Write a summery of your strategy for graphing the function when you are given only the graph of . identify the four key regions.

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11 May 2023

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Related questions

Reflect on the concept of function. What concepts (only the names) did you need to accommodate the concept of function in your mind? What is the simplest function you can imagine? In your day to day, is there any occurring fact that can be interpreted as a function? Is it possible to view a function? What strategy are you using to get the graph of a function?

The Learning Journal entry should be a minimum of 400 words and not more than 750 words.

scarletmink223

(a) Graph several members of the family of the functionsforand look at the regions enclosed by these curves and the x-axis. Make a conjecture about how the areas of these regions are related.

(b) Prove your conjecture in part (a).

(c) Take another look at the graphs in part (a) and use them to sketch the curve traced out by the vertices (highest points) of the family of functions. Can you guess what kind of curve is this?

(d) Find an equation of the curve you sketched in part (c).

fuchsiadolphin670

For what values of c does the polynomial have two inflection points? One inflection point? None? Illustrate by graphing P for several values of c. How does the graph change as c decreases?

Tynup Solution:

Given:

Differentiating

For c ≠ 0, there are two distinct roots for the equation.

Hence, for two inflection points the required condition is.

When c = 0, there is only one root for the equation.

Hence, when c = 0, the function has one inflection point.

Recall that the quadratic equation does not have real roots if its Discrimnant is less than zero.

Hence, for no inflection points

, which is not possible.

Hence, the polynomial function has minimum of one inflection point.

For different values of c