Question

graphing polynomial functions
let a and b be real numbers where a ≠ b ≠ 0. which of the following functions could represent the graph?
$f(x) = (x - a)^5(x - b)$
$f(x) = (x - a)^2(x - b)^4$
$f(x) = x(x - a)^6(x - b)^2$
$f(x) = x(x - a)^3(x - b)^3$

Explanation:

Step1: Analyze the graph's x - intercepts and their multiplicities

The graph touches the x - axis at two points, which means the roots (x - intercepts) have even multiplicities. Also, the leading coefficient's sign: as \(x\to+\infty\) and \(x\to-\infty\), the function \(y = f(x)\) goes to \(+\infty\), so the degree of the polynomial is even (since the ends of the graph go in the same direction) and the leading coefficient is positive.

Step2: Analyze each function

• For \(f(x)=(x - a)^{5}(x - b)\): The degree is \(5 + 1=6\) (even), but the multiplicity of \((x - a)\) is 5 (odd) and \((x - b)\) is 1 (odd). At the roots \(x = a\) and \(x = b\), the graph should cross the x - axis (because of odd multiplicities), but our graph touches the x - axis, so this is not correct.
• For \(f(x)=(x - a)^{2}(x - b)^{4}\): The degree is \(2+4 = 6\) (even). The multiplicities of \((x - a)\) is 2 (even) and \((x - b)\) is 4 (even). Also, when we expand the function, the leading term is \(x^{2}\cdot x^{4}=x^{6}\), and the leading coefficient is 1 (positive). This matches the end - behavior (\(x\to\pm\infty,f(x)\to+\infty\)) and the fact that the graph touches the x - axis (even multiplicities).
• For \(f(x)=x(x - a)^{6}(x - b)^{2}\): The degree is \(1 + 6+2=9\) (odd). For an odd - degree polynomial, as \(x\to+\infty\) and \(x\to-\infty\), the function should go in opposite directions, but our graph has both ends going to \(+\infty\), so this is incorrect.
• For \(f(x)=x(x - a)^{3}(x - b)^{3}\): The degree is \(1+3 + 3=7\) (odd). Also, the multiplicities of \((x - a)\) and \((x - b)\) are 3 (odd), and there is a root at \(x = 0\) with multiplicity 1 (odd). The graph should cross the x - axis at \(x = 0\), \(x = a\), and \(x = b\), which does not match our graph, so this is incorrect.

Answer:

\(f(x)=(x - a)^{2}(x - b)^{4}\)