Question

These 3 points are on a parabola defining the edge of a ski:

Answer 1

The equation of the parabola is y = 0.015x^2 + 0.06x + 1

The linear equations

A parabola is represented as:

y = ax^2 + bx + c

The points are given as:

(-4,1), (-2,0.94) and (0,1)

When these points are substituted in y = ax^2 + bx + c, we have the following linear equations

16a - 4b + c = 1

4a - 2b + c = 0.94

c = 1

The matrix that represent the system of equations

In (a), we have:

16a - 4b + c = 1

4a - 2b + c = 0.94

c = 1

Remove the variables (leaving the coefficients)

a      b       c

16    -4      1        1

4      -2     1        0.94

0      0     1          1

So, the matrix representation of the system of equations is:

$\left[\begin{array}{ccc}16&-4&1\\4&-2&1\\0&0&1\end{array}\right] \left[\begin{array}{c}1&0.94&1\end{array}\right]$

The inverse of the coefficient matrix

Using a graphing calculator, we have the inverse of the coefficient matrix to be

$\left[\begin{array}{ccc}0.125&-0.25&0.125\\0.25&-1&0.75\\0&0&1\end{array}\right]$

The solution to the system of equations

Using a graphing calculator, we have the solution to the system of equations to be

a = 0.015

b = 0.06

c = 1

Recall that:

y = ax^2 + bx + c

So, we have:

y = 0.015x^2 + 0.06x + 1

Hence, the equation of the parabola is y = 0.015x^2 + 0.06x + 1

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