x-1)/(x+2), y = (3x-5)/(2x+4)).
Questions that show two graphs (or multiple graphs) and ask students to identify or state the transformation that maps one to the other, or to state the equation of a transformed graph.
Questions asking students to use the provided sketch to deduce values of x satisfying inequalities (e.g., f(x) > 0, f(x) < 0) by reading directly from the graph.
Questions asking students to solve equations involving transformations of the given function (e.g., f(1/4 x) = 0, f(x-p) = 0) by relating roots to the original sketch.
Questions providing a sketch of y = f(x) and asking students to sketch exactly two transformations on separate diagrams, where both transformations are translations (vertical or horizontal shifts only, e.g. f(x)+a and f(x+b)).
Questions where the function has a quadratic denominator (no vertical asymptotes), requiring proof that y lies within a closed interval using discriminant analysis of the resulting quadratic in x.
Questions asking to show graphically that an equation has a certain number of roots, then use an iterative formula to find a root to specified accuracy.
Questions that give an explicit algebraic function (like f(x) = x³ - 6x² + 5x + 12 or f(x) = ln x) and ask students to sketch the original and/or transformed versions, requiring both algebraic manipulation and transformation application.
Questions that ask for the coordinates of a transformed point (typically a minimum or maximum) after applying a given transformation to a curve.
Questions asking for the equation after applying two or more transformations in a specified order to a given curve.
Rational functions expressed as sum of polynomial and reciprocal terms, where asymptotes are found by considering behavior as x approaches infinity or zero (e.g., y = x/3 + 12/x, y = x - 5 + 1/(x-2), y = 1 + 4/(x(x-3))).
Questions where a quadratic in standard form ax²+bx+c must be converted to completed square form a(x+p)²+q to find the vertex coordinates.
Questions that provide a graph of y = f(x) with key features marked and ask students to sketch a specific transformation (like y = f(x+a) or y = af(x)) on separate axes.
Questions requiring differentiation of a polynomial (including those with exponential or composite factors) to find the equation of a tangent line at a specified point on the curve.
Questions requiring sketching a polynomial alongside a rational function, modulus function, or piecewise function to analyze intersections.
Questions asking students to sketch a composition of multiple transformations applied together (e.g., -4f(x+3), f(|x|+1)) requiring students to apply transformations in sequence.
Questions involving sketching a polynomial and then applying a reflection (such as y → -y or x → -x) or vertical translation, requiring analysis of how the curve changes under these transformations.
Questions showing a sketch with marked features (intercepts, turning points) where students must determine unknown constants in a partially given polynomial equation.
Questions where a cubic is given in factored form, then a horizontal translation f(x-a) or f(x+a) is applied, and students must find the new roots or write the equation of the translated graph.
Questions asking to sketch y = |f(x)| or related absolute value transformations, given the original curve or its equation.
Questions involving sketching a trigonometric curve (sine or cosine) and another curve (typically linear) to determine number of intersections or solutions.
Questions asking for the range of trigonometric functions (sine, cosine) with transformations, using amplitude and vertical shift analysis.
Questions that ask students to determine the new coordinates of specific points after one or more transformations are applied, without necessarily sketching the full curve.
Questions asking to sketch y = f(ax) where a is a constant inside the function argument, involving horizontal stretches or compressions of the given curve.
Questions that ask to sketch the curve and then solve equations involving transformations of the function or intersections with other curves/lines.
Questions that ask to sketch the curve and then manipulate the algebraic form by expanding to standard form or factorising completely.
Questions where the vertex coordinates and another point are given, requiring construction of the quadratic function in vertex form then possibly expanding.
Questions that only require sketching y = |f(x)| without solving any inequality involving a constant k.
Questions providing some roots and asking students to find unknown coefficients in the polynomial by using factor theorem or expanding and comparing coefficients.
Questions providing a sketch of y = f(x) and asking students to sketch two or more transformations on separate diagrams where at least one transformation is a stretch or reflection (e.g. 2f(x), f(2x), -f(x)) rather than a pure translation.
Questions where parameter values (a, b, λ, p) must be determined from given conditions such as known asymptotes, a point on the curve, or a given asymptote equation.
Questions asking to find parameter values or domain restrictions that ensure a function is monotonic (increasing/decreasing), typically using derivative sign analysis.
Questions that sketch a rational function and then sketch y = |f(x)| only, without any squared or reciprocal variations.
Questions asking for the coordinates of a transformed point when a curve undergoes a specified transformation, rather than the equation itself.
Questions asking students to sketch translations of standard algebraic functions given by equation only (e.g., y = (x+3)² or y = (x+3)²+k), without a provided sketch, requiring knowledge of the base function shape.
Questions asking students to prove algebraically whether a given function is even, odd, or neither, sometimes with accompanying sketches.
Questions where the quadratic is already in completed square form a(x+p)²+q and the vertex can be read directly without algebraic manipulation.
Questions where the absolute value is applied to x within the function (e.g., y = (|x|+1)/(|x|-1)) rather than to the entire function output.
Questions requiring product rule differentiation to find stationary points where the function involves a polynomial multiplied by an exponential term such as e^x or e^(-x).
Questions where the polynomial contains unknown constants (a, b, etc.) that must first be determined from given conditions before finding stationary points.
Questions requiring finding the equation of a normal line to a curve, or finding a tangent parallel to a given line or to the tangent at another point, involving an additional step beyond a straightforward tangent calculation.
Questions asking to sketch y = |f(x)| or y = f(|x|), involving modulus/absolute value transformations.
Questions requiring graphical solution of linear programming problems by sketching constraint lines and finding optimal vertices of the feasible region.
Questions where the real-world scenario is modelled by exponential or logarithmic functions, typically involving growth or decay over time.
Questions asking for range or parameter values for rational functions (including reciprocal functions), using asymptote and transformation analysis.
Questions asking to sketch multiple transformations where stretches (vertical or horizontal) are combined with other transformations like reflections or translations, not just pure stretches.