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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.

Conclusion

A parabola opening upwards satisfies both the definition of a function and the Vertical Line Test, proving that it is a function.