Question

Grade 11 High School Functions: Create all three forms of the quadratic equations for the parabola passing through: (-2 , 65) , (0, 165) , and (6 , 225). Two of your quadratic equations must be determined by hand, from scratch, using algebra. State the one unique property that is given from each of your three quadratic equations. Show all of your planning and processing steps. Justify that your three equations all work.

Answer 1

We know a quadratic function in standard form is given by:

$y=ax^2+bx+c$

where a, b and c are constants. To find the value of the constants we can use the points given.

For the point (-2,65) we have:

$\begin{gathered} (-2)^2a-2b+c=65 \\ 4a-2b+c=65 \end{gathered}$

For the point (0,165) we have:

$\begin{gathered} (0)^2a+(0)b+c=165 \\ c=165 \end{gathered}$

For the point (6,225) we have:

$\begin{gathered} (6)^2a+6b+c=225 \\ 36a+6b+c=225 \end{gathered}$

Hence we have the system of equations:

$\begin{gathered} 4a-2b+c=65 \\ 36a+6b+c=225 \\ c=165 \end{gathered}$

Plugging the value of c in the first two equations we have:

$\begin{gathered} 4a-2b+165=65 \\ 36a+6b+165=225 \end{gathered}$

which leads to:

$\begin{gathered} 4a-2b=-100 \\ 36a+6b=60 \end{gathered}$

Multiplying the first equation by 3 we have:

$\begin{gathered} 12a-6b=-300 \\ 36a+6b=60 \end{gathered}$

adding the equations we have that:

$\begin{gathered} 48a=-240 \\ a=-\frac{240}{48} \\ a=-5 \end{gathered}$

Plugging the value of a in the equation above we have:

$\begin{gathered} 4(-5)-2b=-100 \\ -20-2b=-100 \\ 2b=100-20 \\ 2b=80 \\ b=\frac{80}{2} \\ b=40 \end{gathered}$

Once we know the values of the constants we conclude that the quadratic equation in standard form is:

$y=-5x^2+40x+165$

Now, to write the equation in vertex form we need to complete the square on x:

$\begin{gathered} y=-5(x^2-8x)+165 \\ y=-5(x^2-8x+(-\frac{8}{2})^2)+165+5(\frac{8}{2})^2 \\ y=-5(x-4)^2+245 \end{gathered}$

Therefore, the equation written in vertex form is:

$y=-5(x-4)^2+245$

Now, the unique features of the two forms presented are:

Standard form: The equation is expanded and reduced.

Vertex form: The vertex of the parabola is readily shown in the equation.