upwards parabola
An upwards parabola, also known as a “U-shaped” or “concave up” parabola, is a type of graph that represents a quadratic function
An upwards parabola, also known as a “U-shaped” or “concave up” parabola, is a type of graph that represents a quadratic function. The general equation of a quadratic function is written in the form:
y = ax^2 + bx + c
In an upwards parabola, the coefficient ‘a’ in the equation is positive (a > 0). This positive coefficient causes the graph to open upwards, forming a parabolic shape with a vertex at its lowest point.
The vertex of an upwards parabola lies on the axis of symmetry, which is a vertical line that divides the parabola into two symmetrical halves. The equation of the axis of symmetry is given by x = -b/(2a).
The vertex coordinates (h, k) of an upwards parabola can be calculated using the formula:
h = -b/(2a)
k = f(h) = a(h^2) + b(h) + c
The value of ‘a’ determines the steepness or width of the parabola. If ‘a’ is larger than 1, the parabola is relatively narrow, whereas if ‘a’ is a fraction between 0 and 1, the parabola is wider.
The x-intercepts of the parabola represent the points where the graph intersects the x-axis. To find the x-intercepts, you can set y = 0 in the quadratic equation and solve for x using factoring, completing the square, or the quadratic formula:
0 = ax^2 + bx + c
Similarly, the y-intercept represents the point where the parabola intersects the y-axis. To find the y-intercept, you can set x = 0 in the quadratic equation and solve for y:
y = a(0^2) + b(0) + c
Understanding the characteristics and behavior of upwards parabolas is essential in various fields of mathematics, physics, engineering, and even real-life situations like projectile motion or optimization problems.
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