(a) Graph f(x)={ (x-1)^2 if 0 ≤ x<2 -2 x+10 if 2 ≤ x ≤ 6 . (b) Find the domai

(a) Graph f(x)={ (x-1)^2 if 0 ≤ x<2 -2 x+10 ...

Key Concepts

Piecewise-defined Function

A piecewise-defined function is a function that is defined by different expressions depending on the interval of the independent variable. Understanding these functions involves analyzing each segment over its respective domain and ensuring proper identification of the points where the definition changes.

Domain

The domain of a function consists of all the input values for which the function is defined. For piecewise functions, it involves taking the union of the intervals specified for each segment, ensuring that all notation limitations are considered.

Graphing Piecewise Functions

Graphing a piecewise function requires plotting each segment separately on its prescribed interval while noting open and closed endpoints. This approach highlights the overall behavior of the function and any discontinuities or transitions between the segments.

Absolute Maximum and Minimum

Absolute extrema refer to the highest and lowest values a function attains on its domain. Finding these involves evaluating the function at critical points, including endpoints and points where the definition changes, to determine the overall maximum and minimum values.

Quadratic Function

A quadratic function is generally represented by a polynomial of degree two and typically has a parabolic graph. Its vertex can represent a minimum or maximum value depending on the orientation of the parabola, and this is key when analyzing segments of piecewise functions involving quadratic expressions.

Linear Function

A linear function is characterized by a constant rate of change and can be expressed as a first-degree polynomial. Its graph is a straight line, and when used in piecewise definitions, it provides a simple, predictable behavior over the specified intervals.

Transcript

00:01 Hello and welcome to this video where we're going to graph a piecewise function and then we're going to determine its domain and its absolute maximum and minimum if they exist so first thing we want to do is take a gander at these numbers right here this tells me where i need to put up some boundaries on my graph and so one of them i'm seeing is at zero so i'm going to draw some dash lines at zero then at two and at six so that's sort of set in the stage for my graphing.

00:38 Next i got to see what's going to go where.

00:41 Well, now notice i don't have any instructions for numbers that are smaller than six or larger than six.

00:45 And so what that means is that i'm not doing anything in this part of the graph or this part of the graph.

00:51 I'm looking at this narrow window and i have two things that i'm going to be graphing.

00:56 So on the left hand side i'm going to be graphing x minus 1 squared and on the right hand side negative 2x plus 10 and so what i'm going to do is now draw those graphs in that section and i'm going to use a pencil to kind of help me with that so i'm going to do in blue this negative or x the quantity of x minus 1 squared now what i'm doing with x minus 1 squared is i'm going to be shifting my normal parabola to the right one unit and so instead of starting at the origin, it's going to start right here at 1 -0.

01:32 And then it's going to go up 1 over 1 and do that in both directions.

01:38 And that's as far as it's going.

01:39 Like i've already kind of gone through the barriers.

01:42 So, or i've hit the barriers of my graph.

01:45 And so i've got my points.

01:47 I'm going to get rid of this because it's not actually an open circle there.

01:49 It's technically a closed circle.

01:51 All right.

01:52 And then in red, i'm going to do my negative 2x plus 10.

01:57 So i'll start at 10, and i'm going to go down 1, 2 over 1.

02:03 So i have a point there, and now i'm going to graph.

02:05 But here's the thing.