## Reconstructing a Graph From It's Intersections With the Axes

Given the intersections of a graph with the axes, we may be able to construct the equation of the graph. Suppose a quartic polynomial - degree 4 - crosses the\[x\]

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 \[9=k( \times 3 \times \frac{1}{4} \times (- \frac{3}{2})^2 \rightarrow k = \frac{16}{3}\]

.