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. In this situation, we will use the three points to set up and solve a system of linear equations in three variables. Once we've solved for a, b, and c, we can plug them into the equation of the parabola, which will be either y = ax2 + bx + c for a vertical parabola, or x = ay2 + by + c for a horizontal parabola.
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
#1:
Instructions: find the equation of the parabola.
$$y=ax^2+bx+c$$$$a)\hspace{.2em}(-5,-7), (-1,25), (-2,11)$$
$$b)\hspace{.2em}(1,1), (2,-4), (7,1)$$
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#2:
Instructions: find the equation of the parabola.
$$y=ax^2+bx+c$$$$a)\hspace{.2em}(-2,-13), (0,-25), (-4,-9)$$
$$b)\hspace{.2em}(6,43), (2,-5), (1,-2)$$
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#3:
Instructions: find the equation of the parabola.
$$y=ax^2+bx+c$$$$a)\hspace{.2em}(-2,-25), (-5,-7), (-7,-15)$$
$$b)\hspace{.2em}(1,-7), (-2,-13), (4,-37)$$
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#4:
Instructions: find the equation of the parabola.
$$y=ax^2+bx+c$$$$a)\hspace{.2em}(-8,-8), (-4,-120), (-11,-71)$$
$$b)\hspace{.2em}(10,25), (5,50), (11,50)$$
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#5:
Instructions: find the equation of the parabola.
$$x=ay^2+by+c$$$$a)\hspace{.2em}(5,-12), (5,-8), (2,-11)$$
$$b)\hspace{.2em}(23,3), (43,8), (23,7)$$
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Written Solutions:
#1:
Solutions:
$$a)\hspace{.2em}y=2x^2 + 20x + 43$$
$$b)\hspace{.2em}y=x^2 - 8x + 8$$
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#2:
Solutions:
$$a)\hspace{.2em}y=-x^2 - 8x - 25$$
$$b)\hspace{.2em}y=3x^2 - 12x + 7$$
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#3:
Solutions:
$$a)\hspace{.2em}y=-2x^2 - 20x - 57$$
$$b)\hspace{.2em}y=-2x^2 - 5$$
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#4:
Solutions:
$$a)\hspace{.2em}y=-7x^2 - 112x - 456$$
$$b)\hspace{.2em}y=5x^2 - 80x + 325$$
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#5:
Solutions:
$$a)\hspace{.2em}x=y^2 + 20y + 101$$
$$b)\hspace{.2em}x=4y^2 - 40y + 107$$
