Question

Which of the following functions best represents the graph?f(x) = x3 + x2 − 4x − 4f(x) = x3 + 4x2 − x − 4f(x) = x3 + 3x2 − 4x − 12f(x) = x3 + 2x2 − 4x − 8

Answer 1

f(x) = x^3 + x^2 - 4x - 4

Explanation:

Looking at the graph you can notice that its interceptions are at the points (0, -4) on the y-axis, and on the x-axis (-2, 0), (-1, 0) and (2, 0).

To eliminate some possible solutions, start with replacing x by 0, to see if you get -4 (to have the interception with the y-axis)

For x = 0

For the 1st one => y = -4 : Yes

For the 2nd one => y = -4 : Yes

For the 3rd one => y = - 12 : No

For the 4th one => y = -8 : No

Now for the 2 functions left, search for f(x) = 0 (in order words y = 0) to find the intercepts with the x -axis.

$\begin{gathered} x^3+x^2\text{ - 4x -4 = 0 } \\ (x^3+x^2)\text{ + \lparen-4x-4\rparen = 0 } \\ x^2(x+1)-4(x+1)\text{ = 0} \\ (x+1)(x^2-4)\text{ = 0 } \\ (x+1)(x+2)(x-2)\text{ = 0} \\ =>\text{ \lparen x+1\rparen= 0 => x = -1} \\ =>\text{ \lparen x+2\rparen = 0 => x = -2} \\ =>\text{ \lparen x-2\rparen = 0 =>x = 2} \end{gathered}$

Since all 3 x correspond to the intercepts of the graph, you can deduce that the first f(x) is the correct answer.