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Cambridge O Level · Mathematics 4024

Graphs of functions

Construct tables and use formulae, plotted coordinates and written graph features to recognise and interpret linear, polynomial, reciprocal, square-root and exponential functions. This original practice pack targets Cambridge O Level Mathematics 4024 sections 2.10 and 2.11, including roots, intersections, turning points, symmetry, asymptotes, exponential change and tangent gradients. Its text-and-coordinate practice supports but does not replace drawing accurate graphs on axes, so it does not claim complete graphical-skill coverage.

2025 / 2026 / 2027 · Academic review not recorded · Published 2026-09-08

AI-assisted practice — not independently academically reviewed. Answers may contain errors; check important results against your course materials.

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21 lesson sections

Lesson contents · 21 sections

What you will learn

  • Construct accurate tables of values from function rules, including careful use of negative inputs and brackets.
  • Plot and interpret coordinates for sums of up to three terms involving powers x^n for the required integer and half-integer indices.
  • Recognise linear, quadratic and cubic graphs from their equations and explicitly stated features.
  • Interpret reciprocal and inverse-power graphs using domains, signs, branches and vertical or horizontal asymptotes.
  • Construct and interpret exponential graphs of the form ab^x + c, including growth and decay models.
  • Read roots as x-axis intersections and solve equations from explicitly stated curve-line intersection coordinates.
  • Identify turning points and lines of symmetry from equations, completed-square form or stated coordinates.
  • Estimate a curve's gradient from two explicit points on a drawn tangent and attach appropriate units in context.
  • Use graph features to check whether calculated coordinates and interpretations are reasonable.
  • Distinguish the text-and-coordinate reasoning in this pack from the separate practical skill of drawing a graph accurately.

Explanation

Coordinates turn a function rule into a graph

A function rule assigns an output y to each permitted input x. A table of values records ordered pairs (x, y); plotting those pairs on labelled axes and joining them with the appropriate smooth curve or straight line produces the graph. The x-coordinate is always written first. For example, (3, -2) means x = 3 and y = -2, not the reverse.

The y-intercept occurs where x = 0. A root or x-intercept occurs where y = 0. A point belongs to a graph only when its coordinates satisfy the rule. Text coordinates can test these ideas precisely, but learners should also practise choosing scales, plotting points and drawing curves on paper.

Worked example

Worked example: build and read a linear graph

For y = 2x - 3, substitute x = -1, 0, 1 and 3. The outputs are -5, -3, -1 and 3, giving the coordinates (-1, -5), (0, -3), (1, -1) and (3, 3). These points are collinear because the rule is linear.

The y-intercept is (0, -3). Between any two points, y increases by 2 for each increase of 1 in x, so the gradient is 2. To find the root, set y = 0: 2x - 3 = 0, hence x = 3/2 and the x-intercept is (3/2, 0).

Explanation

Construct tables without sign errors

Choose x-values that reveal important features, then evaluate every term before combining them. With negative inputs, use brackets: when x = -2, x^2 means (-2)^2 = 4, while -x^2 means -(4) = -4. A constant can be viewed as a multiple of x^0 because x^0 = 1 for x not equal to zero.

The syllabus power family includes terms based on x^n for n = -2, -1, -1/2, 0, 1/2, 1, 2 and 3, with sums of no more than three such terms. In the specified forms, a and c are rational numbers and the exponential base b is a positive integer. Not every input is permitted for negative or half powers, so mark an undefined entry rather than inventing a coordinate.

Worked example

Worked example: table, roots and turning point of a quadratic

For y = x^2 - 2x - 3, use x = -1, 0, 1, 2, 3 and 4. The corresponding y-values are 0, -3, -4, -3, 0 and 5. Thus the table gives (-1, 0), (0, -3), (1, -4), (2, -3), (3, 0) and (4, 5).

The roots are x = -1 and x = 3. Equal outputs at x = 0 and x = 2 show symmetry about x = 1, and the minimum turning point is (1, -4). Completing the square confirms this: y = (x - 1)^2 - 4.

Explanation

Recognise polynomial families from their features

A linear graph is a straight line with constant gradient. A quadratic graph has one turning point and is symmetric about a vertical line; it opens upward when its x^2 coefficient is positive and downward when that coefficient is negative. Its roots, when real, are the coordinates where the curve meets the x-axis.

A cubic may have zero or two turning points; a stationary point of inflection is not a turning point. It can have one, two or three distinct real roots, where two distinct roots occur when one root is repeated. Its opposite ends ultimately go in opposite vertical directions when the leading cubic coefficient is non-zero. Coordinates and the sign of the leading term are safer evidence than a memorised picture alone.

