Download how to draw a parabola

HOW TO DRAW A PARABOLA
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)
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
What c determines
• Parabola y = ax2 + bx + c has y-intercepts at (0,c)
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.
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)
COORDINATES OF x-INTERCEPTS (2)
• D=0
• D>0
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
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
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: x   b    2  1
E
2a
2.1
2
D
b 2  4ac  2  4.1. 3
yE 


 4
 4a
 4a
 4.1
So the extreme point’s coordinates are (1,-4)
SAMPLE PROBLEM (ctd.)
Step 3:
D = b2 - 4ac = (-2)2 - 4.1.(-3) = 16 >0
As D>0 the parabola has 2 x-intercepts
The parabola intersects the x-axis at (-1,0) and (3,0).
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.
SAMPLE PROBLEM (ctd.)
Step 5:
SAMPLE PROBLEM (ctd.)
Step 6: