Question

replace the values b, c, and d to write function $f$.
$f(x)=(x-b)(x-c)^2(x-d)^3$

Explanation:

Step1: Identify root from cross (odd deg.)

The graph crosses the x-axis at $x=-1$ and $x=4$. The root with degree 1 is $x=-1$, so $b=-1$.

Step2: Identify root from touch (even deg.)

The graph touches the x-axis at $x=2$ (bounces back), so this is the squared root: $c=2$.

Step3: Identify root from cross (odd deg.)

The graph crosses the x-axis at $x=4$ with a steeper curve, matching degree 3, so $d=4$.

Step4: Substitute values into function

Substitute $b=-1$, $c=2$, $d=4$ into $f(x)=(x-b)(x-c)^2(x-d)^3$.
<Expression>
$f(x)=(x-(-1))(x-2)^2(x-4)^3=(x+1)(x-2)^2(x-4)^3$
</Expression>

Answer:

$f(x)=(x+1)(x-2)^2(x-4)^3$