Plotting Quadratic Graphs on Cartesian Axes

In recent examiner reports it is noted how important it is for students to understand the properties of a parabola  when plotting quadratic graphs on Cartesian axes.  Students who have a secure understanding of parabolas can use them to correct miscalculated values in their table of results and are more likely to attain full marks on this Grade 5 topic.

Understanding the Properties of a Parabola

To help students draw a quadratic graph they need to understand the properties of a parabola.  A parabola is a smooth, symmetrical curve with a clearly defined maximum or minimum turning point.

To demonstrate the shape of a parabola I stand at one end of the classroom and ask for a volunteer to stand at the other.  I throw a whiteboard pen up in the air towards the volunteer and ask the students to sketch on their mini-whiteboards the path of the pen as it travels under the force of gravity.

Plotting Quadratic Graphs on Cartesian Axes

Gravity is a force pushing the pen vertically downwards, which is working against the pen moving upwards.  This decelerates the pen until zero velocity at the maximum point and then the pen accelerates back towards the ground. The horizontal motion is due to throwing the pen in that direction.   Because gravity has a constant acceleration of 9.8 m/s2 the pen falls at the same rate as it climbed.  This is what makes the path symmetrical.

Creating a Table of Results from a Quadratic Equation

When plotting quadratic graphs on Cartesian axes students are expected to complete a table of results to help calculate the y value for changing values of x.  I like to include an additional row in the table for each calculation involved in the equation.   The x and calculated y value form a series coordinate pairs which are plotted on a Cartesian grid.   A common mistake when completing a table of results from a quadratic equation is to incorrectly calculate -32 as -9.   In my experience this is more likely to happen if students use a calculator.

Plotting Quadratic Graphs on Cartesian Axes

It is important for students to understand the table of results are used to generate a series of coordinate pairs involving x and y.  A common mistake here is to complete the table but not know how to use it to plot the graph.

Plotting Quadratic Graphs on Cartesian Axes

To help the pace of the lesson students are provided with a pair of correctly scaled axes.  As learning progresses I expect them to choose their own scale and draw the grid on A4 graph paper.

After we have plotted the coordinates I share some tips for drawing the parabola which passes through the points.  It is often easier to draw the parabola with your writing hand inside the curve.  Turning the paper to draw the parabola is more comfortable than rotating their hand.

Common Mistakes when Plotting Quadratic Graphs

Plotting Quadratic Graphs on Cartesian Axes

The parabola does not have a clearly defined minimum or maximum point.

Plotting Quadratic Graphs on Cartesian Axes

A coordinate pair is either calculated or plotted incorrectly.

Plotting Quadratic Graphs on Cartesian Axes

Drawing line segments between each coordinate pair suggest the relationship between them is linear which it is not.

Plotting Quadratic Graphs on Cartesian Axes

The graph is does not pass through each of the coordinate pairs to form a clear defined and smooth parabola.

Using the Parabola to Check for Mistakes

When students have completed plotting the first equation I ask them hold up their graphs for me to check.  To feedback we have a short discussion about whether the graph they have drawn obeys the discussed properties of a parabola.  The students who had made mistakes go back and correct them.

Using Graphical Methods to Solve Quadratic Equations

In the next lesson we progress on to using parabolas to solve quadratic equations graphically.  Here, we talk about whether an equation has one to two solutions depending on where the linear graph cuts the quadratic.  We also look at whether a quadratic and linear will have any solutions depending on the intercept value and gradient of the linear graph.

You can find this lesson in the Key Stage 3 Scheme of Work on Functions, Equations and Graphs.

Plotting Quadratic Graphs on Cartesian Axes

Plotting Quadratic Graphs on Cartesian Axes

Plotting Quadratic Graphs on Cartesian Axes

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Plotting Quadratic Graphs on Cartesian Axes

In recent examiner reports it is noted how important it is for students to understand the properties of a parabola  when plotting quadratic graphs on Cartesian axes.  Students who have a secure understanding of parabolas can use them to correct miscalculated values in their table of results and are more likely to attain full marks […]