Quadratic Graphs: Plotting, Features and Roots

1. 1 A quadratic equation has an x² term as the highest power. y = x², y = x²+2x−3, y = −x²+4 are all quadratics. 'What shape do you think the graph of y = x² makes?' Students sketch a guess.
2. 2 For y = x²−2x−3, complete a table for x = −2 to x = 4. Substitute each x value: x=−2: (−2)²−2(−2)−3 = 4+4−3 = 5. Students complete the full table. Check: y values should show symmetry if you've chosen the right range.
3. 3 Plot all points from the table. 'Join the dots with a smooth curve — not straight lines between points.' Draw the parabola shape. The graph should be a smooth U-shape (or ∩ for negative leading coefficient).
4. 4 'Where does the graph cross the x-axis?' These are the roots (solutions of y=0). Read off: x = −1 and x = 3. Verify: (−1)²−2(−1)−3 = 1+2−3 = 0 ✓. 'What does a root mean algebraically?'
5. 5 'What is the minimum (or maximum) point of the parabola?' Read the turning point from the graph: (1, −4). The x-coordinate of the vertex lies halfway between the two roots: (−1+3)/2 = 1 ✓.
6. 6 The parabola is symmetric about the vertical line through the vertex: x = 1. Draw this dashed line. Students identify the line of symmetry for their graph and for two more displayed graphs.
7. 7 Plot y = −x²+4x on the same axes or a new set. 'What is different?' (∩ shape, has a maximum not minimum.) Identify roots (0 and 4), vertex (2,4), and line of symmetry x=2.

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