Algebra - Graphs — Edexcel GCSE Mathematics

Algebra - Graphs in Pearson Edexcel GCSE Mathematics covers plotting and interpreting linear, quadratic, cubic, reciprocal and exponential graphs, finding gradients and intercepts, solving equations graphically, and identifying key features such as roots, turning points and asymptotes. You must be able to draw these graphs accurately from equations, use them to solve simultaneous equations and inequalities, and apply transformations including translations, reflections and stretches.

Algebra - Graphs is a fundamental topic in Pearson Edexcel GCSE Mathematics that connects algebraic equations with their visual representations. You will learn to plot and recognise linear, quadratic, cubic, reciprocal and exponential graphs, and interpret key features such as gradients, intercepts, roots, turning points and asymptotes. This topic also covers graph transformations, including translations, reflections and stretches, and using graphs to solve equations and inequalities.

Mastering graphs is essential because it appears across both Foundation and Higher tiers and links to many other areas of mathematics, including algebra, geometry and statistics. It is heavily examined in Papers 1, 2 and 3, often with questions that require accurate drawing, interpretation and problem-solving. A strong understanding of graphs will also support your study of calculus at A-level and improve your ability to model real-world situations.

• →Linear graphs: equations of the form y = mx + c, where m is the gradient and c is the y-intercept. Parallel lines have the same gradient; perpendicular lines have gradients that multiply to -1.
• →Quadratic graphs: equations of the form y = ax^2 + bx + c, which produce a U-shaped parabola. The roots are where y = 0, the turning point is the minimum or maximum, and the line of symmetry passes through the turning point.
• →Cubic and reciprocal graphs: cubic graphs have the form y = ax^3 + bx^2 + cx + d and can have up to three roots; reciprocal graphs have the form y = a/x and have asymptotes at x = 0 and y = 0.
• →Graph transformations: y = f(x) + a translates the graph vertically by a; y = f(x + a) translates horizontally by -a; y = -f(x) reflects in the x-axis; y = f(-x) reflects in the y-axis; y = af(x) stretches vertically by scale factor a; y = f(ax) stretches horizontally by scale factor 1/a.
• →Using graphs to solve equations: draw the relevant straight line and find intersection points with the given curve; the x-coordinates of the intersections are the solutions.

