Find the function y=a x^2+b x+c whose graph contains the points (1,-1),(3,-1), and (-2,14)

Name: Problem 76, Chapter 12: Algebra and Trigonometry
Uploaded: 2020-06-22T04:55:18-08:00
Duration: 4 min 57 s
Channel: Christine Anacker
Description: Find the function y=a x^2+b x+c whose graph contains the points (1,-1),(3,-1), and (-2,14)

step 1 We're going to do a curve one for this. So we've got three points. We can create three equations

Find the function y=a x^2+b x+c whose graph contains the...

Key Concepts

Quadratic Function

A quadratic function is a polynomial of degree two typically written in the form ax² + bx + c, where a, b, and c are coefficients. Its graph is a parabola that has distinctive properties such as a vertex, an axis of symmetry, and potentially two x-intercepts, making it a fundamental concept in algebra.

System of Equations

When a quadratic function is required to pass through multiple specific points, each point is substituted into the quadratic equation to generate a system of linear equations in the coefficients a, b, and c. Solving this system ensures that the function fits all the given points.

Solving Linear Systems

Techniques such as substitution, elimination, or using matrices are employed to solve the system of equations obtained from substituting the coordinates of the points. These methods are essential for accurately determining the unique values of the coefficients in the quadratic function.

Transcript

00:01 We're going to do a curve one for this.

00:04 So we've got three points.

00:06 We can create three equations knowing that our points are fx and fx.

00:11 Our first equation is negative 1 is equal to 1 squared, so a plus b and 1 plus b.

00:21 Our second equation would be negative 1 is equal to 3 squared for 9a, 3b plus b.

00:29 And our third equation is 14 is equal to negative 2 squared which is 4a minus kd or fc.

00:39 So from that we can create our matrices.

00:43 So 1 -1 -1 -1, negative 1, 9 3 -1, negative 1, 4 negative 2, 1, 4, 2, 1, 2, 1, 14.

00:56 Alright, we're still right following up.

00:58 So i'm going to use my first row.

01:00 I'm going to do a double step here.

01:02 I'm going to go row 2 minus 9, row 1, and row 3 minus 4.

01:11 Let's get rid of those numbers.

01:13 1 -1 -1, 1 is not changing.

01:19 9 minus 9 is 0, 3 minus 1 is negative 6, 1 minus 9 is negative 8, negative 1 minus negative 9, so negative 1 plus 9.

01:32 4 minus 4 is 0, negative 2 minus 4 is negative 6, 1 minus 4, 1 minus 4.

01:38 Is negative 3.

01:40 14 minus a negative 4, so 14 plus 4 is 18.

01:47 Um, why don't we go quickly and we'll do 3 minus 3 2.

01:59 That'll be 1, 1, 1, 0, negative 6, negative 8.

02:07 0 is 0 minus 0.

02:11 0.

02:11 Negative six minus and negative six zero negative three minus negative eight five eighteen minus eight three i want to do a double step here it doesn't really make a difference but i can do it but i only go one fifth of row three and i'm going to go a half a word two because don't have to but why not one one one maybe one maybe one 0, 3, 4, negative 4, 0 ,0, 1.

02:58 We'll be a double step again.

03:02 I'm going to use row 1, and 02 to get rid of that 1...

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