Will all parabolas of the form y=a(x-h)^2+k, a>0 be...

Key Concepts

Function

A function is a relation between a set of inputs and a set of outputs where each input is related to exactly one output. In the context of the parabola given by y = a(x-h)^2 + k, each x-value produces a unique y-value, thus satisfying the definition of a function.

Vertical Line Test

The vertical line test is a graphical method used to determine if a curve represents a function. If no vertical line intersects the graph at more than one point, the graph represents a function. In the case of y = a(x-h)^2 + k, every vertical line will intersect the parabola at exactly one point, confirming that it is a function.

Domain

The domain of a function is the complete set of possible input values (x-values) for which the function is defined. For a quadratic function in vertex form like y = a(x-h)^2 + k, the domain is all real numbers since any real value can be substituted for x without causing any mathematical issues.

Range

The range of a function is the set of all possible output values (y-values). For the parabola y = a(x-h)^2 + k with a > 0, the graph opens upwards which means that k is the minimum value of y. Therefore, the range is defined as all real numbers greater than or equal to k, i.e., [k, ?).

Vertex Form of a Quadratic Function

The vertex form of a quadratic function, y = a(x-h)^2 + k, clearly reveals the vertex of the parabola, which is the point (h, k). This form also indicates the direction the parabola opens based on the sign of a; a positive a indicates the parabola opens upward and a negative a indicates it opens downward. This form simplifies the determination of the graph's domain and range.

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