Find the equation of the parabola y=a x^2+b x+c that goes through the points (1,6),(2,4), and (3

Find the equation of the parabola y=a x^2+b x+c that goes...

Key Concepts

Quadratic Functions

A quadratic function is a polynomial function of degree two, typically expressed in the form y = ax² + bx + c. Its graph is a parabola, which can open upwards or downwards depending on the coefficient a. Understanding its structure is essential when determining the function's behavior and solving for unknown coefficients.

System of Linear Equations

When a quadratic function is required to pass through specific points, each point provides an equation that relates its x and y coordinates to the coefficients a, b, and c. This set-up leads to a system of linear equations which can be solved simultaneously to find the unique values of the coefficients.

Interpolation

Interpolation involves determining a function that exactly fits a set of given data points. In the context of quadratic functions, this means finding the appropriate coefficients so that the quadratic curve passes through all the specified points, thereby accurately representing the data.

Transcript

00:01 Given the form of an equation, y equals a plus bx plus cx squared, going through the points negative 2, 3, negative 1, 2, and 1 6, then we come up with our equations by substituting x and y.

00:17 So using the point negative 2, 3, it would be 3 equals a plus negative 2b, and then negative 2 squared is 4, so it would be plus 4c.

00:30 A second equation using the second order pair would be 2 equals negative 1a.

00:42 I'm just kidding.

00:44 A and then it would be substituting negative 1 in for x would be minus b, and then negative 1 squared is 1 would be plus c.

00:55 And then from the 3rd coordinate 6 equals substituting 1 in for x would be a plus x plus x plus plus c.

01:06 Sorting these out, we have the, i am going to move all of the a, bs and c's to the, all actually i'm going to do is move the 3, 2, and 6 just to the right hand side.

01:35 So it looks a little more familiar.

01:39 And then we're going to augment this matrix with coefficients 1, negative 2, 4, 1, negative 1, 1, 1, it's actually a b in 1 -1 -1 with the constants 3, 2, and 6.

02:02 And then we'll get this in reduced row form.

02:09 So we already have a 1 in row 1.

02:12 So let's take row 2 and subtract row 1, so that we'll get a 0 in row 2.

02:21 So this will give us a new row 2 of 0 -1 -1 -3 -1.

02:28 1 and 3 stay the same.

02:32 Now we'll get a 0 in row 3 by taking row 3 minus row 1 and putting that in row 3.

02:41 So we'll get a new row 3 of 0 3, negative 3, 3...

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