Graph the quadratic function, which is given in standard form. f(x)=(x-(1)/(3))^2+(1)/(9)

Name: Problem 19, Chapter 2: Precalculus
Uploaded: 2020-06-26T23:22:28-08:00
Duration: 4 min 20 s
Channel: Michael Jacobsen
Description: Graph the quadratic function, which is given in standard form. f(x)=(x-(1)/(3))^2+(1)/(9)

Graph the quadratic function, which is given in standard...

Key Concepts

Quadratic Function

A quadratic function is a polynomial function of degree two, typically expressed in the form ax²+bx+c. Its graph is a parabola, which can open upward or downward depending on the coefficient a. This concept is fundamental in algebra and calculus, as quadratic functions model a variety of real-world phenomena and are key in understanding polynomial behavior.

Vertex Form

The vertex form of a quadratic function is expressed as f(x) = a(x - h)² + k, where (h, k) is the vertex of the parabola. This form makes it straightforward to identify the vertex and to understand the transformations applied to the basic quadratic function y = x². Recognizing and using the vertex form is crucial for efficiently graphing quadratic functions.

Graph Transformations

Graph transformations refer to the methods used to shift, stretch, compress, or reflect the graph of a basic function, such as y = x². In the context of quadratic functions in vertex form, transformations include horizontal shifts (determined by h in the term (x - h)), vertical shifts (determined by k), and vertical stretches or compressions (indicated by the coefficient a). Understanding these transformations is essential for accurately graphing quadratic functions and interpreting their behavior.

Transcript

00:01 For this exercise, we have a quadratic that has been provided, and our job is to give a sketch for what its graph would look like.

00:08 We start by determining the vertex, which is a fast process, since this quadratic is already written in standard form.

00:16 So to get the x coordinate, we go here and take the opposite.

00:20 So we have positive a third, and to obtain the y coordinate, we directly copy this constant one -ninth.

00:27 So now we have that the vertex is at a third and a ninth.

00:31 So let's go a distance of a third in the x -axis direction.

00:35 I'll let these tick marks represent 1 3rd.

00:41 So we go 1 3rd in this direction, and then we have to go a distance of positive 1 3rd in the vertical direction.

00:50 So if this is 1 3rd, a 9th is a 3rd of a 3rd, which means this would be roughly 1 9th.

00:58 So this is where our vertex ends up at, and for the further analysis of this particular question, quadratic, notice that the leading coefficient is a positive one.

01:11 That means that our graph is going to be opening upwards.

01:14 But the next step to take is we're going to obtain a table of values, we'll call it x and y, and use this table to determine two more points.

01:26 Already we have the point one -third for x, one -ninth for y, so now our strategy is going to be pick a value here and here that's symmetrically the same distance away from one -third so that the values here and here will be identically equal.

01:44 One strategy here is if we go a distance of one away from the x value of one -third, we'll be adding one -third to obtain a value of four -thirds.

01:56 And if we go a distance of negative one -third away from one -thirds, we would be taking one -third minus one, which would become negative 2 thirds.

02:08 Let's evaluate say 4 thirds...

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