Graph the family of polynomials in the same viewing rectangle, using the given values of c . Exp

step 1 So, in this question we have a 4 degree polynomial given to us. In the first case, if C is 0, th

Graph the family of polynomials in the same viewing...

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Key Concepts

Polynomial Functions

Polynomial functions are algebraic expressions consisting of variables and coefficients combined using only the operations of addition, subtraction, multiplication, and non-negative integer exponents. They are classified by their degree, which affects the overall shape of the graph, the number of turning points, and the end behavior. These functions are fundamental in algebra and calculus due to their smooth and continuous nature.

Parameter Variation in Functions

Introducing a parameter, such as the constant 'c' in a polynomial, creates a family of functions where each function’s behavior can be analyzed by varying the parameter. This concept is crucial for understanding how altering a coefficient affects the characteristics of the graph, including its intercepts, slopes, and overall shape.

Graphical Transformations of Polynomial Functions

Graphical transformations involve changes to a function's graph through shifts, stretches, or compressions as parameters are modified. In the context of polynomial functions, varying a parameter can result in the movement of the graph’s critical points, changes in symmetry, and alterations in the curvature, which helps in visualizing how the graph reacts to changes in the function's structure.

Transcript

00:02 So, in this question we have a 4 degree polynomial given to us.

00:05 In the first case, if c is 0, then my y is simply x -res to the power 4, which is a simple graph.

00:12 If this is y -axis and this is the x -axis, x -raise to the power 4 graph will be appearing something like this, symmetrically along the y -axis.

00:22 In the second case, when i say c is equal to 1, my graph will be y is equal to x -res to the power 4 minus x, which is equal to x taken common, x.

00:32 X cube minus 1.

00:33 So this x cube minus 1 will have only single root.

00:37 So x equal to 1 will be one of the root and it will gradually come.

00:42 The other root is 0.

00:44 So it will be coming down just normally.

00:49 So it will come like this, pass through the origin for sure because origin is one of the root.

00:56 Go down, cut the x -axis again at 1 and then it will raise up like this.

01:02 So that is the second part.

01:03 In the third part, if i observe, c is equal to 8, so y is equal to x4 minus 8x.

01:11 Here if i see, x taken common, x cubed minus 8, so here the root is x equal to 2.

01:16 I have to mark here clearly that this is 1 .0.

01:20 Next case, i will have to make the graph pass through 2 .0 and the origin.

01:25 So let us see how the graph is going to come...

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