Write the quadratic equation
represented by the graph shown.
We’re told that this is going to be
a quadratic equation. And in fact, observing the general
shape of the graph, it’s a parabola. So, we know it’s definitely the
graph of a quadratic function. This is of the form 𝑓 of 𝑥 equals
𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐, where 𝑎, 𝑏, and 𝑐 are real numbers and 𝑎 is not
equal to zero.
Now, in fact, our job is to
identify the values of 𝑎, 𝑏, and 𝑐. And we can quite quickly determine
the value of 𝑏. The function of the form 𝑎𝑥
squared plus 𝑐, or indeed 𝑘𝑥 squared plus 𝑐, has a line of symmetry 𝑥 equals
zero; it’s symmetrical about the 𝑦-axis. We see our graph is indeed
symmetrical about the 𝑦-axis, so we can deduce that the value of 𝑏 must be equal
to zero. And so, our function is of the form
𝑓 of 𝑥 equals 𝑎𝑥 squared plus 𝑐.
And next, we can use information
about its 𝑦-intercept to work out the value of 𝑐. Given a function of the form 𝑓 of
𝑥 equals 𝑎𝑥 squared plus 𝑐, its 𝑦-intercept is zero, 𝑐. Now, our 𝑦-intercept is zero,
two. It passes through the 𝑦-axis at
two. And so, 𝑐 must be equal to two,
meaning we can write our function as 𝑓 of 𝑥 equals 𝑎𝑥 squared plus two.
But how do we find the value of
𝑎? Well, we can choose a point that
lies on the given curve and substitute the 𝑥- and 𝑦-values of this coordinate into
the function. Let’s choose the point with
coordinates two, negative two. As an ordered pair, this is 𝑥, 𝑦
or 𝑥, 𝑓 of 𝑥. And so, 𝑓 of 𝑥 equals 𝑎𝑥
squared plus two becomes negative two equals 𝑎 times two squared plus two. Two squared is four. So, the equation becomes negative
two equals four 𝑎 plus two. Then subtracting two from both
sides of this equation, we get negative four equals four 𝑎. And then we divide through by four,
giving us 𝑎 is equal to negative one.
And so, we could write our function
as 𝑓 of 𝑥 equals negative one 𝑥 squared plus two, or we could write it as 𝑦
equals negative 𝑥 squared plus two. And in fact, since the parabola in
our diagram is n shaped, we were expecting the coefficient of 𝑥 squared to be
negative. The equation of the curve is 𝑦
equals negative 𝑥 squared plus two.