Write the quadratic equation
represented by the graph shown.
We have the general form for a
quadratic equation 𝑓 of 𝑥 equals 𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐 and the vertex
form 𝑎 times 𝑥 minus ℎ squared plus 𝑘. To find the equation, we’ll
consider some features of the graph. The shape of this graph opens
downward. And that means we know that our
𝑎-value will be less than zero. It will be negative. We have a 𝑦-intercept at the point
zero, two. When it comes to the roots, we
can’t know with a great deal of accuracy what the roots are. So we’ll leave them there for
now. The vertex here is a maximum. And we do know where it’s located,
at the point one, three.
Because we know the vertex, we can
start with our vertex form. Our vertex is ℎ, 𝑘. And so we plug in one for ℎ and
three for 𝑘. The only thing we’re missing now is
this 𝑎-variable. We know that it’s less than zero,
but we don’t know exactly what it is. To find it, we can plug in another
point from the graph that we already know. If we plug in zero for 𝑥 and two
for 𝑓 of 𝑥, we’ll be able to solve for our 𝑎-value. Zero minus one squared is one; one
times 𝑎 is 𝑎, which means 𝑎 plus three equals two. And two minus three is negative
one, so we can say that 𝑎 equals negative one.
And we’ll go back and plug that
in. Instead of having negative one, we
can just write the negative sign and say that 𝑓 of 𝑥 equals negative 𝑥 minus one
squared plus three. This is the vertex form of the
graph we have. If we wanted to write this in the
general form, we could expand this 𝑥 minus one squared, which would give us the
negative of 𝑥 squared minus two 𝑥 plus one plus three. We distribute the negative. And when we combine like terms, we
have the general form of negative 𝑥 squared plus two 𝑥 plus two. Both of these forms are the
quadratic represented by this graph.