To write a quadratic function given three points, we can use the general form of a quadratic function:
Given the points , and , we can substitute these points into the general form of the quadratic function to find the values of , , and .
Let’s substitute the points into the general form of the quadratic function:
For the point :
For the point :
For the point :
Now, we have a system of three equations with three unknowns. We can solve this system to find the values of , , and and then write the quadratic function.
Solve the system of equations by following steps:
step0: Solve using the substitution method:
step1: Solve the equation:
step2: Substitute the value of
step3: Simplify:
step4: Solve the equation:
step5: Substitute the value of
step6: Multiply the terms:
step7: Swap the sides:
step8: Multiply both sides of the equation by LCD:
step9: Simplify the equation:
step10: Divide both sides:
step11: Divide the numbers:
step12: Substitute the value of
step13: Substitute back:
step14: Calculate:
step15: Substitute the values of
step16: Substitute back:
step17: Calculate:
step18: Calculate:
step19: Check the solution:
step20: Rewrite:
The solution to the system of equations is , , and .
Therefore, the quadratic function that passes through the points , and is:
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Bonus Knowledge
To find the quadratic function given the points, we can use the fact that a quadratic can be expressed in the form , where and are the x-intercepts. Given the roots at and , we can write:
We can substitute the point to find the value of . Plugging in: results in , therefore, , giving us . The final function becomes:
Which simplifies to:
.
This equation will give you a beautiful parabola passing through the given points!
Now, to ensure everything fits nicely, you can double-check by substituting back the points , , and into your final equation. If they all satisfy the equation, congratulations, you’ve successfully derived the quadratic function!