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When plotting curved graphs there are several misconceptions students have with quadratic, cubic and reciprocal functions. These include:

- Joining coordinates with straight lines instead of a smooth curve;
- Plotting the points but not joining them together;
- Drawing a line of best fit through the points;
- Students often have a preconceived idea of what the graph should look like and will attempt to draw a parabola for a cubic or reciprocal function;
- When drawing reciprocal graphs students will often connect the two vertical asymptotes to create one continuous line;
- When asked to match a sketched graph to its correct function students often fail to identify how the properties of a curve relate to its equation. For instance how y = x(x – 2) has two roots rather than one.

Here are three videos for teaching students about plotting curved graphs to address the misconceptions above.

In the run up to their exams I help students remember how to plot and interpret quadratic, cubic and reciprocal graphs through the Plotting Curved Graphs revision lesson. The lesson includes five questions that address each of the misconceptions listed above and a student worksheet. In this blog I would like to share three of the five questions.

In this question students are asked to plot the graph of y = x² – 5x + 8 between the range x = 0 and x = 6.

I encourage students to use the bracket symbols on their calculators when working out the y values. This helps them to break down the calculation into three separate steps. For instance, when x = 2, (2²) – (5 × 2) + 8

When plotting the coordinate points I remind students that a parabola is a smooth and symmetrical curve. This helps the class to identify any incorrect values.

To solve the equation in question d I ask students to draw the line y = 5. This can lead to a correct method mark in exam papers even if the x values are incorrect.

To revise plotting cubic graphs students are asked to complete the table of values and plot y = x³ + 4.

Here, I introduce negative values of x while keeping the calculations fairly simple. In my experience, students are likely to struggle cubing negative x values correctly.

To check their points are correctly plotted we discuss the rotational symmetry properties of cubic graphs in a mini-plenary.

This is the final question of five in the lesson and meant to challenge the most able students.

Because of the squared denominator some students incorrectly end up squaring the entire fraction. To help with this, I encourage students to use the fraction button on their calculator. This way their calculation looks more like the form given in the question.

Before students plot the remaining coordinates I ask them to consider why there are no negative values of y and where the vertical asymptotes are likely to be. Prompting students to consider these points helps them to visualise what the graph should look like. This way they are more likely to spot any incorrect y values.

My name is Jonathan Robinson and I am passionate about teaching mathematics. I am currently Head of Maths in the South East of England and have been teaching for over 15 years. I am proud to have helped teachers all over the world to continue to engage and inspire their students with my lessons.

A conceptual understanding of the properties of parabolic graphs.

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