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**Graphs of quadratic functions**

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**Plotting graphs of quadratic functions**

y = ax2 + bx + c (where a ≠ 0) A quadratic function in x can be written in the form: We can plot the graph of a quadratic function using a table of values. For example: Plot the graph of y = x2 – 4x + 2 for –1 < x < 5. x x2 – 4x + 2 y = x2 – 4x + 2 –1 1 2 3 4 5 1 1 4 9 16 25 Talk through the substitution for each value of x to give the corresponding value of y. Ask students if they can tell you between which values of x the roots will be. + 4 + 0 – 4 – 8 – 12 – 16 – 20 + 2 + 2 + 2 + 2 + 2 + 2 + 2 7 2 –1 –2 –1 2 7

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**Plotting graphs of quadratic functions**

x –1 1 2 3 4 5 7 –2 y = x2 – 4x + 2 The points given in the table are plotted … y 6 … and the points are then joined together with a smooth curve. 5 4 3 The shape of this curve is called a parabola. Point out that, at this level, graphs are rarely plotted in this way but are usually sketched to show their shape relative to the x- and y-axes and their general features. When a sketch is required we only find the coordinates of the points where the function crosses the axes and the coordinates of any turning points. 2 1 It is characteristic of a quadratic function. –1 1 2 3 4 5 x –1

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**Parabolas Parabolas have a vertical axis of symmetry …**

…and a turning point called the vertex. When the coefficient of x2 is positive the vertex is a minimum point and the graph is -shaped. When the coefficient of x2 is negative the vertex is a maximum point and the graph is -shaped.

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**Graphs of the form y = ax2 + bx + c**

Change the values of a, b and c to observe how each one affects the shape and position of the parabola. In particular, draw students’ attention to the fact that when a is positive the parabola is -shaped and when a is negative the parabola is -shaped. Changing the value of a stretches or squeezes the graph. Note, too, that when a = 0 the function is no longer quadratic but linear.

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