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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: