Question

Which graph best represents the function f(x) = (x + 1)(x − 1)(x − 4)? Graph of a cubic polynomial that falls to the left and rises to the right with x intercepts negative 3, 2, and 3. The graph intersects the y axis at a point between 10 and 15. Graph of a cubic polynomial that falls to the left and rises to the right with x intercepts negative 3, 2, and 3. The graph intersects the y axis at a point between 15 and 20. Graph of a cubic polynomial that falls to the left and rises to the right with x intercepts negative 3, 1, and 3. The graph intersects the y axis at a point between 5 and 10. Graph of a cubic polynomial that falls to the left and rises to the right with x intercepts negative 1, 1, and 4. The graph intersects the y axis at a point between 0 and 5.

Answer 1

Answer:

Graph of a cubic polynomial that falls to the left and rises to the right with x intercepts negative 1, 1, and 4. The graph intersects the y axis at a point between 0 and 5.

Step-by-step explanation:

$f(x) = (x + 1)(x- 1)(x-4)$

The x-intercepts are when $f(x)=0$

$\implies (x+1)=0 \implies x=-1$

$\implies (x-1)=0 \implies x=1$

$\implies (x-4)=0 \implies x=4$

Therefore, the x-intercepts are -1, 1 and 4

The y-intercept is when $x=0$

$\implies f(0) = (0 + 1)(0- 1)(0-4)=4$

Therefore, the graph intersects the y-axis at (0, 4)

$f(x) = (x + 1)(x- 1)(x-4)$

$\implies f(x)=x^3-4x^2-x+4$

As the degree of the polynomial is odd (3 = cubic) and the leading coefficient is positive, the end behavior of the function is:

$f(x) \rightarrow - \infty, \textsf {as } x \rightarrow - \infty$

$f(x) \rightarrow + \infty, \textsf {as } x \rightarrow + \infty$

So this graph falls to the left and rises to the right.

Answer 2

$\\ \rm\rightarrowtail y=(x+1)(x-1)(x-4)$

• x intercepts=-1,1,4

Simplify

$\\ \rm\rightarrowtail y=(x^2-1)(x-4)$

$\\ \rm\rightarrowtail y=x^3-4x^2-x+4$

Graph it

behaviour:-

• Left:-Fall
• Right:-rise