Drawing Quadratic Graphs
A quadratic graph is the U-shaped or n-shaped curve produced by an equation with an x^2 term, such as y = x^2 - 4x + 3. In GCSE questions you complete a table of values, plot the points as a smooth curve (never straight lines), then use the graph to estimate solutions, intercepts or turning points of the equation.
Before you start
Make sure you're comfortable with these topics first:
Method
- Substitute each x-value into the equation to complete the table of y-values, working through BIDMAS carefully with negative x-values.
- Choose sensible scales for both axes so all your points fit on the grid, then plot each (x, y) pair accurately.
- Join the points with a single smooth curve, not a series of straight lines; the curve should be symmetrical about its turning point.
- To find where the curve crosses the x-axis, read off the x-values where y = 0; these are the solutions of the quadratic equation.
- To solve a different equation using the same graph, draw the appropriate straight line (e.g. y = k or y = mx + c) and read off where it crosses the curve.
- Check your curve makes sense: a positive x^2 coefficient gives a u-shape with a minimum; a negative coefficient gives an n-shape with a maximum.
Worked example
Draw the graph of y = x^2 - 2x - 3 for values of x from -2 to 4, then use it to find the solutions of x^2 - 2x - 3 = 0.
- Substitute each x-value into y = x^2 - 2x - 3: x = -2 gives y = 4 + 4 - 3 = 5; x = -1 gives y = 1 + 2 - 3 = 0; x = 0 gives y = -3; x = 1 gives y = 1 - 2 - 3 = -4; x = 2 gives y = 4 - 4 - 3 = -3; x = 3 gives y = 9 - 6 - 3 = 0; x = 4 gives y = 16 - 8 - 3 = 5.
- Plot the points (-2, 5), (-1, 0), (0, -3), (1, -4), (2, -3), (3, 0) and (4, 5) on the grid.
- Join the points with a single smooth u-shaped curve, since the coefficient of x^2 is positive.
- Read off the x-values where the curve crosses the x-axis, since these are the solutions of x^2 - 2x - 3 = 0.
- The curve crosses the x-axis at x = -1 and x = 3, so the final answer is x = -1 and x = 3.
Practice questions
Try each question, then tap to reveal the answer.
Exam-style questions
Written in the style of a GCSE Maths exam paper, with a full mark scheme.
The graph of y = x^2 - 4x + 1 is to be drawn for -1 to 5. (a) Find the value of y when x = -1. (b) Find the value of y when x = 5. (c) State whether the graph is u-shaped or n-shaped, giving a reason.
The grid shows an accurately drawn graph of y = x^2 - 3x - 4 for -2 to 5. (a) Use the graph to write down the solutions of x^2 - 3x - 4 = 0. (b) By drawing the line y = 6, use the graphs to find the solutions of x^2 - 3x - 4 = 6.
A stone is thrown from a cliff. Its height above sea level, h metres, after t seconds is given by h = 15 + 10t - 5t^2 for 0 to 4. (a) Find the value of h when t = 1. (b) Find the value of h when t = 3. (c) Using a graph of h against t, explain how you could find the time at which the stone lands in the sea (h = 0), and state how many such times there are in the given range.
Free printable worksheet
Want more practice on paper? Download the drawing quadratic graphs worksheet pack - 25 pages of exam-style questions with a full mark scheme. No sign-up, no email wall - just the PDF, free for personal and classroom use.
Build a full practice pack.
This topic is one of hundreds in the library - pick the ones a student needs and generate a printable PDF in minutes.