Understanding the Orientation of Quadratic Functions: Upward and Downward Parabolas

Upward/Downward quadratic

The terms “upward” and “downward” in the context of a quadratic function refer to the orientation of the parabola that represents the equation

The terms “upward” and “downward” in the context of a quadratic function refer to the orientation of the parabola that represents the equation.

A quadratic function is an equation of the form y = ax^2 + bx + c, where a, b, and c are constants. The coefficient “a” determines whether the parabola opens upwards or downwards.

1. Upward Quadratic: If “a” is positive:
– When the coefficient “a” is positive, the parabola opens upwards.
– The vertex, which is the lowest point on the parabola, will occur at the minimum value of the function.
– The graph of the quadratic will be concave up.
– As x approaches positive or negative infinity, the function values (y-values) will also tend towards positive infinity.

Example:
Consider the quadratic function y = x^2 – 2x + 1.
– The coefficient of x^2 is 1, which is positive.
– Therefore, the parabola opens upwards.
– The minimum value (vertex) of the function can be found by using the equation x = -b/2a.
– In this case, x = -(-2)/(2*1) = 1.
– Substitute x = 1 into the equation to find the corresponding y-value: y = 1^2 – 2(1) + 1 = 1.
– The vertex is located at (1, 1).
– The graph of this quadratic will open upwards and have a minimum value at (1, 1).

2. Downward Quadratic: If “a” is negative:
– When the coefficient “a” is negative, the parabola opens downwards.
– The vertex, which is the highest point on the parabola, will occur at the maximum value of the function.
– The graph of the quadratic will be concave down.
– As x approaches positive or negative infinity, the function values (y-values) will also tend towards negative infinity.

Example:
Consider the quadratic function y = -x^2 + 4x – 3.
– The coefficient of x^2 is -1, which is negative.
– Therefore, the parabola opens downwards.
– The maximum value (vertex) of the function can be found by using the equation x = -b/2a.
– In this case, x = -4/(2*(-1)) = 2.
– Substitute x = 2 into the equation to find the corresponding y-value: y = -(2^2) + 4(2) – 3 = 1.
– The vertex is located at (2, 1).
– The graph of this quadratic will open downwards and have a maximum value at (2, 1).

Remember, the “a” coefficient determines whether the parabola opens upwards (positive) or downwards (negative), and the vertex is the highest or lowest point on the parabola, respectively.

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