What can you say about the range of a quadratic polynomial with a positive leading coefficient?

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A quadratic polynomial can be expressed in the standard form as $$ f(x) = ax^2 + bx + c $$, where $$ a $$, $$ b $$, and $$ c $$ are constants, and $$ a $$ is the leading coefficient. When the leading coefficient $$ a $$ is positive, the parabola opens upwards.

The key characteristics of the range of a quadratic polynomial with a positive leading coefficient are as follows:

1. **Vertex**: The vertex of the parabola represents the minimum point of the function. The x-coordinate of the vertex can be found using the formula $$ x = -\frac{b}{2a} $$. The y-coordinate of the vertex, which gives the minimum value of the function, can be calculated by substituting this x-coordinate back into the polynomial.

2. **Minimum Value**: Since the parabola opens upwards, the minimum value of the quadratic function occurs at the vertex. This minimum value is the lowest point on the graph of the function.

3. **Range**: The range of the quadratic polynomial is all real numbers greater than or equal to the minimum value. In interval notation, if the minimum value at the vertex is $$ k $$, the range can be expressed as $$ [k, \infty) $$.

In summary, for a quadratic polynomial with a positive leading coefficient, the range is $$ [k, \infty) $$, where $$ k $$ is the minimum value of the polynomial, occurring at the vertex.
The range of a quadratic polynomial with a positive leading coefficient is all real numbers greater than or equal to the minimum value, which occurs at the vertex.

What can you say about the range of a quadratic polynomial with a positive leading coefficient?

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