## 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
$9=k( \times 3 \times \frac{1}{4} \times (- \frac{3}{2})^2 \rightarrow k = \frac{16}{3}$
.

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