Select The Correct Graph Of The Function F(x) = 4x²

Select The Correct Graph Of The Following Functionfx 53 4x2 8x

The provided instructions involve selecting the correct graph of a function, specifically of the form fx 53 4x2 8x. However, the exact explicit form of the function is unclear due to formatting issues. Presuming the intended function is f(x) = 53 + 4x^2 + 8x, the goal is to analyze this quadratic function, graph it correctly, and understand its properties such as vertex, intercepts, and shape.

Paper For Above instruction

The function under consideration appears to be a quadratic function of the form f(x) = 53 + 4x^2 + 8x. To identify its correct graph, it is essential to analyze its key features: vertex, y-intercept, and its parabola's orientation. The coefficient of x^2 is positive (4), indicating a parabola opening upward. The function can be expressed in vertex form to make this analysis more straightforward.

To convert f(x) = 53 + 4x^2 + 8x into vertex form, we perform completing the square:

f(x) = 4x^2 + 8x + 53
= 4(x^2 + 2x) + 53

Next, complete the square inside the parentheses:

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

Therefore:

f(x) = 4[(x + 1)^2 - 1] + 53 = 4(x + 1)^2 - 4 + 53 = 4(x + 1)^2 + 49

The vertex of the parabola is at (-1, 49), indicating the minimum point because the parabola opens upward. The y-intercept occurs at x=0:

f(0) = 4(0 + 1)^2 + 49 = 4(1)^2 + 49 = 4 + 49 = 53

The parabola is symmetric about the vertical line x = -1, and the positive leading coefficient indicates the parabola opens upward.

Choosing the correct graph involves looking for these features: a parabola with vertex at (-1, 49), passing through the y-axis at (0, 53), and opening upward. Any graph matching these characteristics is the correct choice.

Finding Factors and Fully Factoring the Polynomial

The second part involves factoring a cubic polynomial:

Given f(x) = x^3 + 2x^2 - 14x - 24, and that (x - 4) and (x + 2) are factors, we can determine the remaining factors through polynomial division or synthetic division.

Using synthetic division with x=4:

4 | 1  2  -14  -24
|     4   24    40
-------------------
1  6    10  16

Resulting quotient: x^2 + 6x + 10, with a remainder of 16, indicating an inconsistency. Alternatively, performing polynomial division confirms that (x - 4) is a factor. Similarly, applying synthetic division with x = -2:

-2 | 1  2  -14  -24
|   -2    0    28
------------------
1  0  -14     4

This suggests reevaluation because the remainder is not zero, indicating that (x + 2) is not a factor unless there was an error. However, assuming the initial factors are correct, the remaining quadratic factor after dividing out (x - 4) and (x + 2) can be found accordingly.

Performing polynomial division with (x - 4):

4 | 1  2  -14  -24
|     4   24   40
-------------------
1  6   10   16

The quotient is x^2 + 6x + 10 with a remainder of 16, indicating a miscalculation or that (x - 4) isn't a factor. Clarification suggests that perhaps the factors provided were (x - 4) and (x + 2), with the polynomial in question being f(x) = x^3 + 2x^2 - 14x - 24, which should be factorable accordingly. After double-checking, dividing f(x) by (x - 4):

Using synthetic division:

4 | 1  2  -14  -24
|   4   24   40
----------------
1  6   10    16

Since the remainder is 16, (x - 4) isn't a root, conflicting with the initial statement. Alternatively, substituting x=4 into f(x):

f(4) = 64 + 216 - 144 - 24 = 64 + 32 - 56 - 24 = 96 - 80 = 16

Since f(4)=16 ≠ 0, (x - 4) isn't a root; hence, it is not a factor. Similarly, evaluate x = -2:

f(-2) = -8 + 2*4 + 28 -24 = -8 + 8 + 28 -24= 4

Again, not zero, so neither of these are roots. Therefore, the initial information must be corrected or clarified. For the purpose of the paper, assume the factors are correctly provided but only (x + 2) is a factor; thus, dividing to find other factors would proceed accordingly.

Analyzing Polynomial Functions and Roots

The polynomial function y = x^5 - 5x^3 + 3x, for example, exhibits characteristics of polynomial behavior that can be understood through its derivatives, end behavior, and zeros. Its degree is five, leading to potential for up to five real roots by the Fundamental Theorem of Algebra.

Using Descartes' Rule of Signs, the number of positive roots corresponds to the number of sign changes in f(x). The sequence of coefficients (+, -, +, +) indicates one sign change, so one positive root or none. For negative roots, consider f(-x) and count sign changes accordingly.

Developing a Mathematical Model and Polynomial Factoring

The problem involving the statement "P varies directly as x and inversely as the square of y" can be modeled as:

P = k * x / y^2

Given P = 3/2 when x=25 and y=unknown, we can solve for k if the y value is known. If y can be determined or specified, then caclulate k accordingly. This is an example of formulating and solving direct-inverse variation models.

To express a polynomial like f(x) = x^3 + 7x^2 + 22 in the form f(x) = (x - k) q(x) + r with k = -1, synthetic division or polynomial division can be performed. The process involves dividing f(x) by (x + 1) to find q(x) and r, which yields insights into the roots and factors of the polynomial.

Applying Descartes' Rule to Polynomial Roots

For the equation 4x^7 + 9x^2 + 5x + 10 = 0, Descartes' rule of signs helps to determine an estimate of the possible positive and negative roots. Counting sign changes in the polynomial's coefficients informs us about the maximum number of positive roots. For negative roots, substitute x with -x and analyze sign changes again.

The table below summarizes the possible counts of positive, negative, and zero real roots based on sign changes:

In conclusion, analyzing functions graphically and algebraically, factoring polynomials, and applying rules like Descartes' Rule of Signs are fundamental skills for understanding the behavior of polynomial functions. These techniques allow us to interpret complex polynomial equations and select accurate graph representations based on their features.

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