Video Transcript
Write the quadratic equation
represented by the graph shown. Give your answer in factored
form.
To do this, let’s consider some
features of the graph of quadratic functions like shape, 𝑦-intercept, roots, and
vertex. The shape of this parabola opens
upward. And that tells us that our 𝑎-value
will be positive, greater than zero. We have a 𝑦-intercept located at
the point zero, zero. To find the roots, we look for the
place where the graph crosses the 𝑥-axis. Here we have a root at zero, zero
and at five, zero. The vertex of this graph has been
labeled. It’s a minimum coordinate located
at two and a half, negative 6.25.
What we wanna do now is think a
little bit more about the roots. The roots are sometimes called
solutions. They’re the place where our
function equals zero. We have solutions when 𝑥 equals
zero and when 𝑥 equals five. We can take these solutions and
turn them into factors. Because 𝑥 is the same thing as 𝑥
minus zero, 𝑥 equals zero is a factor. And for factor two, we would want
to say 𝑥 minus five equals zero. Here are two factors. However, we still don’t know what
our 𝑎-value is because there are many equations that have the factors 𝑥 and 𝑥
minus five. And that means we should have 𝑓 of
𝑥 equals 𝑎 times 𝑥 times 𝑥 minus five.
To solve for 𝑎, we can plug in
coordinates of points we know fall on this graph. We know that when 𝑥 equals 2.5, 𝑓
of 𝑥 equals negative 6.25. Two and a half minus five is
negative two and a half. Two and a half times negative two
and a half equals negative 6.25. And if we divide both sides of the
equation by negative 6.25, we see that 𝑎 equals one. And if 𝑎 equals one, then our
factored form is 𝑓 of 𝑥 equals 𝑥 times 𝑥 minus five. We always want to check for this
𝑎-value because as I’ve just drawn on top of this graph — here’s an equation that
has the same factors. However, in this case, the 𝑎 would
have to be less than zero because the graph opens downward, which means it’s not
enough just to know the factors. You need to also check for the
𝑎-value.