• each point plotted in the right place to within half a square, even if a later part of the answer is wrong
• a coordinate pair written in the correct order, in brackets and separated by a comma
• using the scale on the axes when the squares are worth more than one unit
• a missing vertex or a midpoint that follows correctly from the points already plotted
• a correct table of values, or for three correct points, even if the line is then drawn badly
• one straight ruled line through the points, drawn across the range of x the question gives
• rearranging an equation so that y is on its own before the gradient is read off
• the gradient as a change in y over a change in x, even if the division slips
• substituting a point to find the intercept, and one for the finished equation
• using a given gradient with a given point to find c, as in 7 = 3 × 2 + c giving c = 1
• plotting a straight-line graph by substituting x-values, plotting the points and joining them with a ruled line
• reading the gradient from y = mx + c only after rearranging so that y is on its own
• identifying parallel lines by equal gradients, and perpendicular lines by gradients multiplying to −1
• using the negative reciprocal as the gradient of a perpendicular line, even if later work is wrong
• finding the gradient through two points as the change in y divided by the change in x, then substituting one point to find c
• the finished equation, written in whichever form the question asks for
• a correct rise and run taken from the graph using two points on grid intersections
• the gradient as a number, carrying a minus sign when the line falls from left to right
• the intercept read from the point where the line meets the y axis
• interpreting the gradient in the units of the context, such as pounds per mile or metres per second
• setting y = 0 as the first step towards the roots
• a correct factorisation, even if the roots are then written with the wrong signs
• give both roots when the equation has two solutions and check each in the original equation
• the y intercept obtained by substituting x = 0
• a smooth curve drawn through the plotted points across the range of x given
• halving the coefficient of x and writing the bracket that comes from it
• adjusting the constant outside the bracket by the square that was added
• the coordinates of the turning point read from the completed square, even if the earlier algebra slipped
• saying whether the turning point is a minimum or a maximum, with a reason from the sign of the x² term
• using the completed square to argue that a curve has no real roots when that is asked for
• Generating a correct table of values for the given function.
• Plotting calculated points accurately on the coordinate grid.
• Drawing a single, smooth curve of the correct characteristic shape through the points.
• Correctly identifying the type of function from its graph (e.g., linear, quadratic, cubic).
• Reading a value from a graph, or stating the coordinates of the y-intercept.
• Recognising that the reciprocal graph y = 1/x has two separate branches and asymptotes at x = 0 and y = 0.
• the correct general shape of the named curve
• the curve passing through its key point, such as (0, 1) on an exponential graph
• the right period on a trigonometric sketch, drawn across the whole range of angles asked for
• a value or an angle read off the curve to the accuracy the grid allows
• a transformed curve that keeps the same shape as the original
• the correct direction of a translation, even if the distance moved is wrong
• the image of a named point, such as a turning point or an intercept, given as coordinates
• describing a transformation fully, naming a reflection with its mirror line or a translation with its vector
• Plotting points accurately from a table of values.
• Joining the points with a suitable line or curve based on the context.
• Reading a value correctly from a graph, with clear construction lines shown.
• Correctly interpreting a feature of a distance-time graph, e.g., identifying the steepest section as the highest speed.
• Calculating acceleration from the gradient of a section on a speed-time graph.
• Describing the relationship shown by the graph in the context of the problem, using correct units.
• Calculating speed as the gradient of a distance-time graph, and acceleration as the gradient of a velocity-time graph, using change in y divided by change in x.
• Estimating displacement as the signed area under a velocity-time graph, and distance travelled as the total area ignoring sign (e.g. using triangles and trapeziums).
• Determining the multiplier for exponential growth or decay, such as 1.05 for 5% growth or 0.95 for 5% decay.
• Plotting points accurately for a given function and joining them with a suitable line or smooth curve, keeping reciprocal branches separate.
• Finding an approximate solution by drawing a suitable line on the graph and reading the x-coordinate of the intersection, e.g. drawing y = 3 to solve f(x) = 3.
• Interpreting a reading from a graph in context, stating correct units and explaining what the value represents.
• a tangent drawn touching the curve at the point the question names
• the rise and the run taken from that tangent, even if the division goes wrong
• the gradient interpreted in context, such as an acceleration in metres per second squared
• splitting the region into strips of equal width with heights read off the curve
• the total area, and one for saying what it represents, such as the distance travelled
• the radius found from the equation, or for the equation written from a given radius
• showing a point lies on the circle by substituting both of its coordinates
• the gradient of the radius drawn to the point of contact
• the negative reciprocal taken as the gradient of the tangent
• the finished tangent equation, in whichever form the question asks for