Worked example

Worked example: interpret a cubic from explicit coordinates

For y = x^3 - 4x, factorise to y = x(x - 2)(x + 2). The curve therefore has roots at x = -2, 0 and 2, giving x-axis intersections (-2, 0), (0, 0) and (2, 0). Further table points include (-3, -15), (-1, 3), (1, -3) and (3, 15).

The coordinates occur in opposite pairs: replacing x by -x changes y to -y, so the graph has rotational symmetry about the origin. The positive x^3 term also explains why the values eventually decrease to the left and increase to the right.

Explanation

Negative powers create restrictions and asymptotes

The rule x^(-1) means 1/x and x^(-2) means 1/x^2, so x = 0 is excluded. For y = a/x or y = a/x^2 with a non-zero, the line x = 0 is a vertical asymptote: the curve approaches it but the function has no point there. The line y = 0 is also a horizontal asymptote because the output approaches zero as the magnitude of x grows.

For x^(-1/2) = 1/sqrt(x), real outputs require x > 0; zero is excluded because it would be in the denominator. For x^(1/2) = sqrt(x), the real domain is x >= 0. Domains and asymptotes must be stated rather than hidden by a calculator error message.

Worked example

Worked example: a reciprocal graph

For y = 6/x, choose non-zero inputs x = -6, -3, -2, 2, 3 and 6. The outputs are -1, -2, -3, 3, 2 and 1, so the coordinates are (-6, -1), (-3, -2), (-2, -3), (2, 3), (3, 2) and (6, 1).

Positive x-values give positive y-values and negative x-values give negative y-values, so the two branches lie in quadrants I and III. Neither axis is crossed: x = 0 and y = 0 are asymptotes. The coordinate products xy all equal 6, which is a useful check.

Worked example

Worked example: an inverse-square graph

For y = 4/x^2, use x = -4, -2, -1, 1, 2 and 4. The y-values are 1/4, 1, 4, 4, 1 and 1/4. Both branches are above the x-axis because x^2 is positive for every permitted x.

The equal outputs for x and -x show symmetry about the y-axis. The vertical asymptote is x = 0 and the horizontal asymptote is y = 0. Unlike y = 4/x, the branches are in quadrants I and II because the outputs never become negative.

Worked example

Worked example: square-root and reciprocal-square-root coordinates

For y = 3sqrt(x), convenient inputs 0, 1, 4, 9 and 16 give coordinates (0, 0), (1, 3), (4, 6), (9, 9) and (16, 12). The domain is x >= 0, and the graph begins at the origin rather than extending to negative x-values in the real-number system.

For y = 6/sqrt(x), the same positive square inputs 1, 4, 9 and 16 give (1, 6), (4, 3), (9, 2) and (16, 3/2). Here the domain is x > 0, and x = 0 and y = 0 are asymptotes.

Explanation

Exponential graphs change by a constant factor

In y = ab^x + c with a not equal to zero and b > 1, multiplying b^x by a sets the scale and adding c makes y = c the horizontal asymptote as x decreases without bound. When a is positive, an increase of 1 in x multiplies the distance y - c by b. This repeated multiplication is the signature of exponential growth.

Decay can be written with a negative exponent, such as y = a2^(-x) + c with a not equal to zero. Increasing x by 1 then divides y - c by 2, and y approaches c as x increases without bound. These non-degenerate exponential values do not equal c for finite x. If b = 1, the rule is constant rather than a growth or decay curve.

Worked example

Worked example: exponential growth with a vertical shift

For y = 5(2^x) + 1, inputs x = -1, 0, 1, 2 and 3 give y = 7/2, 6, 11, 21 and 41. After subtracting 1, the values 5/2, 5, 10, 20 and 40 double whenever x increases by 1.

The horizontal asymptote is y = 1 because 5(2^x) approaches zero as x becomes very negative. The y-intercept is (0, 6), and every output is greater than 1. These checks catch a common error of adding the shift before multiplying.

Worked example

Worked example: exponential decay

A quantity follows Q = 64(2^(-t)), where t is time in hours. At t = 0, 1, 2, 3 and 4, the coordinates are (0, 64), (1, 32), (2, 16), (3, 8) and (4, 4). Each extra hour halves the quantity.

The graph remains above the t-axis and approaches Q = 0 as time increases. In this model, a negative time may be algebraically possible but is outside the stated measurement period t >= 0. The domain restriction comes from the context, not from the exponential rule itself.

