let a and b be real numbers, where a ≠ b ≠ 0. which of the following functions could represent the graph on…

let a and b be real numbers, where a ≠ b ≠ 0. which of the following functions could represent the graph on the right?\n○ f(x)=x(x - a)(x - b)^2\n○ f(x)=(x - a)(x - b)^2\n○ f(x)=x(x - a)^3(x - b)\n○ f(x)=x^2(x - a)^2(x - b)^2\ndone

let a and b be real numbers, where a ≠ b ≠ 0. which of the following functions could represent the graph on the right?\n○ f(x)=x(x - a)(x - b)^2\n○ f(x)=(x - a)(x - b)^2\n○ f(x)=x(x - a)^3(x - b)\n○ f(x)=x^2(x - a)^2(x - b)^2\ndone

Answer

Explanation:

Step1: Analyze root - behavior

The roots of a polynomial function (y = f(x)) are the (x) - values for which (f(x)=0). If a factor ((x - r)) has an odd power (n) in the polynomial, the graph of the function crosses the (x) - axis at (x = r). If a factor ((x - r)) has an even power (n), the graph of the function touches the (x) - axis at (x = r).

Step2: Analyze the given graph

The graph crosses the (x) - axis at (x = 0) and touches the (x) - axis at two non - zero points (corresponding to (x=a) and (x = b)). So, the factor corresponding to (x = 0) should have an odd power and the factors corresponding to (x=a) and (x = b) should have even powers.

Step3: Check each option

  • For (f(x)=x(x - a)(x - b)^{2}), the factor (x) has power 1 (odd), ((x - a)) has power 1 (odd), and ((x - b)) has power 2 (even).
  • For (f(x)=(x - a)(x - b)^{2}), there is no factor of (x) corresponding to the root (x = 0).
  • For (f(x)=x(x - a)^{3}(x - b)), the factor ((x - a)) has power 3 (odd) and ((x - b)) has power 1 (odd).
  • For (f(x)=x^{2}(x - a)^{2}(x - b)^{2}), the factor (x) has power 2 (even), ((x - a)) has power 2 (even), and ((x - b)) has power 2 (even).

Answer:

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