find the polynomial function of degree 3 whose graph is shown in the figure. y =

Explanation:

Step1: Determine the roots

From the graph, the x - intercepts (roots) are (x = 2) and (x = 4). Since the graph touches the x - axis at (x = 2), the root (x = 2) has a multiplicity of 2. A cubic polynomial with roots (r_1,r_2,r_3) can be written in factored form as (y=a(x - r_1)(x - r_2)(x - r_3)). Here, the polynomial is (y=a(x - 2)^2(x - 4)).

Step2: Find the value of (a)

We know that the point ((1,-3)) lies on the graph. Substitute (x = 1) and (y=-3) into (y=a(x - 2)^2(x - 4)). [ \begin{align*} -3&=a(1 - 2)^2(1 - 4)\ -3&=a(-1)^2(-3)\ -3&=a\times1\times(-3)\ a& = 1 \end{align*} ]

Step3: Write the polynomial function

Substitute (a = 1) into (y=a(x - 2)^2(x - 4)). [ \begin{align*} y&=(x - 2)^2(x - 4)\ &=(x^{2}-4x + 4)(x - 4)\ &=x^{3}-4x^{2}-4x^{2}+16x+4x - 16\ &=x^{3}-8x^{2}+20x - 16 \end{align*} ]

Answer:

(y=x^{3}-8x^{2}+20x - 16)