• 💡Say "along the corridor, up the stairs" to yourself so the x coordinate always comes first.
• 💡Check the numbers on both axes before plotting, and write the value of one square in the margin if it is not one unit.
• 💡Mark a point with a sharp pencil cross so its position can be read accurately against the scale.
• 💡Draw the line right across the grid the question gives, because a short line can miss the points an examiner checks.
• 💡Put x = 0 in your table of values, since it hands you the y intercept with no extra work.
• 💡When two points are given, mark the change in y and the change in x on the diagram before you divide.
• 💡When one point and a gradient are given, write y = mx + c with m filled in, then substitute the point to find c.
• 💡Write the perpendicular gradient as a fraction and cancel it, so −4/8 becomes −1/2 before you substitute.
• 💡Check the pair by multiplying the two gradients together, because anything other than −1 means one of them is wrong.
• 💡If a particular form of the equation is asked for, rearrange as the very last step so the method marks are already safe.
• 💡Choose two points that are far apart and land on grid intersections, so the arithmetic stays exact.
• 💡Give the gradient its units whenever the question is set in a context, because the interpretation mark needs them.
• 💡If y is not on its own in the equation, rearrange before you name the gradient.
• 💡Plot every value from the table, negative x values included, before deciding where the curve turns.
• 💡Draw the curve in one movement with your wrist inside the bend, and go over the line again only to darken it.
• 💡Label roots, intercepts and the turning point on the sketch with their coordinates, because the marks are for the labels.
• 💡Check a completed square by expanding it again, since it should return the quadratic you started with.
• 💡The turning point of y = (x + p)² + q is at (−p, q), so watch the sign inside the bracket change as it comes out.
• 💡When the question says show that, finish in the exact form it asks for rather than a rounded decimal.
• 💡Memorise the basic shapes. If a plotted point from your table of values breaks the expected pattern, re-check your calculation as it is likely an error.
• 💡For quadratics, check the sign of the x² term. A positive coefficient gives a U-shape ('smile'), a negative coefficient gives an n-shape ('frown').
• 💡When calculating values for a cubic function, be careful with negative numbers: for example, (-2)³ = -8, not 8.
• 💡Mark the horizontal scale in steps of 90° before sketching a trigonometric graph, so the wave lands on the right values.
• 💡For y = kˣ, plot the point where x = 0 first, because the curve passes through a height of 1 there for every positive k.
• 💡When a question asks for all the solutions in a range, read every crossing off the graph, not only the first one.
• 💡Track three points from the original curve through the change, then draw the new curve through them.
• 💡Say it as inside or outside: outside the bracket moves the curve vertically, inside moves it horizontally the opposite way.
• 💡Label the image of the turning point with its coordinates, because that is often where the accuracy mark sits.
• 💡Always draw construction lines (e.g., from an axis value to the curve and then to the other axis) on the graph. They show your method and can earn marks even if your final reading is slightly out.
• 💡When asked to interpret a graph, write a sentence that uses the words from the axis labels to describe what is happening in the real-world context, e.g., 'The car is accelerating between 10 and 20 seconds'.
• 💡To estimate the area under a curved graph, split it into several trapeziums; using more strips generally improves the estimate.
• 💡To find the gradient of a curve at a point, draw a tangent there and calculate the gradient of that straight line.
• 💡For approximate solutions, draw the required horizontal or slanted line accurately with a ruler and read the intersection carefully against the grid.
• 💡Extend the tangent right across the grid before measuring, because a short tangent gives a small rise and run and a rough answer.
• 💡Say whether your estimate is too big or too small and why, since that reason is often worth a mark.
• 💡Build the units of a gradient from the axes: the vertical unit per horizontal unit.
• 💡Take the radius as the square root of the number on the right of the equation, so x² + y² = 49 has radius 7.
• 💡Sketch the circle and mark the point of contact, and the right angle between radius and tangent becomes obvious.
• 💡Keep fractions as fractions all the way through, because a rounded gradient makes the final equation wrong.
• 💡Always use a sharp pencil and a ruler for straight lines. Plot at least three points for any straight line and check that they lie on a straight line before drawing. Label your graphs clearly with their equations.
• 💡When reading values from a graph, show your working by drawing lines to the axes. If the question asks for a solution to a certain degree of accuracy, ensure your answer is within the acceptable range, typically +/- half a small square.
• 💡For transformation questions, describe the transformation fully: state the type (translation, reflection, stretch), the direction and the amount. For example, 'translation by vector (0, 3)' or 'stretch parallel to the y-axis with scale factor 2'.

