4 high-yield, syllabus-aligned questions on Graphs of functions, covering 11 individually creditable mark points. Each one shows the answer that scores full marks under our guide, which phrase earns which mark, and a common incomplete answer — so you can see the difference rather than guess at it.
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Every question and solution is reviewed by examiners versed in WAEC and JAMB, each with more than thirty years of experience.
Mark allocations are TopMarks’ own, written against the published syllabus and Chief Examiners’ reports. They are a guide to how these answers are usually credited, not an official marking scheme.
State3 marks
The graph of y = x² - 2x - 3 is drawn for -2 ≤ x ≤ 4. State the values of x at which the curve crosses the x-axis, and explain what these values represent.
TopMarks model answer
The curve crosses the x-axis where y = 0, that is where x² - 2x - 3 = 0. Factorising gives (x - 3)(x + 1) = 0, so x = 3 or x = -1. These values are the roots, or solutions, of the equation x² - 2x - 3 = 0.
Suggested marking guide
1Recognises that y = 0 on the x-axis
2Both values found
3Identifies them as the roots of the equation
Why it scores
The link between the picture and the algebra is the mark: where the curve meets the x-axis, y is zero, and the x-values there are the roots of the equation. Students who can factorise perfectly still lose this mark because they never say that y = 0 on the axis.
A common incomplete answer
“x = 3 and x = -1”
What it costs: Two marks of three. Both values are correct, but nothing says why they are found where y = 0, and the question explicitly asked what they represent.
Key ideas to include: roots, x-axis, y = 0, factorise, solutions of the equation.
Explain3 marks
Explain how the graph of y = x² - 2x - 3 can be used to solve the equation x² - 2x - 5 = 0.
TopMarks model answer
Subtract the equation to be solved from the equation of the curve: (x² - 2x - 3) - (x² - 2x - 5) = y - 0, which gives y = 2. So the solutions of x² - 2x - 5 = 0 are the values of x where the curve y = x² - 2x - 3 meets the line y = 2. Draw the straight line y = 2 on the same axes as the curve, and read off the x-coordinates of the two points of intersection.
Suggested marking guide
1Derives the correct straight line
2Says the line must be drawn on the same axes
3Read off the x-coordinates of the points of intersection
The working, line by line
1Write the curve: y = x² - 2x - 3 — The curve is already drawn; you must not redraw it for a different equation.
2Subtract the required equation: y - 0 = (x² - 2x - 3) - (x² - 2x - 5) = 2 — Line up the two expressions and subtract term by term. Whatever is left over is the equation of the line to draw.
3Draw y = 2 on the same axes — A horizontal straight line through 2 on the y-axis.
4Read the x-values where the line cuts the curve — Those x-values are the solutions. Give them to the accuracy the graph allows, usually one decimal place.
Why it scores
This question type appears almost every year and it always follows the same recipe: subtract the equation you must solve from the equation of the curve, and whatever remains is the line to draw. Learn the recipe and the whole family of questions becomes routine.
A common incomplete answer
“Draw the line y = 2 on the graph.”
What it costs: One mark of three. The line is correct and the instruction to draw it is credited, but nothing shows how y = 2 was obtained, and nothing says to read off the x-coordinates of the intersections.
Key ideas to include: points of intersection, straight line, read off, same axes.
Explain2 marks
From a distance-time graph, explain how the speed of a body is found and state what a horizontal section of the graph represents.
TopMarks model answer
The speed is found from the gradient of the distance-time graph, that is the change in distance divided by the corresponding change in time. A horizontal section has zero gradient, so the distance is not changing; it represents a period during which the body is at rest.
Suggested marking guide
1Speed is the gradient — distance change over time change
2A horizontal section means the body is at rest
Why it scores
The word “gradient” alone is not the mark — the mark is for what the gradient IS on this particular graph: change in distance over change in time. On a velocity-time graph the same gradient would be the acceleration, so the quantity has to be named.
A common incomplete answer
“The speed is the gradient of the graph.”
What it costs: One mark of two. The gradient is credited only loosely, and nothing says what a horizontal section means, which was half the question.
Key ideas to include: gradient, distance-time graph, at rest, zero gradient.
State3 marks
State three precautions to be observed when drawing the graph of a function.
TopMarks model answer
Use a scale that allows the whole of the given range of values to fit on the graph paper and that occupies most of the page, and state the scale you have used. Label both axes with the variable they represent and mark the origin clearly. Plot the points accurately and join them with a smooth free-hand curve rather than a series of straight line segments when the function is not linear.
Suggested marking guide
1Suitable scale, stated
2Both axes labelled
3Points joined with a smooth curve
Why it scores
The smooth curve is worth a mark of its own and is thrown away constantly. Joining plotted points with a ruler turns a parabola into a chain of straight lines, and any value read off between the points will then be wrong — which is exactly what the later parts of the question ask you to do.
A common incomplete answer
“Label the axes and plot the points accurately.”
What it costs: Two marks of three. The axis labels are credited, but nothing is said about the scale and nothing about joining the points with a smooth curve.
Reading a full-mark answer is the first half. Writing one under time is the other. A free account opens exam-standard practice in Mathematics with the full solution on every question.