Maximum and Minimum Values A quadratic function f is given. (a) Express f in standard form. (b)

step 1 We want to convert this general form quadratic to standard form. So first thing we're going to d

Maximum and Minimum Values A quadratic function f is given....

Key Concepts

Quadratic Functions

A quadratic function is a second-degree polynomial typically expressed in the form ax² + bx + c. These functions produce parabolic curves when graphed, and their properties such as the vertex, axis of symmetry, and the direction of opening depend on the coefficients. Quadratic functions are fundamental in algebra and calculus due to their predictable behavior and wide applications.

Standard Form vs Vertex Form

Quadratic functions can be expressed in standard form (ax² + bx + c) or vertex form (a(x - h)² + k), where (h, k) represents the vertex of the parabola. Converting from standard form to vertex form makes it easier to identify the vertex and understand the transformations applied to the basic parabolic graph.

Completing the Square

Completing the square is a method used to convert a quadratic function from standard form to vertex form. This technique involves manipulating the equation so that a perfect square trinomial is formed, revealing the vertex parameters. It is a powerful tool in understanding and solving quadratic equations as well as in graphing parabolas.

Vertex of a Parabola

The vertex of a parabola is the point that represents the maximum or minimum value of a quadratic function. It is found by transforming the quadratic into vertex form or using formulas derived from the coefficients in the standard form. The vertex provides crucial insight into the function’s behavior, especially regarding optimization problems.

Graph of a Parabola

The graph of a quadratic function is a parabola, which is symmetric about a vertical line known as the axis of symmetry. The shape and orientation of the parabola are determined by the coefficient of the x² term: if positive, the parabola opens upward indicating a minimum point; if negative, it opens downward indicating a maximum point.

Maximum and Minimum Values of Quadratic Functions

The maximum or minimum value of a quadratic function is given by the y-coordinate of its vertex. If the parabola opens upward, the vertex is the minimum point; if it opens downward, it is the maximum point. Identifying these extreme values is essential in various optimization problems and analyses within algebra and calculus.

Transcript

00:01 We want to convert this general form quadratic to standard form.

00:06 So first thing we're going to do is rearrange it because it's not exactly general form.

00:12 We really want the x squared term up front, then the constant, excuse me, then the x to the first, and then the constant.

00:20 Now, in order to make it a perfect square, we're going to have to complete the square.

00:24 So we're going to factor out this minus one so that the x squared term has positive one as its coefficient.

00:34 And from there, we're going to add, we'll leave room, we're going to leave a placeholder for that thing.

00:40 We're going to add the special term.

00:42 There's our constant.

00:43 And since we're adding a term, we also have to subtract it back out.

00:48 Don't want to forget the negative one.

00:49 We factored out.

00:51 So here we go.

00:52 We are taking the six and dividing it by two and squaring that.

00:58 So six over two is three.

01:00 So we're adding three squared.

01:03 Do that here as well.

01:05 Direction.

01:06 So now we've made ourselves a perfect square, which means we can factor it.

01:13 And it's going to have x.

01:14 That's from here.

01:16 This is a positive.

01:18 So the sign between them is positive.

01:20 And the three is what's being squared.

01:22 So three is what goes here.

01:25 And then what do we have over here? we have three squared is nine, negative nine.

01:29 One minus negative nine is positive 10.

01:34 So that right there, is our standard form, or vertex form, if we want to call it that.

01:46 So the next order of business is to graph this.

01:50 So let's start with the vertex, since we have it easily available.

01:54 The vertex, its x coordinate is this three, but we multiply it by negative one.

02:00 And the y coordinate is the shift right here.

02:02 So negative three, 10 is our vertex.

02:05 Negative 3 .10...

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