Graphs: sketch translations and reflections of a given function (Higher tier only)

The idea

This Higher tier skill asks you to sketch, not plot point by point, the graph of a function after a translation or a reflection. Start from the given function y = f(x). A translation shifts the whole curve without turning it over: y = f(x) + a moves it a units up (a > 0) or down (a < 0), while y = f(x + a) moves it a units left when a > 0 and right when a < 0. A reflection flips the curve: y = −f(x) reflects it in the x-axis, and y = f(−x) reflects it in the y-axis. For example, if f(x) = x², then y = f(x) + 3 is the same parabola moved 3 units up, and y = f(−x) is unchanged because x² is even. You should mark key features such as intercepts, turning points and asymptotes, and label the new position clearly.

What earns the marks

• Recognise that y = f(x) + a translates the graph vertically by a units, with a > 0 moving it up and a < 0 moving it down.
• Recognise that y = f(x + a) translates the graph horizontally by −a units, so a > 0 moves it left and a < 0 moves it right.
• Recognise that y = −f(x) reflects the graph in the x-axis, changing the sign of every y-coordinate.
• Recognise that y = f(−x) reflects the graph in the y-axis, changing the sign of every x-coordinate.
• Sketch the transformed curve with the correct shape and position, marking key points such as intercepts, turning points or asymptotes where relevant.

Common mistakes and how to put them right

• Thinking that y = f(x + a) moves the graph right when a is positive.

  Put it right: It moves left, because the input x must be smaller to produce the same output.

• Confusing y = −f(x) with y = f(−x).

  Put it right: y = −f(x) reflects in the x-axis (vertical flip) while y = f(−x) reflects in the y-axis (horizontal flip).

• Plotting many individual points instead of sketching the transformed shape.

  Put it right: Move or flip the whole curve and then mark the key features.

Exam tips

• Identify the type of transformation first: a change outside the function affects y-values, while a change inside the brackets affects x-values.
• Use a few key points from the original graph, transform their coordinates, and check that the new curve passes through them.
• For reflections, remember that the shape stays the same but is turned over; check the direction of opening or the sign of the gradient.