Explanation

Roots and intersections solve equations graphically

A root of y = f(x) is read where the curve crosses or touches y = 0. To solve f(x) = g(x) graphically, draw both y = f(x) and y = g(x) on the same axes and read the x-coordinates of their intersection points. The corresponding y-coordinate shows the shared output.

A graph-based answer is normally an estimate unless the coordinates are stated exactly. In this pack, every reading task supplies explicit coordinates or a complete textual description so no unseen graph is required. Algebra may be used afterward as an independent check, but the graphical meaning is the intersection.

Worked example

Worked example: line-curve intersections

The curve y = x^2 - 1 and the line y = x + 1 intersect at exactly (-1, 0) and (2, 3). Therefore the graphical solutions of x^2 - 1 = x + 1 are x = -1 and x = 2. Each intersection supplies one shared x-value.

As a check, rearrange to x^2 - x - 2 = 0 and factorise: (x - 2)(x + 1) = 0. Substitution also confirms both coordinate pairs satisfy both rules. If the coordinates had been estimated from a drawing, the answers should be reported to justified accuracy.

Explanation

Exponential models connect graph shape to context

A growth model has outputs that rise by a constant multiplier over equal time intervals; a decay model falls by a constant multiplier between zero and one. State the initial value at time zero, identify the multiplier, calculate the requested coordinate and interpret both axes with units.

A model should not be extended beyond its sensible context without comment. Population counts may need whole-number interpretation, and a decay model can approach zero without predicting a negative amount. These contextual checks are part of interpreting the graph rather than merely evaluating a formula.

Worked example

Worked example: interpret a growth scenario

A laboratory culture is modelled by N = 250(2^t), where t is the number of hours after observation begins. The coordinates (0, 250), (1, 500), (2, 1000), (3, 2000) and (4, 4000) show that the model doubles each hour.

At t = 3, the model predicts 2000 organisms. The initial value is 250 because 2^0 = 1. The graph is increasing and has horizontal asymptote N = 0 when considered for all real t, but the stated context uses only t >= 0.

Explanation

A tangent estimates instantaneous gradient

The gradient of a curve changes from point to point. To estimate it at one point, draw a tangent that touches the curve there and follows its local direction, then choose two well-separated readable points on the tangent. The points used for the calculation lie on the tangent and need not both lie on the curve.

Calculate gradient = change in y / change in x. Keep the subtraction order consistent in numerator and denominator, include a negative sign for a falling tangent, and attach compound units such as metres per second when y and x represent measured quantities.

Worked example

Worked example: estimate a tangent gradient from coordinates

A tangent to a curve at x = 3 passes through the clearly read points (1, 2) and (5, 14). Its gradient is (14 - 2)/(5 - 1) = 12/4 = 3. Using a wide separation reduces the relative effect of coordinate-reading error.

If x is time in seconds and y is distance in metres, the estimated instantaneous speed at x = 3 is 3 metres per second. This is only a supporting illustration of units for a tangent gradient, not a claim to cover kinematics or practical travel graphs. The value describes the curve's rate of change at the contact point, not necessarily the average gradient of the curve from x = 1 to x = 5.

Explanation

Sketch from decisive features

A useful sketch begins with intercepts, turning points, symmetry, asymptotes and end behaviour. Plot these features in proportion, then connect them with the correct general shape. A vertical asymptote is a domain boundary for the reciprocal examples here. A horizontal asymptote may be crossed by some functions, so check the equation; the specific reciprocal and exponential examples in this pack do not cross theirs.

For a quadratic, completed-square form reveals the turning point and symmetry line. For reciprocal and exponential graphs, asymptotes organise the branches. For a cubic, roots and the sign of the leading term organise crossings and end behaviour. A written feature list is a planning aid, not a substitute for practising a hand-drawn sketch.

Worked example

Worked example: plan a quadratic sketch from text

For y = (x - 2)^2 - 9, the minimum turning point is (2, -9) and the line of symmetry is x = 2. Setting y = 0 gives (x - 2)^2 = 9, so the roots are x = -1 and x = 5, at (-1, 0) and (5, 0).

Setting x = 0 gives the y-intercept (0, -5). The positive squared term means both arms rise. These stated coordinates and features determine a reliable sketch plan: plot the intercepts and turning point, respect symmetry about x = 2, and draw one smooth upward-opening curve.

Original practice, not an official examination paper. Readnary is not affiliated with the awarding body. Prepared with AI assistance.