A) Find the equation of the function shown in the graph: B) State the domain, Range, and End behavior for the function:

9514 1404 393

Step-by-step explanation:

The graph appears to show zeros at x=-3, x=1 and x=2. The zero at x=1 has multiplicity 2, so the factored form of the function's equation is ...

  y = (x +3)(x -1)^2(x -2)

When expanded, the equation becomes ...

  y = x^4 -x^3 -7x^2 +13x -6

This polynomial has a domain of all real numbers.

The degree is even, and the leading coefficient is positive, so this function will have a minimum value. The graph shows it as about -36. Evaluating the function in that neighborhood, we find the minimum to be an irrational number with a value of approximately -36.100456736.

The range is from about -36.1 to infinity.

Since the degree is even the end behavior will be the same for both positive and negative values of x. Since the leading coefficient is positive, the function value will tend toward positive infinity for either positive or negative large values of x.

  |x| → ∞, y → ∞

_____

Additional comment

The derivative is a cubic whose root is (-1-√209)/8 at the minimum of the function. This means the function's minimum value is (-9419 -627√209)/512 ≈ -36.100456736016019747. The range can only be stated in symbolic form (using a radical, infinity symbol), since the actual numerical values of the limits can never be stated exactly.

y = x^4 -x^3 -7x^2 +13x -6

domain: all real numbers; range: -36.1 ≤

y < infinity

• y +∞o for Ixl → co

Step-by-step explanation:

The graph appears to show zeros at x=-3, x=1 and x=2. The zero at x=1 has multiplicity 2, so the factored form of the function's equation is...

y = (x+3)(x-1)^2(x-2)

When expanded, the equation becomes ...

y=x^4 -x^3-7x^2 +13x -6

This polynomial has a domain of all real numbers.

The degree is even, and the leading coefficient is positive, so this function will have a minimum value. The graph shows it as about -36. Evaluating the function in that neighborhood, we find the minimum to be an irrational number with a value of approximately -36.100456736.

The range is from about -36.1 to infinity.

Since the degree is even the end behavior

will be the same for both positive and negative values of x. Since the leading coefficient is positive, the function value will tend toward positive infinity for either positive or negative large values of x.

Ixl → ∞, y → ∞

Additional comment

The derivative is a cubic whose root is (-1-√209)/8 at the minimum of the function. This means the function's minimum value is (-9419 -627√209)/512= -36.100456736016019747. The range can only be stated in symbolic form (using a radical, infinity symbol), since the actual numerical values of the limits can never be stated exactly.

region b will make the inequalities y < 2x + 1 and y > -x - 2 true ⇒ answer a

step-by-step explanation:

* lets study the graph and the inequality to solve the question

- the inequality y > -x - 2

∵ the form of the inequality is y > mx + c , m is the slope of the line

  and c is the y-intercept (the line intersect y-axis at point (0 , c)

∴ c = -2 that means the line intersect the y-axis at point (0 , -2)

∵ the sign of the inequality is >

∴ we shad the part over the line

  region a and b

- the inequality y < 2x + 1

∵ the form of the inequality is y > mx + c , m is the slope of the line

  and c is the y-intercept (the line intersect y-axis at point (0 , c)

∴ c = 1 that means the line intersect the y-axis at point (0 , 1)

∵ the sign of the inequality is <

∴ we shad the part under the line

  region b and c

∵ the common region between the two lines is region b

∴ region b will make the inequalities y < 2x + 1 and y > -x - 2 true