Finding the turning point and the line of symmetry - Higher
The coordinates of the turning point and the equation of the line of symmetry can be found by writing the quadratic expression in completed square form.
Example
Find the equation of the line of symmetry and the coordinates of the turning point of the graph of \(y = x^2 鈥 6x + 4\)
The coefficient of \(x^2\) is positive, so the graph will be a positive U-shaped curve.
Writing \(y = x^2 鈥 6x + 4 \) in completed square form gives \(y = (x 鈥 3)^2 鈥 5\)
Squaring positive or negative numbers always gives a positive value. The lowest value given by a squared term is 0, which means that the turning point of the graph \(y = x^2 鈥6x + 4\) is given when \(x = 3\)
\(x = 3\) is also the equation of the line of symmetry
When \(x = 3\), \(y = -5\) so the turning point has coordinates (3, -5)
Question
Sketch the graph of \(y = x^2 鈥 2x 鈥 3\), labelling the points of intersection and the turning point.
The coefficient of \(x^2\) is positive, so the graph will be a positive U-shaped curve.
Factorising \(y = x^2 鈥 2x 鈥 3\) gives \(y = (x + 1)(x 鈥 3)\) and so the graph will cross the \(x\)-axis at \(x = -1\) and \(x = 3\).
The constant term in the equation \(y = x^2 鈥 2x 鈥 3\) is -3, so the graph will cross the \(y\)-axis at (0, -3)
Writing \(y = x^2 鈥 2x 鈥 3\) in completed square form gives \(y = (x 鈥 1)^2 鈥 4\), so the coordinates of the turning point are (1, -4).
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