1. HOW TO DRAW A PARABOLA

2. PARABOLA EQUATIONS General form: y = ax2 + bx + c ; a ≠ 0
Example 1: y = -2x2 + 9x – 7 (a=-2, b = 9, c = -7)
Example 2: y = x2 + 2 (a = 1, b = 0, c = 2)

3. Algebraic Signs of a
If a < 0 the parabola is concave downwards and has a maxiumum turning point
If a > 0 the parabola is concave upwards and has a minimum turning point

4. What c determines
Parabola y = ax2 + bx + c has y-intercepts at (0,c)

5. Algebraic Signs of D = b2 – 4ac
If D > 0 the parabola intersects the x axis at two different points
If D = 0 the parabola intersects the x axis at only one point; the parabola is tangent to the x axis
If D < 0 the parabola has no x intercept.

6. COORDINATES OF x-INTERCEPTS (1)
If D = 0 the parabola intersects the x-axis at (x*,0)
If D > 0 the parabola intersects the x-axis at (x1,0) and (x2,0)

7. COORDINATES OF x-INTERCEPTS (2)

8. 2 KINDS OF EXTREME POINTS
maximum turning point (if a < 0)
minimum turning point (if a > 0)
(xE,yE) is the coordinate of the extreme point

9. STEPS TO DRAW A PARABOLA
Determine the coordinates of the y-intercept
Determine the coordinates of the extreme point
Determine the coordinates of the x-intercept(s), if there exists
If needed, substitute some values of x to the equation, so that we have the coordinates of some points which are on the parabola.
Plot the points whose coordinates have been obtained in the preceding steps
Passing through the points, draw a smooth curve

10. SAMPLE PROBLEM Sketch the graph of y = x2 - 2x - 3! Answer:
In this case, a = 1, b = -2, c = -3
Step 1: coord. of y-intercept (0,-3)
Step 2:
So the extreme point’s coordinates are (1,-4)

11. SAMPLE PROBLEM (ctd.)
Step 3: D = b2 - 4ac = (-2) (-3) = 16 >0 As D>0 the parabola has 2 x-intercepts
The parabola intersects the x-axis at (-1,0) and (3,0).

12. SAMPLE PROBLEM (ctd.) Step 4:
To have a better result, we can add several additional points which are on the parabola. In this example, substitute x = -2, x = 2, and x = 4 into the parabola equation.

13. SAMPLE PROBLEM (ctd.)
Step 5:

14. SAMPLE PROBLEM (ctd.)
Step 6: