Question

which of the following graphs could represent a cubic function? graph d graph b graph a graph c

Explanation:

Step1: Recall the definition of a function

A function is a relation where each input (x - value) has exactly one output (y - value). This is checked using the vertical line test. If a vertical line intersects the graph at more than one point, it is not a function.

Step2: Analyze Graph A

A vertical line can be drawn that intersects Graph A at more than one point. So, it is not a function.

Step3: Analyze Graph B

Graph B is a vertical line. For a vertical line \(x = a\) (where \(a\) is a constant), there are infinitely many \(y\) - values for a single \(x\) - value. So, it is not a function.

Step4: Analyze Graph C

For every \(x\) - value in Graph C, there is exactly one \(y\) - value. A cubic function has the form \(y=ax^{3}+bx^{2}+cx + d\) (\(a
eq0\)), and its graph is a smooth curve. Graph C is a smooth curve and passes the vertical line test.

Step5: Analyze Graph D

Graph D is a straight line (a linear function \(y = mx + b\) with \(m
eq0\)). While it is a function, it is a linear function, not a cubic function. A cubic function has a degree of 3 (\(y=ax^{3}+bx^{2}+cx + d,a
eq0\)) and its graph is not a straight line.

Answer:

C. Graph C