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

A quadratic function is a polynomial function of degree two, typically expressed in the form ax² + bx + c, where a, b, and c are constants and a ? 0. This type of function produces a parabolic graph that can open upwards or downwards depending on the sign of the coefficient a. Understanding quadratic functions is fundamental in algebra, as they appear in many real-world problems and mathematical models.

Standard (Vertex) Form

Expressing a quadratic function in standard or vertex form involves rewriting it as a(x - h)² + k, where (h, k) represents the vertex of the parabola. This form directly reveals the location of the maximum or minimum value of the function, as well as the axis of symmetry. Converting to vertex form is useful for both analytical and graphical analyses of the function.

Completing the Square

Completing the square is an algebraic technique used to transform a quadratic expression into the vertex form by adding and subtracting the appropriate constant term. This process not only simplifies finding the vertex and extrema of the quadratic function, but it also aids in understanding the structure of the quadratic equation by isolating the perfect square term.

Graphing Quadratic Functions

Graphing a quadratic function involves plotting its vertex, determining the axis of symmetry, and identifying other key points along the parabola. The shape and orientation of the parabola are dictated by the coefficient of the quadratic term, while the vertex serves as the point where the function reaches its maximum or minimum value. This visual representation reinforces understanding of the function's behavior over its domain.

Extrema of Quadratic Functions

The maximum or minimum value of a quadratic function, known as the function's extremum, occurs at the vertex of the parabola. For a quadratic function in vertex form a(x - h)² + k, if a is positive, the function attains a minimum value at k; if a is negative, it attains a maximum value at k. Identifying this extremum is crucial in many optimization problems and practical applications.

Transcript

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

00:03 And the way we do that is we need to complete the square.

00:06 First step to complete the square is making sure that our x squared coefficient is one.

00:10 And in this case, it's not one.

00:11 It's actually three.

00:12 So in order to amend that, we just need to take out a factor of three.

00:17 So that leaves us with x squared minus 4x inside the bracket.

00:21 And we can keep the plus 13 on the outside for now.

00:25 All right.

00:26 So then the next step is that we need to figure out something that squares in order to get x squared minus 4x.

00:32 That's going to give us an x for the x squared, and it's always going to be half of whatever this value is, so it's going to be x minus two.

00:38 So if you imagine squaring that out, now just to check, you get x squared minus 2x minus 2x for the minus 4x, and then you also get an extra plus 4.

00:45 So the way we cancel the plus 4 out, it's going to get times by 3.

00:49 So the way we cancelled that out is we minus 4 times by 3, plus 13 of course.

00:58 And that leaves 3, x minus 2 all squared, minus 12 plus 13, so that's going to be minus plus 1.

01:07 Okay.

01:08 So that is it in standard form.

01:12 And then part b asked for us to sketch a graph of f of x.

01:18 In order to sketch a graph, we're first going to need to find the x and the y intercepts.

01:23 I'll start off by finding the x intercept, so we need to set f of x equal to zero...

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