\[x\]
axis at \[x=-3, \: x=- \frac{1}{4}\]
and touches at \[x= \frac{3}{2}\]
. The graph crosses the \[y=9\]
.Each crossing of the
\[x\]
axis results in a single linear factor and each touching results in a repeated linear factor.\[f(x)=k(x+3)(x+ \frac{1}{4})(x- \frac{3}{2})^n\]
.The powers of the factors add up to 4 since
\[f(x)\]
is a quartic, so \[n=2\]
.\[f(x)\]
passes through the point \[(0,9)\]
so \[9=k( \times 3 \times \frac{1}{4} \times (- \frac{3}{2})^2 \rightarrow k = \frac{16}{3}\]
.