AQA · GCSE · Mathematics
Course version: 8300 · Version from Sept 2015
Graphs: identify and interpret roots, intercepts and turning points of quadratic functions graphically deduce roots algebraically
A quadratic function has the form y = ax² + bx + c and its graph is a parabola.
- 4 key terms
Study notes
From the checked teaching notes for this specification point.
The idea
A quadratic function has the form y = ax² + bx + c and its graph is a parabola. The roots are the x-values where y = 0, so they are the x-intercepts of the curve. The y-intercept is the value of y when x = 0, which is c. The turning point is the vertex where the curve changes direction; it is a minimum when a is positive and a maximum when a is negative. Graphically, read roots where the curve crosses the x-axis, read the y-intercept where it crosses the y-axis, and read the turning point at the peak or trough. Algebraically, deduce roots by solving ax² + bx + c = 0, for example by factorising x² − 5x + 6 = 0 as (x − 2)(x − 3) = 0, giving roots x = 2 and x = 3.
What earns the marks
- Roots of a quadratic are the x-values where the graph crosses the x-axis, that is where y = 0.
- The y-intercept of y = ax² + bx + c is the constant term c, at x = 0.
- The turning point is the vertex where the curve changes from decreasing to increasing or vice versa.
- A positive coefficient of x² gives a minimum turning point; a negative coefficient gives a maximum turning point.
- Roots can be deduced algebraically by solving the quadratic equation, for example by factorising.
Common mistakes and how to put them right
Confusing roots with the y-intercept.
Put it right by remembering roots are x-values where y = 0 and the y-intercept is where x = 0.
Reading the turning point as a root.
Put it right by identifying the vertex as the maximum or minimum point, not necessarily on the x-axis.
Forgetting that a quadratic may have no real roots if the graph does not cross the x-axis.
Put it right by checking whether the curve meets the x-axis at all.
Assuming the turning point always lies on the y-axis.
Put it right by locating the vertex from the symmetry of the curve or by completing the square.
Exam tips
- Sketch the parabola to see where it crosses the axes and where the turning point lies.
- When factorising to find roots, set each factor equal to zero and solve for x.
- Check whether the turning point is a maximum or minimum by looking at the sign of the x² coefficient.
Try the lesson
The start of the Graphs: identify and interpret roots, intercepts and turning points of quadratic functions graphically deduce roots algebraically lesson, made from checked teaching notes for this specification point.
Have a go first
A quadratic equation y = f(x) is plotted on a graph. The curve intersects the x-axis at the points (−4, 0) and (1, 0), and intersects the y-axis at (0, −4). What are the roots of the equation f(x) = 0?
Listen
A quadratic function has the form y = ax² + bx + c and its graph is a parabola. The roots are the x-values where y = 0, so they are the x-intercepts of the curve. The y-intercept is the value of y when x = 0, which is c. The turning point is the vertex where the curve changes direction; it is a minimum when a is positive and a maximum when a is negative. Graphically, read roots where the curve crosses the x-axis, read the y-intercept where it crosses the y-axis, and read the turning point at the peak or trough. Algebraically, deduce roots by solving ax² + bx + c = 0, for example by factorising x² − 5x + 6 = 0 as (x − 2)(x − 3) = 0, giving roots x = 2 and x = 3.
See it
Key words
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Roots
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y-intercept
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Turning point
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Quadratic function
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See every definition
- Roots
- The x-values where y = 0, so they are the x-intercepts of the curve.
- y-intercept
- The value of y when x = 0, which is c in y = ax² + bx + c.
- Turning point
- The vertex where the curve changes direction; it is a minimum when a is positive and a maximum when a is negative.
- Quadratic function
- A function of the form y = ax² + bx + c whose graph is a parabola.
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