Question

which of the following graphs could represent a quartic function?
a. graph a
b. graph b
c. graph c
d. graph d
(options: graph c, graph b, graph a, graph d)

Explanation:

Step1: Recall the properties of a quartic function

A quartic function is a polynomial function of degree \(4\). The general form is \(y = ax^{4}+bx^{3}+cx^{2}+dx + e\), where \(a
eq0\). The graph of a quartic function is a smooth curve. The maximum number of turning points of a polynomial function of degree \(n\) is \(n - 1\). For a quartic function (\(n = 4\)), the maximum number of turning points is \(4-1=3\).

Step2: Analyze each graph

• Graph A: It is a smooth curve with \(3\) turning points. A quartic function can have \(3\) turning points.
• Graph B: It is a straight - line graph. The equation of a straight line is of the form \(y=mx + c\), which is a linear function (\(n = 1\)).
• Graph C: It is not a function. By the vertical line test, if a vertical line intersects the graph at more than one point, it is not a function.
• Graph D: It is a vertical line \(x = k\) (where \(k\) is a constant). By the vertical line test, it is not a function.

Answer:

A. Graph A