Graph the function f by starting with the graph of y=x^2 and using transformations, compressing,

Graph the function f by starting with the graph of y=x^2 and...

Key Concepts

Completing the Square

Completing the square is an algebraic method used to rewrite a quadratic function in vertex form. This technique involves adding and subtracting the appropriate constant term to form a perfect square trinomial, which then can be factored into the form (x - h)². This method is essential for revealing the geometric transformations of the quadratic function.

Graph Transformations

Graph transformations involve shifting, reflecting, stretching, or compressing the graph of a basic function to obtain the graph of a more complex function. In the context of quadratics, this includes vertical and horizontal translations (shifts), reflections over the axes, and changes in the width of the parabola through vertical stretching or compressing. These transformations are applied to the graph of the basic parabola y = x² to yield the desired graph.

Vertex Form of a Quadratic Equation

The vertex form of a quadratic function is written as y = a(x - h)² + k, where (h, k) is the vertex of the parabola and a describes the vertical stretch or compression as well as the direction in which the parabola opens. Rewriting a quadratic function in vertex form makes it easier to identify the vertex and to apply transformations to graph the function.

Quadratic Functions

A quadratic function is a second degree polynomial of the form y = ax² + bx + c, where a, b, and c are real numbers and a is nonzero. Quadratic functions represent parabolas in the Cartesian coordinate system, and their graphs can open upward or downward depending on the sign of the coefficient a.

Transcript

00:01 So we have the function here, f of x is equal to negative x squared minus 2x.

00:05 So we could write this in vertex form, which is in the form a times x minus h quantity squared plus k.

00:13 Well, then we could see what the transformation is on just the graph of f of x equals x squared.

00:17 Because then while we would shift h units to the, well, to the right if h is positive, or to the left if we have x plus a squared, and then we shift while k units up.

00:30 Or down, depending on if k is positive or negative.

00:33 So if we can go ahead and take this function here and put it in the vertex form, we could see what the transformation is.

00:41 So to do so, we want to complete the square.

00:44 So to complete the square, we first, well, we don't have any constants here.

00:47 If we had it plus a constant, we want to put that to the other side of the equation.

00:51 Basically, think of the equation set equal to zero, and then move the constant over.

00:55 We don't have any constants.

00:56 So we just have negative x squared minus 2x.

01:00 Equal to 0.

01:02 Now to complete the square, we have to have the reading coefficient or the coefficient on the quadratic term has to be equal to 1.

01:09 Here it's equal to negative 1.

01:11 So we just factor out the negative.

01:13 So we have negative x squared plus 2x is equal to 0.

01:20 Now complete the square, we look at in parentheses here and we first, we copy it down, and then we take half of the coefficient on the linear term.

01:31 So half of two is one, and then we square that, and that's what we add.

01:35 So half of two, one, one squared is one.

01:38 So we add a one, and then we have to add the same thing to the left side.

01:42 We have zero.

01:43 Then, well, we really didn't add a one.

01:45 We really added a negative one, because that negative sign distributes, right? so we have a negative one, so we have to add the same thing to the right side, so we have plus a negative one or minus one...

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