VERTEX FORM The vertex form of a quadratic function is...

VERTEX FORM The vertex form of a quadratic function is $y=a(x-h)^{2}+k$. Its graph is a parabola with vertex at $(\boldsymbol{h}, \boldsymbol{k})$. Use completing the square to write the quadratic function in vertex form. Then give the coordinates of the vertex of the graph of the function. $$y=-x^{2}-5 x+6$$

Key Concepts

Quadratic Function

A quadratic function is a second-degree polynomial function typically expressed in standard form as y = ax² + bx + c, where a, b, and c are constants. Its graph is a parabola which can open upwards or downwards depending on the sign of a.

Vertex Form

The vertex form of a quadratic function is written as y = a(x - h)² + k, where (h, k) represents the vertex of the parabola. This form clearly shows how the graph is shifted horizontally and vertically from the parent function y = ax².

Completing the Square

Completing the square is an algebraic method used to transform a quadratic equation from standard form into vertex form. This technique involves rewriting the quadratic and linear terms as a perfect square trinomial plus a constant, thus facilitating the identification of the vertex of the parabola.

Vertex

The vertex of a parabola is the point where it attains its maximum or minimum value. In the vertex form y = a(x - h)² + k, the vertex is immediately given by the coordinates (h, k), making it easier to determine the position and the direction of the parabola's opening.

Transcript

00:01 Quadratic here, y is equal to negative x squared minus 5x plus six.

00:05 So to put this into vertex form, well, we are going to complete the square.

00:11 We're going to first take our constant and basically move it to the other side of the equation.

00:17 So basically think of this as being set equal to zero and then move the constant over.

00:22 So we have negative x squared minus 5x is then equal to a negative six.

00:30 Now, to complete the square, well, we have to have the coefficient on our quadratic term, on our x -square term, has to be one.

00:38 Well, right now it's negative 1, right? so we can factor out a negative.

00:43 So we have negative x squared plus 5x, okay, is equal to, well, negative 6, right? that doesn't change.

00:52 Okay, and then we can go ahead and complete the square here.

00:55 So we have, well, we have negative, and we have the x squared plus 5x.

01:01 Now we take half of the coefficient on our linear term.

01:05 So our linear term is, well, 5x.

01:08 The coefficient of 5.

01:09 So what's half of 5? well, it's just 5 over 2, right? half of any number is just divided by 2.

01:15 So half of 5 is 5 over 2, but then we square it.

01:21 Right? we take half of the coefficient, which is 5, so it's not a perfect, you know, it's odd.

01:26 We get a fraction, 5 over 2.

01:28 Then we square it, so we get, well, do you square to top? let's go to the bottom.

01:32 We have 25 over 4.

01:34 So we're adding here, we're adding 25 over 4.

01:44 Okay.

01:45 And then that's equal to, well, a negative 6, we have to add the exact same thing we added to this side.

01:52 Now, we added a 25 over 4, but wait a minute, we factored out a negative.

01:57 And that negative is going to distribute.

01:59 So we really added a negative 25 4ths.

02:02 So we have negative six.

02:05 Let's go off the side here.

02:06 We have negative six.

02:08 And then we're adding a negative 25 force.

02:10 That's negative 6 minus 25 force.

02:14 What is that? well, it's a common denominator by multiplying my 4 over 4.

02:18 6 times 4 is 24.

02:20 This is minus 24 force, right, which is negative 6.

02:28 And then minus 25 force.

02:36 Okay, what's that? well, that's 20, 30, 40, 45 minus 49 force...

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