2nd-Quarter-Week-6_Session-3.ddndndndndndndn

This lesson introduces the concept of a circle—its definition, parts, properties, and real-life applications. Students will explore the relationships between the radius, diameter, chord, tangent, secant, and arc. The lesson also covers how to find the circumference and area of a circle and solve problems involving these measures. Interactive discussions and visual examples help students understand the importance of circles in both geometry and everyday life.

Presentation Summary:
The presentation begins with a short review of basic geometric figures, followed by an introduction to the circle as a set of points equidistant from a fixed point called the center. Each slide explains and labels the essential parts of a circle with diagrams. Example problems demonstrate how to compute circumference and area using formulas. The session ends with a short quiz and real-world applications, such as circular objects and motion, to deepen student understanding and engagement.

1. Polynomial Function and Equation Patterns and Algebra

2. SESSION 2 Week 6 At the end of the lesson, the students can: a. graph polynomial functions.

3. Graphing Polynomials

4. GRAPHING POLYNOMIALS In this section, you will apply everything we learned about polynomials to sketch the graph of some polynomial functions. The following are the things you need to consider when graphing polynomial functions: Zeros or Multiplicity of zeros End behavior of the graph.  

5. GRAPHING POLYNOMIALS Zeros or The zeros or x intercepts of P(x) are the intersection points of the graph of P(x) on then x-axis while the y-intercept is the intersection point of the graph of P(x) on the y-axis. Multiplicity is the number of times a factor or roots occurred. If the multiplicity is odd (1, 3, 5, 7, …), then the graph will cross the x-axis, while if it is even (2,4,6,8,…), the graph will be tangent to the x-intercepts.  

6. Example: Sketch the graphs of the following polynomial functions.   Zeros or   and   These are the points where the graph will intersect the : and .   Multiplicity of Zeros: From the given, all zeroes are of multiplicity 1, which means that they are of odd multiplicity, thus the graph will cross the x-axis on these points     End Behavior: is of degree 3 and the leading term is positive.   Zeros or   y   End Behavior: increasing End Behavior: increasing

7. Example: Sketch the graphs of the following polynomial functions.   Zeros or   and   These are the points where the graph will intersect the : and .   Multiplicity of Zeros: 1 is of even multiplicity (cross x-axis) -2 is of odd multiplicity (touch only x-axis)     End Behavior: is of degree 3 and the leading term is negative.   Zeros or   y   End Behavior: decreasing End Behavior: decreasing Multiplicity Turning point at the  

8. Example: Sketch the graphs of the following polynomial functions.   Zeros or   and   These are the points where the graph will intersect the : and .   Multiplicity of Zeros: is of odd multiplicity (cross x-axis) 2 is of odd multiplicity (cross x-axis)       End Behavior: is of degree 4 and the leading term is negative.   End Behavior: Increasing End Behavior: Decreasing Zeros or     Multiplicity Inflection point at the  

9. Example: Sketch the graphs of the following polynomial functions.   Zeros or   and   These are the points where the graph will intersect the : and .   Multiplicity of Zeros: 2 and -1 have an even multiplicity (touch only x-axis) 1 has an odd multiplicity (cross x-axis)     End Behavior: is of degree 5 and the leading term is positive.   End Behavior: Increasing End Behavior: Increasing Zeros or     Multiplicity 2: Turning points at  

10. Example: Sketch the graphs of the following polynomial functions. z   Zeros or   and   These are the points where the graph will intersect the : and .   Multiplicity of Zeros: you see that and are of multiplicity which is an odd multiplicity, therefore the graph will cross the x-axis at these intercepts.       Sometimes, the given is not in factored form such as this. This means that you need to factor completely the polynomial to get the roots.  

11. Example: Sketch the graphs of the following polynomial functions. z   Zeros or   and   These are the points where the graph will intersect the :   Multiplicity of Zeros: 3 has an even multiplicity (touch only x-axis) 2 has an odd multiplicity (cross x-axis)     Sometimes, the given is not in factored form such as this. This means that you need to factor completely the polynomial to get the roots.  

12. PRACTICE NOW For each of the following polynomial functions, determine the end behavior base on the degree and the sign of the leading term, identify the zeros or (indicating multiplicity of zeros), and and then sketch the graph.   Graph End Behavior: _______________________ Sign of the Leading Coefficient: _________ _______________ Graph Increasing on both sides. positive    

13. PRACTICE NOW For each of the following polynomial functions, determine the end behavior base on the degree and the sign of the leading term, identify the zeros or (indicating multiplicity of zeros), and and then sketch the graph.   Graph End Behavior: _______________________ Sign of the Leading Coefficient: _________ ______________________ Graph Increasing on both sides Positive    

14. PRACTICE NOW 13 For each of the following polynomial functions, determine the end behavior base on the degree and the sign of the leading term, identify the zeros or (indicating multiplicity of zeros), and and then sketch the graph.   Graph End Behavior: ________________________ ________________________ Sign of the Leading Coefficient: _________ ______________________ Graph Decreasing on the left side and increasing on the right side Positive