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

Graph of a Parabola

The graph of a quadratic function is a parabola that either opens upward or downward. Understanding its shape, axis of symmetry, and vertex is essential for sketching the function accurately, as well as for solving problems related to maximum and minimum values.

Quadratic Function

A quadratic function is a polynomial function of degree two, typically expressed in the form ax² + bx + c. Its graph is a parabola, which can open upward or downward depending on the sign of the coefficient a, and it is fundamental in algebra and calculus for modeling various phenomena.

Completing the Square

Completing the square is an algebraic method used to rewrite a quadratic function in vertex form. This process involves creating a perfect square trinomial from the quadratic expression, allowing for easier identification of the vertex and analysis of the function's properties.

Vertex Form of a Quadratic

The vertex form of a quadratic function is written as a(x - h)² + k, where (h, k) represents the vertex of the parabola. This form makes it straightforward to determine the maximum or minimum value of the function, as it directly reveals the vertex which is the point of the extremum.

Maximum and Minimum Values

The maximum or minimum value of a quadratic function is found at its vertex. When the parabola opens upward (a > 0), the vertex represents the minimum value; when it opens downward (a < 0), the vertex represents the maximum value. This concept is crucial in optimization problems and in understanding the behavior of quadratic functions.

Transcript

00:03 We're going to convert this quadratic into standard form, also known as vertex form.

00:07 And to do that, we need to factor out the three so that the x squared term has a coefficient of positive one.

00:17 So like that.

00:20 And then we're going to add a special term.

00:24 And in order to not actually change the value of the function, we're going to have to subtract that special term.

00:29 And we don't want to forget the three that we factored out because it certainly will apply to the value.

00:34 The number in here.

00:36 So how do we get that number? we take whatever this is, so it's two, and we divide it by two, and then square it.

00:45 So what we end up doing is we're adding one squared and over here, subtracting it.

00:50 Now, this part here is a perfect square binomial.

00:54 So we've got f of x is equal to three times.

01:03 All right, we've got an x.

01:05 This is a minus sign, which means this is a minus sign.

01:09 And whatever is being squared here is what goes here.

01:12 So we have x minus one, the quantity squared.

01:15 And over here, what do we have? one squared is one.

01:18 Negative three times one is negative three plus one is negative two.

01:23 So there we have it.

01:24 F of x is equal to three times x minus one, the quantity squared minus two.

01:30 Our next task is to sketch this.

01:32 So we're going to want to put the vertex on there.

01:34 And the vertex is x coordinate is this right here, negative one, times negative one.

01:41 So it's positive one.

01:43 And this here, as is, is the y coordinate.

01:46 So one comma negative two.

01:49 So there's our vertex.

01:51 And we're going to need to plot a few more points so we can get the curvature.

01:57 Y intercept is always easy enough to find.

01:59 So let's do that.

01:59 We'll go back to the general form.

02:01 When x is zero, this is zero.

02:04 This is zero and all we have left is one.

02:06 So my y intercept is one...

Please give Ace some feedback