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

Completing the Square

Completing the square is a method used to transform a quadratic equation from its general form (ax² + bx + c) into the vertex form. This technique involves adding and subtracting an appropriate constant term to form a perfect square trinomial, which then reveals the vertex of the parabola. It is an essential algebraic tool for solving quadratic equations, optimizing functions, and analyzing their graphs.

Maximum and Minimum Values

For a quadratic function, the maximum or minimum value is found at the vertex of the parabola. If the parabola opens upward (a > 0), the vertex represents the minimum value; if it opens downward (a < 0), the vertex represents the maximum value. Understanding how to locate the vertex and interpret its coordinates is crucial in optimization problems and in determining the behavior of quadratic functions on a graph.

Quadratic Function

A quadratic function is a second-degree polynomial function that can be expressed in the form ax² + bx + c, where a, b, and c are constants and a is non-zero. Its graph is a parabola, which can either open upward (if a > 0) or downward (if a < 0). This type of function is fundamental in algebra and calculus for modeling various phenomena and understanding key concepts such as symmetry and extremum values.

Standard (Vertex) Form

The standard or 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 is particularly useful for quickly identifying the vertex, which is the highest or lowest point of the graph, and for understanding the transformation from the basic quadratic function x². Converting a quadratic into this form is a critical skill for analysis and graphing.

Transcript

00:01 Okay, so the part of this question asks us to put our equation h of x into standard form.

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

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

00:12 It's not in this case, it's actually negative 1.

00:14 So we need to take out a factor of negative 1.

00:17 And that i'll leave x and plus 1, x squared, sorry, plus x.

00:24 And that's going to leave a plus 1 on the outside.

00:28 Okay.

00:30 Our next step is we need to figure out.

00:31 Out something that squares to get x squared plus x again leaving the plus one on the outside so we're going to need an x to get the x squared we're also going to need a half to get the x it's always going to be half or whatever this value is here this value here is one so it's just going to be a half and if you expand that out you'd get x squared plus the x that we need so we've got that we also have an extra quarter as well so to cancel that out we need to minus a half and we also need to times that by whatever here and that's a minus 1.

01:03 So it's going to be minus a quarter, sorry, not a minus a half.

01:07 So minus a quarter, it's times by minus 1.

01:10 And then we just simplify that to make it very clear.

01:16 So that's 1 minus minus course, plus a quarter, so that's plus 5 over.

01:24 All right.

01:25 So that's the equation in standard form.

01:27 And then part b, that's just a sketch of graph of h of x.

01:31 And to do that, we're going to need to first find the the y and the x intercepts.

01:38 To start off we'll find the x intercepts so we'll do that by setting y equal to zero.

01:45 So we've got minus x plus a half all squared here plus five over four equals zero.

01:55 We'll take the five over to four and i will take the this all of this over to the other side so it's five over four and equals x plus a half all squared.

02:04 I'm going to need to square both sides.

02:06 So we're going to get the square root of 5 over 2 here, and that's going to be plus or minus.

02:13 And we're left with just the x plus a half on this side.

02:17 So then x is equals plus or minus the square root of 5 over 2 minus a half.

02:27 All right.

02:29 And then we'll find our y intercept.

02:36 And to do that, we'll subbing x equals zero.

02:41 So x equals zero here.

02:43 We'll get us anything with an x, and it's just going to turn to zero...