Vertical Line Test and Parabola Function

Explanation

Based on the Vertical Line Test, the graph that represents a function is:

The parabola opening upwards. Here's the reasoning again for clarity:

Parabola opening upwards: Passes the vertical line test because any vertical line intersects the graph at most once.

Hence, it is a function.

Sideways parabola opening to the right: Fails the vertical line test because vertical lines can intersect the graph at more than one point. Not a function.

Sideways parabola opening to the left: Fails the vertical line test for the same reason as above. Not a function.

Two intersecting lines forming an "X" shape: Fails the vertical line test because vertical lines can intersect the graph at two points where the lines cross. Not a function.

Thus, only the parabola opening upwards is a function.To determine which of the following graphs represents a function, we can use the Vertical Line Test. A graph represents a function if and only if no vertical line intersects the graph at more than one point.

Test Results

The parabola opening upwards: Passes the vertical line test because any vertical line intersects the graph at most once. Hence, it is a function.
The sideways parabola opening to the right: Fails the vertical line test because vertical lines can intersect the graph at more than one point. Not a function.
The sideways parabola opening to the left: Fails the vertical line test for the same reason as above. Not a function.
Two intersecting lines forming an "X" shape: Fails the vertical line test because vertical lines can intersect the graph at multiple points where the lines cross. Not a function.

Proof That a Parabola Opening Upwards is a Function

Definition of a Function

A relation \( f(x) \) is a function if each input \( x \) from the domain is mapped to exactly one output \( y \).

For a parabola opening upwards, the general equation is:

\( y = ax^2 + bx + c \quad (a \neq 0) \)

For any given \( x \), substituting into the equation produces a single value of \( y \), because there is no ambiguity or additional solutions for \( y \). Hence, every \( x \) has exactly one \( y \), satisfying the definition of a function.

Vertical Line Test

The Vertical Line Test states that a graph represents a function if any vertical line intersects the graph at most one point.

For a parabola opening upwards:

The graph is symmetric about its axis of symmetry \( x = -\frac{b}{2a} \).
Any vertical line either does not intersect the parabola (outside the domain) or intersects it at exactly one point.
No vertical line can intersect the parabola more than once.

Thus, the parabola passes the vertical line test and is a function.