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

step 1 First asked us to find the standard form equation of f of x. So the way we do that is we need to

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

Key Concepts

Quadratic Functions

Quadratic functions are polynomial functions of degree two, typically expressed in the form ax² + bx + c, where a, b, and c are constants and a is nonzero. They produce parabolic graphs that can open upward or downward depending on the sign of a, and are a fundamental part of algebra and calculus.

Standard Form

The standard form of a quadratic function is written as f(x) = ax² + bx + c. This form clearly shows the coefficients that determine the parabola’s width, orientation, and position, and serves as the starting point for many algebraic manipulations and analyses.

Vertex Form

The vertex form of a quadratic function is written as f(x) = a(x - h)² + k, where (h, k) represents the vertex of the parabola. This form is particularly useful for identifying the vertex and understanding the transformation of the graph from the parent function, making it easier to find the maximum or minimum value directly.

Completing the Square

Completing the square is a method used to convert a quadratic from standard form to vertex form. This technique involves creating a perfect square trinomial from the quadratic terms, which then allows the function to be rewritten in a way that clearly highlights the vertex and aids in graphing and optimization tasks.

Graphing Parabolas

Graphing parabolas involves plotting points, locating the vertex, determining the axis of symmetry, and understanding the direction in which the parabola opens. This process helps visualize the behavior of the quadratic function, including the location of its maximum or minimum value.

Maximum and Minimum Values

The maximum or minimum value of a quadratic function is found at its vertex. For a quadratic that opens upward (a > 0), the vertex represents the minimum value, and for one that opens downward (a < 0), it represents the maximum value. This concept is vital in optimization problems across various fields of mathematics and applications.

Transcript

00:01 First asked us to find the standard form equation of f of x.

00:06 So the way we do that is we need to complete the square.

00:08 The first step to complete the square is making sure that our x squared coefficient is 1.

00:12 And in this case, it's not.

00:13 So we're actually going to have to factorize out a number here to make the x squared coefficient 1.

00:19 And the number we need to factor out here is going to be 3.

00:22 And we're going to be left with x squared minus 2x inside of the racket and a plus 1 on the outside.

00:32 Right and then to complete the square here we need to figure out something that squares to get x squared minus 2x keeping the plus one on the outside so we're going to need an x to get the x squared and to get the minus 2x we're going to need a minus 1 it's always going to be half of this value right here so if we imagine swearing this out what we're going to get is we're going to get x squared minus 2x but we're also going to get an extra plus 1 so we need to take that away in order to make sure that the equation is still the same.

01:04 But since that's getting times by three, we're actually going to need to take away minus one times by three.

01:10 So that's going to be, and then if we just simplify that there to get our equation in standard form, so that's plus one minus three.

01:18 So that's going to be minus two.

01:21 All right.

01:21 And then for part b, it asks us to sketch the graph.

01:24 In order to sketch the graph, we're going to need to find our x and y intercepts.

01:32 So we'll start off with finding our x and susceptibles.

01:34 To do that, we're going to need to set our f of x or our y to zero.

01:39 And that equals 3x minus 1 or squared minus 2.

01:43 We'll take our minus 2 to the other side here.

01:47 So we get 3x minus 1 r squared equals 2.

01:54 We divide that by 2, so we get 2 thirds equals x minus 1 r squared.

01:59 We're then going to need to square root both sides.

02:01 So we get plus or minus square root of 2 over 3 equals x minus 1.

02:08 So then we get x equals plus or minus the square root of 2 over 3 plus 1.

02:17 All right.

02:19 So that's going to be our x intercepts.

02:21 And to find our y intercepts, we're just going to need to sub in x equals 0 into our equation.

02:29 So we need to sub in f of 0.

02:33 Anything with an x and it's just going to turn to zero because anything ties by zero is zero.

02:39 And that's going to leave our y intercept being one...

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