### About Finding the Equation of a Parabola:

In some cases, we will be asked to find the equation of a parabola given three points on the parabola. To perform this action, we will use the three points given to set up a system of linear equations in three variables. To obtain our three equations, we will simply plug in for x and y in the equation y = ax2 + bx + c. This will allow us to solve for the three unknowns: a, b, and c. Once we have solved for a, b, and c, we can plug into the standard form of a parabola: y = ax2 + bx + c and obtain our equation.

Test Objectives
• Demonstrate the ability to solve a system of equations in three variables
• Demonstrate the ability to find the equation of a parabola given three points
Finding the Equation of a Parabola Practice Test:

#1:

Instructions: find the equation of the parabola.

$$y=ax^2+bx+c$$

$$a)\hspace{.2em}(-5,-7), (-1,25), (-2,11)$$

$$b)\hspace{.2em}(1,1), (2,-4), (7,1)$$

#2:

Instructions: find the equation of the parabola.

$$y=ax^2+bx+c$$

$$a)\hspace{.2em}(-2,-13), (0,-25), (-4,-9)$$

$$b)\hspace{.2em}(6,43), (2,-5), (1,-2)$$

#3:

Instructions: find the equation of the parabola.

$$y=ax^2+bx+c$$

$$a)\hspace{.2em}(-2,-25), (-5,-7), (-7,-15)$$

$$b)\hspace{.2em}(1,-7), (-2,-13), (4,-37)$$

#4:

Instructions: find the equation of the parabola.

$$y=ax^2+bx+c$$

$$a)\hspace{.2em}(-8,-8), (-4,-120), (-11,-71)$$

$$b)\hspace{.2em}(10,25), (5,50), (11,50)$$

#5:

Instructions: find the equation of the parabola.

$$x=ay^2+by+c$$

$$a)\hspace{.2em}(5,-12), (5,-8), (2,-11)$$

$$b)\hspace{.2em}(23,3), (43,8), (23,7)$$

Written Solutions:

#1:

Solutions:

$$a)\hspace{.2em}y=2x^2 + 20x + 43$$

$$b)\hspace{.2em}y=x^2 - 8x + 8$$

#2:

Solutions:

$$a)\hspace{.2em}y=-x^2 - 8x - 25$$

$$b)\hspace{.2em}y=3x^2 - 12x + 7$$

#3:

Solutions:

$$a)\hspace{.2em}y=-2x^2 - 20x - 57$$

$$b)\hspace{.2em}y=-2x^2 - 5$$

#4:

Solutions:

$$a)\hspace{.2em}y=-7x^2 - 112x - 456$$

$$b)\hspace{.2em}y=5x^2 - 80x + 325$$

#5:

Solutions:

$$a)\hspace{.2em}x=y^2 + 20y + 101$$

$$b)\hspace{.2em}x=4y^2 - 40y + 107$$