✅👉 Graficar Funciones Polinomiales de Grado 3

Graphing a Cubic Polynomial Function

Introduction to Polynomial Functions

• The video begins with an introduction to graphing a cubic polynomial function, emphasizing the importance of factoring the polynomial correctly.

• The speaker notes that the number of terms in the polynomial influences how it can be factored, highlighting a polynomial with four terms.

Factoring Process

• The first step involves grouping terms for factorization; specifically, the first two and last two terms are grouped separately.

• A common factor is extracted from each group: x^2 from the first group and 4 from the second group, adjusting signs accordingly.

• After extracting common factors, both groups yield identical parentheses, allowing further simplification by combining them into one expression.

Completing Factorization

• Since it's a cubic function requiring three zeros, additional factorization is necessary. The speaker applies the difference of squares method to continue factoring.

• This results in a fully factored form of the polynomial which includes identifying all roots or zeros.

Finding Zeros of the Function

• Each factor is set to zero to find its corresponding root: x + 3 = 0, leading to x = -3.

• Continuing this process reveals additional zeros at x = 2 and x = -2, completing the identification of all roots.

Plotting Zeros on Cartesian Plane

• The identified zeros are plotted on the x-axis: -3, -2, and +2.

• A discussion arises about whether these points indicate where the graph crosses or touches the x-axis, prompting further analysis.

Testing Intervals for Graph Behavior

• To determine how the graph behaves around these zeros (whether it goes above or below), test values within intervals created by these roots are chosen.

• Values such as -4, -2.5, 1, and 3 are selected as test points for evaluating function behavior between roots.

Evaluating Test Points

• Each test point will be substituted back into the original polynomial function to assess their impact on determining graph shape.

• For example, substituting -4: calculations show how this value affects overall output when plugged into factored expressions.

Graphing Polynomial Functions

Evaluating Function Values

• The function is evaluated at x = -4 , yielding a value of -12 . This serves as a test point for further calculations.

• For x = -2.5 , the calculation results in 0.5 . Further evaluations show that when substituting into the polynomial, it leads to a product of approximately 1.13 .

• At x = 1 , the evaluations yield values of 4, -1, and 3 . The multiplication of these results gives a final output of -12 .

• When evaluating at x = 3 , the calculations lead to positive outputs, specifically resulting in a total of +30 .

Graphing Points on Cartesian Plane

• The graph will represent points based on calculated values:

• At x = -4, the corresponding y-value is approximately -12.

• At x = -2.5, it is slightly above zero at around +1.13.

• At both points where x = 1 and where it exceeds typical graph limits (like at 3 with +30).

Identifying Key Features

• The independent term from the polynomial function indicates that it intersects the y-axis at exactly (-12). This provides another critical point for plotting.

Drawing the Graph

• A line is drawn through identified points, illustrating how it passes through zeros and intercepts while maintaining an upward trend towards higher values despite limitations in scale on the Cartesian plane.