Question

which of the following graphs does not represent a function?
a. a cubic curve.
b. a straight line.
c. a circle.
d. a parabola opening upwards.

Explanation:

To determine if a graph represents a function, we use the vertical line test: a graph is a function if no vertical line intersects it more than once.

• A cubic curve: Passes the vertical line test (each \( x \) has one \( y \)).
• A straight line (non - vertical): Passes the vertical line test (linear function \( y = mx + b \) has one \( y \) per \( x \)).
• A circle: Its equation is \( (x - h)^2+(y - k)^2 = r^2 \). Solving for \( y \), we get \( y=k\pm\sqrt{r^{2}-(x - h)^{2}} \). For a given \( x \) (within the domain of the circle), there are two \( y \)-values, so it fails the vertical line test.
• A parabola opening upwards: Its equation is \( y = ax^{2}+bx + c(a>0) \), and it passes the vertical line test (one \( y \) per \( x \)).

So the graph of a circle does not represent a function.

Answer:

c. A circle.