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 Functions

Quadratic functions are second-degree polynomial functions typically expressed in the form f(x) = ax² + bx + c, where a, b, and c are constants and a ? 0. They graph as parabolas, which open upward when a is positive (indicating a minimum vertex) and downward when a is negative (indicating a maximum vertex).

Standard Form of a Quadratic Function

The standard, or vertex, form of a quadratic function is f(x) = a(x – h)² + k, where (h, k) represents the vertex of the parabola. This form is particularly useful because it directly reveals the vertex and makes it easier to determine key properties of the function such as its maximum or minimum value.

Completing the Square

Completing the square is a method used to transform a quadratic function from its general form into its standard form. The process involves rearranging the terms and adding and subtracting a constant to form a perfect square trinomial, which then facilitates the identification of the vertex of the parabola.

Vertex and Extremum

The vertex of a quadratic function is the point at which the function attains its maximum or minimum value. For a quadratic function, if the coefficient a is positive, the vertex represents the minimum value; if a is negative, it represents the maximum value. This point is critical in understanding the function's overall behavior.

Graphing Parabolas

Graphing parabolas involves plotting the vertex, drawing the axis of symmetry, and using additional points to sketch the curve accurately. Recognizing whether the parabola opens upward or downward, along with identifying intercepts and the vertex, are all important steps in creating an accurate graph of the quadratic function.

Transcript

00:01 Okay, so this question first asks us to put our equation into standards form.

00:05 And the way we do that is we just need to complete the square.

00:08 Now to start off, completing the square, we just need to make sure that our x squared coefficient is 1.

00:13 And in this case, it already is so we don't have to do anything with that.

00:16 Then our next step is we need to find out something that squares to get x squared plus 2x.

00:21 We can leave the minus 1 on the outside for now.

00:23 So we're going to need our x to get our x squared.

00:25 And to get our plus 2x, we're going to need a plus 1.

00:28 It's always going to be half of this value here.

00:30 Checking that, you can imagine squaring this out, you're going to get an x squared plus 2x plus 1.

00:37 So then, but then you're also going to need to get rid of that plus 1 as well, just so that the equation stays the same.

00:46 So we're going to minus an extra 1 here as well.

00:49 So you get x plus 1, all squared minus 2.

00:54 Okay, so that's that in completed the square form.

00:58 And then it asks us to sketch a graph of f of x.

01:05 Okay, so to sketch a graph, we're going to need to find a couple of things, first of all.

01:12 We're going to need to find our x and our y intercepts.

01:18 So to find our x and y intercepts, we're going to need to set x equal to zero and y equal to zero respectively.

01:25 First off, we're going to find out what x intercepts, so we'll set x, we'll set y equal to zero.

01:32 So we've already completed the square here, so we can just set y equals zero by doing this.

01:37 We'll get x plus one, all squared minus two equals zero.

01:41 We'll take the two over to the other side, x plus one or squared equals two.

01:46 We'll square root both sides, so you get square root of two here, but it's plus or minus, always plus or minus if you're square rooting, x plus one.

01:54 And then the x values, again, and i use, again, and then the x, and i use, going to be plus or minus square of two minus one.

02:02 All right.

02:04 Okay.

02:05 So then our x intercepts are actually going to be plus or minus the square root of two minus one, comma zero.

02:16 So that's two intercepts for our x here.

02:18 And then to find our y intercepts, we're going to need to set our x equal to zero or solving x equals zero...

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