Question

Find the function that has the following graph (not the x-intercepts and given point) Answer choices: a. f(x) = 3(x - 2)(x + 1)^2b. f(x) = (x - 2)(x + 1)(x - 3)c. f(x) = 2(x - 1)(x + 1)(x - 2)d. f(x) = 4(x - 3)(x - 2)(x + 1)

Answer 1

Before finding the function that describes the graph we can at a glance identify the following:

Y-intercepts

`\begin{gathered} (0,4) \\ y=4\to x=0 \end{gathered}`

X-intercepts or the roots

`\begin{gathered} x=-1 \\ x=1 \\ x=2 \end{gathered}`

Other point given

`(3,16)`

with the given points we will replace in each function to identify the correct one. we will start with (0,4)

`\begin{gathered} a) \\ f(0)=3(0-2)(0+1)^2 \\ f(0)=3(-2)(1) \\ f(0)=-6_{}\to\text{Not correct} \end{gathered}`
`\begin{gathered} b) \\ f\mleft(0\mright)=(0-2)\mleft(0+1\mright)\mleft(0-3\mright) \\ f(0)=(-2)(1)(-3) \\ f(0)=6\to\text{Not correct} \end{gathered}`
`\begin{gathered} c) \\ f\mleft(0\mright)=2\mleft(0-1\mright)\mleft(0+1\mright)\mleft(0-2\mright) \\ f(0)=2(-1)(1)(-2) \\ f(0)=4\to is\text{ Possible} \end{gathered}`
`\begin{gathered} d) \\ f\mleft(0\mright)=4\mleft(0-3\mright)\mleft(0-2\mright)\mleft(0+1\mright) \\ f(0)=4(-3)(-2)(1) \\ f(0)=24\to\text{Not correct} \end{gathered}`

After analyzing the y-intercept we can see that the function it fulfills is letter C

To verify it, we will replace the additional point and see if it gives the expected result:

`\begin{gathered} f(3)=2(3-1)(3+1)(3-2) \\ f(3)=2(2)(4)(1) \\ f(3)=16\to\text{Correct} \end{gathered}`

With this we can confirm that the function of the letter C describes the figure