Write the quadratic function in standard form and sketch its graph. Identify the vertex, axis of

Write the quadratic function in standard form and sketch its...

Key Concepts

Quadratic Function

A quadratic function is a second-degree polynomial that can be expressed in the standard form f(x) = ax^2 + bx + c. Its graph is a parabola that can open upwards or downwards depending on the leading coefficient a. This concept is fundamental in algebra and is used to model various phenomena in mathematics and real-life situations.

Completing the Square

Completing the square is a technique used to transform a quadratic expression from its standard form into its vertex form, f(x) = a(x - h)^2 + k, where (h, k) represents the vertex of the parabola. This method not only simplifies the process of graphing the function by identifying the vertex but also aids in solving quadratic equations and analyzing their properties.

Vertex and Axis of Symmetry

The vertex of a parabola is the point where it attains its maximum or minimum value. It can be found by using the formula h = -b/(2a) from the quadratic equation in standard form and then computing k = f(h). The axis of symmetry is a vertical line that passes through the vertex, given by x = h, and it divides the parabola into two mirror-image halves.

X-intercepts

X-intercepts are the points where the graph of the quadratic function crosses the x-axis, which occur when f(x) = 0. They can typically be found by factoring the quadratic, completing the square, or using the quadratic formula. The discriminant, calculated as b^2 - 4ac, determines the nature and number of the x-intercepts.

Transcript

00:01 Okay, here we have a problem that asks us to take a function of x and get it into a standard form function of the form of m of x that i have written above.

00:11 Now, the first thing we want to do is look at the x squared term.

00:15 We see a negative one there.

00:16 So whenever we see the x squared term multiplied by something, we need to factor that out if you want to get the equation into standard form.

00:25 So let's factor out a negative one equals x squared minus 2x minus 5.

00:32 So all the signs will change.

00:34 Now, we need to get this into the form of a complete square, this section here.

00:42 So x squared minus 2x, that would be x minus 1 squared, right? so if we had x minus 1 quantity squared, we'd have x squared minus 2x plus 1.

00:56 But here we have minus 5.

00:59 So we need to get that to be a plus 1.

01:02 So what i'm going to do is i'm going to add 4.

01:10 What i'm going to do first, actually, is move this negative 1 over.

01:14 I'm going to divide everything by negative 1.

01:17 And so this is negative of x over there.

01:20 And so now we just need to add 4, or excuse me, add 6, if we want positive 1, to both sides.

01:32 So we've got 6 minus f of x over here is equal to x.

01:37 Minus 1.

01:39 Now, all we need to do is move that 6 over and then multiply everything by negative 1 again.

01:46 So i'm going to just erase the 6 from here and then put it on the other side and then put this negative sign.

01:53 All i have to do is remove it from here and put it on this side because that's the same thing as multiplying by negative 1 or dividing by negative 1.

02:00 Either one will work.

02:01 So that's our equation in standard form.

02:03 So we've got that done.

02:04 The good thing about standard form is that it gives us our vertex very easily, because the form v is hk, which means that our axis of symmetry is that x equals 1, and our intercepts are the points where the function takes the value of 0.

02:23 So i'm going to plug in 0 for f of x and just solve...

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