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

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

Key Concepts

Quadratic Function

A quadratic function is a second-degree polynomial function of the form ax² + bx + c, where a, b, and c are constants and a ? 0. This type of function produces a parabola when graphed, and its properties—including the direction of opening (upward or downward), vertex position, and axis of symmetry—are essential in understanding its behavior.

Standard Form

The standard form of a quadratic function is expressed as f(x) = ax² + bx + c. This format is useful because it directly shows the coefficients that determine the shape and position of the parabola and provides the starting point for further manipulations, such as converting to vertex form.

Vertex Form

The vertex form of a quadratic function is written as f(x) = a(x - h)² + k, where (h, k) is the vertex of the parabola. Converting a quadratic from standard form to vertex form makes it easier to identify the vertex, which represents the maximum or minimum point of the function, as well as the line of symmetry.

Completing the Square

Completing the square is an algebraic technique used to transform a quadratic function from standard form into vertex form. This method involves rewriting the quadratic expression as a perfect square trinomial plus or minus a constant, thereby revealing the vertex and facilitating the determination of the function's maximum or minimum value.

Graphing Parabolic Functions

Graphing a quadratic function involves plotting its vertex, axis of symmetry, and intercepts to visualize the parabola's shape. Understanding the vertex form helps in accurately sketching the curve, as it clearly indicates the direction in which the parabola opens and the location of its maximum or minimum value.

Transcript

00:01 All right.

00:02 So we have a quadratic, and our first task is to convert it to standard form, also known as vertex form.

00:10 So we're going to complete the square.

00:11 And the way we're going to do that is we're going to add a term that will turn this part into a perfect square binomial.

00:24 And then we're also going to subtract it back out so we don't actually change the value.

00:29 And the way we determine it is we take whatever the coefficients of the x is, eight in this case, divide it by two, and then square the whole thing.

00:40 So we're going to add four squared and we're going to subtract four squared.

00:45 So now this has become a perfect square binomial and it is x because this is negative, this is negative and whatever's being squared is what goes here so four.

00:59 Then we add eight and we subtract 16.

01:02 So we're really 8 minus 16 is negative 8.

01:06 So there we have it.

01:07 That's our standard form.

01:11 And from here, we could pretty easily sketch a graph.

01:15 We've got our vertex.

01:18 X is this.

01:19 The x coordinate of the vertex is negative 1 times that.

01:23 So negative negative 4 is positive 4.

01:26 And the y coordinate is exactly this.

01:29 So negative 8.

01:31 So 4, comma, negative 8 is our vertex.

01:36 So here's 4.

01:37 Negative 8.

01:39 Put that right there.

01:41 And since this is positive, we know it's going to be concave upward, so it's looking good.

01:46 Now we need to find, hmm, how about we need a few more points.

01:51 The y intercept is best found using the general form because the first two terms, if x is zero, they go away.

02:00 So eight.

02:02 And this is one, two, three, four.

02:04 To the left, so equally as far to the right, one, two, three, four, which is eight, will also have the same value because parabolas are symmetric and they're symmetric about the vertex, specifically about a line drawn straight through the vertex.

02:18 So x equals four.

02:21 Okay, then if we wanted, we could find when x is three, what is the value? so we're going to figure out just a couple more values.

02:31 When x is three, we have three minus four is negative one squared, is 1, 1 minus 8 is negative 7.

02:39 So when x is 3, we have negative 7, and that's 1 to the left.

02:44 So one to the right.

02:45 We also have that value...

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