Quadratic Curved Lines: Studying Parabolas that Open Upward and Downward

University:
University of Michigan
Course:
ECON 613 | Advanced Topic in Mathematics for Economists
Academic year:

2024
Views:

347

Pages:

2
Author:

Alvinska H.

Non-Linear Functions: Quadratic Functions (Parabolic Functions) And Power-Of-Three Functions Name: Alvinska Hebat Grade: 910-Math Quadratic Function General Equation Form of Quadratic Function: Y= f(X) = aX2 + bX + c a ≠ 0 Y = dependent variable X = independent variable a,b,c = coefficient/constant • • • • • • • When drawn, the function will be a symmetrical parabola. In drawing a quadratic function, it is necessary to note certain requirements that a quadratic function must fulfill whether it is open upwards or downwards, intersecting or tangent to each coordinate axis. The six drawings differ in the value of "a" and the discriminant "D" of the equation Y = f(X) = aX2 + bX + c. With D = b2 - 4ac. In drawing a quadratic function, it is necessary to note certain requirements that a quadratic function must fulfill whether it opens up/down, intersects or is tangent to each coordinate axis. The six drawings differ in the value of "a" and the discriminant "D" of the equation Y = f(X) = aX2 + bX + c. With a value of D = b 2 - 4ac. Figures (a), (c) and (e) are "symmetrical parabolas" that open downwards, with point V as the vertex. • • • • • • • Figures (b), (d) and (f) are "symmetrical parabolas" that open upwards, with point V as the vertex. Vertex: V = (-b/2a; -D/4a); D = Discriminant; D = b2 - 4ac. In determining whether the Vertex is a maximum or minimum turning point will be learned when discussing the Extreme Value of Functions on the material Next. Determining the tipot (intersection point) with the Y axis then X = 0 Determine the tipot with the X axis by using the Quadratic Formula (ABC Formula) as follows: X 1,2 = (-b ± √D): 2a When graphing a quadratic function, first determine the turning point, then determine the other points around the turning point. The best way to determine the points around the turning point is to determine one price of Y for two prices of X. Example: Determine the turning point of the function Y = f(X) Y = X2 - 10 X + 12 and then draw the function: D = b 2 - 4ac = (-10) 2 - (4 x 1x 12) = 100 - 48 = 52 Turning point = (-b/2a ; -D/4a) = (10/(2x1) ; -52/(4x1)) = (5, -13) Tipot (intersection point) with Y axis then X = 0, then Y = 12 Tipot (intersection point) with the X-axis: X 1,2 = (-b ± √D) : 2a X 1 = (10 + √52): (2x1) = (10 + 7,2) : 2 = 8,6 X 2 = (10 √52): (2x1) = (10 - 7,2) : 2 = 1,39 ≈ 1,4