• writing the pair the wrong way round, so plotting (3, −4) when the question says (−4, 3)
• ignoring the minus sign, so putting a point with a negative y coordinate above the x axis
• assuming each square is one unit when the axis is labelled in twos or fives
• adding the two coordinates instead of averaging them when finding a midpoint
• working out the gradient as the change in x over the change in y; it is the change in y divided by the change in x
• giving a positive gradient for a line that falls from left to right; a falling line has a negative gradient
• joining the plotted points with several short segments instead of one ruled line
• deciding lines are parallel from the intercepts rather than the gradients, so treating y = 3x + 2 and y = 5x + 2 as parallel; equal gradients, not equal intercepts, make lines parallel
• substituting the given point into the wrong place when finding c, such as swapping x and y
• turning the gradient upside down but leaving its sign alone, so pairing gradient 5 with 1/5 instead of −1/5
• changing the sign but not turning the gradient upside down, so pairing gradient 5 with −5
• reading the gradient straight off an equation such as 2y = 8x + 3 without dividing through first
• keeping the intercept of the original line instead of substituting the point the question gives
• joining plotted points freehand or with too few points, so the line is not straight
• counting squares instead of units, so the gradient is out by the scale of the axis
• dividing the run by the rise
• giving a positive gradient for a line that slopes downwards
• reading the intercept from the bottom left corner of the grid when the axes do not meet at the origin
• reading the roots straight off the brackets, so giving 2 and 3 as the roots of (x + 2)(x + 3)
• drawing the turning point as a sharp corner instead of a smooth minimum or maximum
• confusing the y intercept with the turning point
• reading only one root from a graph that crosses the x axis twice
• forgetting to subtract the square of the halved coefficient, so leaving the constant untouched
• reading the turning point of (x − 3)² + 2 as (−3, 2) rather than (3, 2)
• completing the square without taking out the coefficient of x² first when it is not 1
• assuming every parabola has roots, so hunting for crossings on a curve that never reaches the x axis
• Joining plotted points with straight line segments instead of a smooth curve. Correction: Draw a single continuous curve that passes through the points.
• Drawing a reciprocal graph (y = 1/x) that crosses the y-axis. Correction: The graph has two separate branches and never touches either axis.
• Assuming a cubic graph has the same line symmetry as a quadratic parabola. Correction: Cubic graphs have rotational symmetry about a point, not the reflective symmetry of a parabola.
• Forgetting that the reciprocal function is undefined at x = 0. Correction: State x ≠ 0 and show the two branches approaching but never meeting the axes.
• drawing an exponential curve that crosses or touches the x axis
• letting a sine or cosine curve rise above 1 or drop below −1
• giving the tangent graph the same period as the sine graph
• starting the cosine curve at 0 rather than at 1, which draws the sine graph twice
• sliding y = f(x + 2) two units to the right because the sign inside the bracket is positive
• confusing y = −f(x) with y = f(−x), so reflecting in the wrong axis
• changing the shape of the curve as well as its position
• moving only the turning point and leaving the rest of the sketch where it was
• Confusing a horizontal line on a distance-time graph (which means stationary) with a horizontal line on a speed-time graph (which means constant speed). Correction: Check the vertical axis label. If it's distance, no change means no movement. If it's speed, no change means speed is steady.
• Misreading the scale on an axis, particularly when each square represents more than one unit. Correction: Before reading any values, determine what one small square is worth on each axis.
• Calculating speed from a speed-time graph instead of acceleration from its gradient. Correction: The gradient of a speed-time graph is acceleration (change in speed ÷ time). The speed at any instant is read directly from the vertical axis.
• Treating the area under a velocity-time graph as distance even when velocity is negative. Correction: area gives displacement; for distance, add the absolute areas of each section separately.
• Using straight line segments for an exponential curve, giving inaccurate estimates. Correction: exponential functions produce smooth curves, so join plotted points with a curve rather than a ruler.
• Joining the two branches of a reciprocal graph across x = 0. Correction: y = k/x has separate branches with asymptotes at x = 0 and y = 0; draw each branch separately.
• Forgetting to split the area under a curve into regular shapes. Correction: divide into strips of equal width and sum the trapezium or triangle areas.
• drawing a chord between two points on the curve instead of a tangent at one point
• counting squares under a curve without allowing for what one square is worth on each axis
• giving the area as a bare number with no units, which loses the interpretation mark
• treating the gradient of a distance-time graph as an acceleration
• giving the radius as 25 rather than 5 for the circle x² + y² = 25
• using the gradient of the radius as the gradient of the tangent
• turning the gradient upside down without changing its sign, or changing the sign without turning it upside down
• assuming a centre somewhere other than the origin, when this content covers only circles about the origin
• Students often think that the gradient of a line is always positive or that it is simply the change in x divided by the change in y. Correction: gradient = change in y divided by change in x, and it can be negative, zero or undefined.
• Many students confuse the roots of a quadratic with the y-intercept. Correction: roots are where the graph crosses the x-axis (y = 0), while the y-intercept is where it crosses the y-axis (x = 0).
• Students frequently misapply graph transformations, especially horizontal translations and stretches. Correction: y = f(x + a) moves the graph left by a, and y = f(ax) stretches horizontally by scale factor 1/a, not a.

1. 1Week 1: Revise the equations and shapes of linear, quadratic, cubic and reciprocal graphs. Practise plotting each type using a table of values, and identify key features such as gradient, intercepts, roots and turning points.
2. 2Week 1: Learn and apply graph transformations. Create a summary sheet for translations, reflections and stretches, and practise sketching transformed graphs from a given original graph.
3. 3Week 2: Focus on solving equations graphically. Practise drawing straight lines on given curves to find approximate solutions, and check your answers by substitution. Work through past paper questions on this topic.
4. 4Week 2: Complete a full past paper question set on Algebra - Graphs under timed conditions. Mark your work using the official mark scheme and note any recurring errors. Review these areas and re-attempt similar questions.
5. 5Ongoing: Use flashcards for key formulae and transformations, and use online graphing tools to visualise how changes to equations affect the graph.

• 📋Plotting and drawing graphs: You may be asked to complete a table of values and plot a graph, or to draw a straight line given its equation. Advice: take care with negative values and always use a ruler for straight lines.
• 📋Interpreting graphs: Questions may ask you to find the gradient, intercepts, roots or turning points from a given graph. Advice: read the question carefully and show your method, especially for gradient calculations.
• 📋Solving equations graphically: You will be asked to draw a suitable straight line on a given curve and use it to solve an equation. Advice: ensure the line is drawn accurately and extend it across the full grid; read solutions to the required accuracy.
• 📋Graph transformations: You may be asked to sketch or describe transformations of a given graph. Advice: learn the transformation rules and always describe transformations fully, including direction and scale factor.