PARABOLA (CH 9)

PARABOLA (CH 9)

1. Plotting

   • TP: the point where the graph changes direction or turns.
     • Axis of Symmetry: the line that divides the parabola exactly in half.
       • TP: maximum if it is the highest point on the graph and a minimum if it is the lowest point on the graph.
   • Can be plotted using a table

2. Basic graph: y = x2

   • This parabola can be dilated (made wider or narrower), translated (moved horizontally and/or vertically) and reflected (turned upside down), to form other parabolas.
     • Dilation
       • As the coefficient of x2 increases, the graph becomes narrower and closer to the y-axis
       • As the coefficient of x2 decreases (but remains positive), the graph becomes wider or closer to the x-axis.
     • Vertical Translation
       • The whole graph has been moved or translated 2 units upwards. The turning point has become (0, 2).
       • The whole graph has been moved or translated 3 units downwards. The turning point has become (0, -3)
     • Reflection
       • Any quadratic graph where x2 is positive has a shape and is said to be upright. Conversely, if x2 is negative the graph has a shape and is said to be inverted.

3. Turning Point Form

   • y = a(x - h) 2 + k.
     • (i) the turning point (h, k) (ii) whether the graph is upright with a minimum turning point (a > 0) or the graph is inverted with a maximum turning point (a < 0)(iii) whether it is wider (|a| > 1), narrower (|a| < 1) or the same width (|a| = 1) as the graph of y = x2.

4. Standard Form

   • y = ax2 + bx + c
     • As seen previously, when an equation is written in turning point form the coordinates of the turning point can be read from the equation. That is, the coordinates of the turning point for y = a(x - h) 2 + k are (h, k). To change to turning point form we use the completing the square method.
     • One way to find the x-coordinate of the turning point is to calculate the average of the two x-intercepts. In this case, 1 1 2 + − = 0 is the x-coordinate of the turning point. The y-coordinate of the turning point can then be found by substituting the x-coordinate into the equation.

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