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

step 1 So part A here asks us to express F of X in standard form. And the way we do that is we need to

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

Key Concepts

Quadratic Functions

A quadratic function is a second-degree polynomial function generally expressed in the form ax² + bx + c, where a, b, and c are constants and a is non-zero. These functions are fundamental in algebra and exhibit a parabolic graph, which can either open upward or downward based on the sign of the coefficient a.

Standard Form

The standard form of a quadratic function is written as f(x) = a(x - h)² + k, where (h, k) is the vertex of the parabola. This form makes it easier to identify key features such as the vertex and to determine whether the function has a maximum or minimum value.

Completing the Square

Completing the square is a technique used to convert a quadratic function from its general form into its standard form. This method involves rearranging and grouping terms to form a perfect square trinomial, thereby exposing the vertex of the parabola and simplifying the process of graphing.

Vertex and Extremum

The vertex of a quadratic function is the point on the graph where the function reaches its maximum or minimum value. For a quadratic function, if a > 0, the vertex is the minimum point, and if a < 0, the vertex is the maximum point. The vertex is crucial for understanding the function's behavior and is easily identified when the function is in standard form.

Graphing Parabolic Functions

Graphing a quadratic function involves plotting the vertex, determining the axis of symmetry, and identifying key intercepts. The shape of the parabola, determined by the leading coefficient a, along with its vertex form, assist in accurately sketching the curve and understanding its overall behavior.

Transcript

00:01 So part a here asks us to express f of x in standard form.

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

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

00:12 In this case, it's not.

00:14 It's actually minus 1.

00:15 So we're going to need to factorise out minus 1 to make sure that our x squared coefficient is 1.

00:22 So that's going to do this.

00:24 We can keep the plus 3 on the outside for now.

00:28 Okay, so now the next step to completing the square is we need to figure out what do we put in this bracket to get x squared plus 3x? and the answer to that is that we're going to need an x to get the x squared.

00:42 And to get the plus 3x, we're going to do half that.

00:45 So we're going to need plus 3 over 2.

00:48 If we imagine expanding this out here, you're going to get an x squared.

00:51 You're going to get a plus 3 over 2x plus another 3 over 2x.

00:54 So that gives us that.

00:55 We're also going to get an extra plus 9 over 2.

00:58 Four if you want to imagine squaring three over two but that's actually going to be negative because of this negative here because we factorized out the minus one and so to cancel that out we're actually going to plus nine over four plus the original three just using our calculator here we can simplify that if you don't have a calculator you can use various methods of figuring out of factions or if you're very clever you can do it in your head i'm not that clever so that's plus 21 over four all right.

01:35 Okay.

01:36 And then part c, app b, sorry, asked us to sketch the graph.

01:41 So to sketch the graph, at a minimum, we're going to at least need the x end to the y intercept.

01:48 So to get the x intercepts, we're going to need to still be in y equals zero to solve the equation.

01:52 We've already completed the square here so we can do it by doing that.

01:55 So we've got minus x plus 3 over 2 all squared, plus 21 over 4.

02:02 So we're going to take the x plus 3 over 2 or square to the other side, just so we get both our things as positive.

02:11 Okay.

02:13 So we're going to be square root both sides here.

02:14 So we're going to plus on minus the square root of 21 over 4, x plus 3 over 2.

02:20 So our x is going to equal plus or minus the square root of 21 over 4 minus 3 over 2.

02:29 So our x intercept here is going to be plus or minus square of 21 over 4.

02:33 Minus 3 over 2.

02:36 To find our y intercept, we're actually going to need to sub in x equals 0 into our equation.

02:43 So when we sub in x equals 0 into our equation, anything with an x, it turns to 0, because anything times by 0 is 0...

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