Question

If the points in the table lie on a parabola, write the equation whose graph is the parabola. x|-3 |-2 |-1 |0

y|-1 |-4 |-1 |8

Answer 1

Given the table of points that lie on a parabola.


`\begin{center} \begin{tabular} x&-3&-2&-1&0\\ y&-1&-4&-1&8 \end{tabular} \end{center}`

The equation of a parabola is given by
`y=ax^2+bx+c`, where a, b and c are constants. Since the equation of a data point has three constants, three data points/equations will be needed to obtain the value of the constants and hence the required equation.

when x = 0, y = 8 and we have:

`8=a(0)^2+b(0)+c \\ \\ \Rightarrow c=8`

When x = -1, y = -1 and we have:


`-1=a(-1)^2+b(-1)+c \\ \\ \Rightarrow-1=a-b+8 \\ \\ \Rightarrow a-b=-1-8=-9`

When x = -2, y = -4 and we have:


`-4=a(-2)^2+b(-2)+c \\ \\ \Rightarrow-4=4a-2b+8 \\ \\ \Rightarrow4a-2b=-4-8=-12`

Solving the two equations simultaneously, we have:


`a-b=-9\Rightarrow a=b-9 \\ \\ 4a-2b=-12 \\ \Rightarrow4(b-9)-2b=-12 \\ \Rightarrow4b-36-2b=-12 \\ \Rightarrow2b-36=-12 \\ \Rightarrow2b=-12+36=24 \\ \Rightarrow b=24/2=12. \\ \\ a=b-9=12-9=3.`

Thus, a = 3, b = 12 and c = 8

Therefore, the required equation of the parabola is given by:


`y=3x^2+12x+8`