Which relations are functions?

Question

Select Function or Not a function for each graph.



Function Not a function
A circle on a coordinate plane centered at begin ordered pair 1 comma 1 end ordered pair. The circle passes through begin ordered pair 4 comma 1 end ordered pair.

Graph of a line on a coordinate plane. The horizontal x axis ranges from negative 5 to 5 in increments of 1. The vertical y axis ranges from negative 5 to 5 in increments of 1. A line passes through the points begin ordered pair negative 2 comma 0 end ordered pair and begin ordered pair 0 comma 1 end ordered pair.

A parabola graphed on a coordinate plane. The x axis ranges from negative 5 to 5 in increments of 1. The y axis ranges from negative 5 to 5 in increments of 1. The vertex of the parabola is at the origin. The parabola opens downward and passes through the points begin ordered pair negative 4 comma 2 end ordered pair and begin ordered pair negative 4 comma negative 2 end ordered pair.

A parabola graphed on a coordinate plane. The x axis ranges from negative 5 to 5 in increments of 1. The y axis ranges from negative 5 to 5 in increments of 1. The vertex of the parabola is at the origin. The parabola opens to the left and passes through the points begin ordered pair negative 4 comma 2 end ordered pair and begin ordered pair negative 4 comma negative 2 end ordered pair.

Answer 1

To determine if a relation is a function, we can use the Vertical Line Test. This test states that if a vertical line intersects a graph at more than one point, the graph does not represent a function.

Let's analyze each case:

1. A circle centered at (1, 1) passing through (4, 1):

   • A circle will intersect a vertical line at two points unless the line is tangential at a single point.
   • Not a function.

2. Graph of a line passing through (-2, 0) and (0, 1):

   • A straight line will intersect any vertical line at exactly one point.
   • Function.

3. A parabola opening downward with a vertex at the origin:

   • Since this parabola will also intersect a vertical line at two points for values of x where there are two corresponding y-values,
   • Not a function.

4. A parabola opening to the left with a vertex at the origin:

   • Similar to the previous case, this parabola will intersect a vertical line at two points for the x-values corresponding to those y-values.
   • Not a function.

Summary:

• Circle: Not a function
• Line: Function
• Downward Opening Parabola: Not a function
• Left Opening Parabola: Not a function