Video Transcript
Write the quadratic equation
represented by the graph shown.
First, notice that the graph is
symmetrical about the 𝑦-axis. The general equation of a quadratic
that is symmetrical about the 𝑦-axis is 𝑦 equals 𝑘𝑥 squared plus 𝑐. To determine the values of 𝑘 and
𝑐, we can use the graph to find pairs of 𝑥- and 𝑦-values that satisfy the
equation at various points. This will give us a set of
simultaneous equations, which can be solved to find 𝑘 and 𝑐. Since we have two unknown
coefficients, we will need no more than two equations and therefore two pairs of 𝑥-
and 𝑦-values.
Firstly, the point where the curve
intercepts the 𝑦-axis is at 𝑥 equals zero and equals 𝑦 equals one. Secondly, at 𝑥 equals one, the
corresponding 𝑦-value is two. Substituting in the first pair of
𝑥- and 𝑦-values into the general equation gives us one equals 𝑘 times zero
squared plus 𝑐. 𝑘 times zero squared is just zero,
so this immediately tells us that the value of 𝑐 is one.
Now, substituting the second pair
of 𝑥- and 𝑦-values into the general equation gives us two equals 𝑘 times one
squared plus 𝑐. Since we already know the value of
𝑐 to be one, this gives two equals 𝑘 plus one. And this can be simplified by
subtracting one from both sides to give one equals 𝑘. We can now place these values of 𝑘
and 𝑐 into the general quadratic equation. This gives us the quadratic
equation represented by the graph shown: 𝑦 equals 𝑥 squared plus one.
