51-54 : Local Extrema The graph of a polynomial function is given. From the graph, find (a) the

51-54 : Local Extrema The graph of a polynomial function is...

Key Concepts

Graph Analysis

Graph analysis involves interpreting the visual representation of a function to identify its key characteristics such as intercepts, turning points, and overall behavior. This includes determining where the function increases or decreases, locating potential local extrema, and understanding how these features relate to the algebraic form of the function.

Local Extrema

Local extrema refer to the points on a graph where the function attains a local minimum or maximum value compared to neighboring points. These points are critical in understanding the behavior of a function and are often found by analyzing the derivative of the function to determine where it is zero or undefined, indicating potential changes in the function’s increasing or decreasing behavior.

Polynomial Functions

A polynomial function is an expression consisting of variables and coefficients combined using only addition, subtraction, multiplication, and non-negative integer exponents. Understanding the general shape and behavior of these functions, such as smoothness and end behavior, is critical for analyzing their graphs and finding key features like intercepts and extrema.

Intercepts

Intercepts are the points where the graph of a function crosses the coordinate axes. The x-intercepts are found by setting the function equal to zero and solving for x, while the y-intercept is found by evaluating the function at x = 0. This concept is fundamental in graphing functions as it provides key information about the location of the graph in the coordinate plane.

Transcript

00:01 Okay, so this question first asked us to find the coordinates of the x and the y intercept.

00:05 So the way we're going to do that is we're going to set x equal to zero and y because zero respectively to find the y and the x intersets.

00:13 Okay, so we'll start off by setting y to zero to find the x intercept.

00:17 So we'll have minus x squared plus 4x equals zero.

00:22 We can factorise out on x here.

00:24 Not canceling out completely because that's actually going to be one of our solutions.

00:28 Equals 0.

00:30 And since we've got two things, times them together to get 0 here, one of them's going to have to be 0.

00:34 So we'll have x equals 0 or minus x plus 4 equals 0, which simplifies to x equals 4.

00:44 So we have our 2 x intercept here.

00:46 And now we're going to find our y intercept by subbing in x equals 0.

00:50 So subbing x equals 0 here, anything with an x and it's just going to turn to 0.

00:54 So that's just going to be 0.

00:55 Again, the origin here is one of our x intercept.

00:57 So therefore it's going to be one of our y intercept as well.

01:00 So we have our two x intercepts.

01:03 We'll just make it extra clear here.

01:05 So our x intercept are going to be 0 .0 and 4 .0.

01:15 And then our y intercept similarly will be the origin again, so 0 .0.

01:23 Okay.

01:24 And then the second part of our question here, ask us to find all of the extremer.

01:29 So the way i'm going to do this is i'm going to complete the square...

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