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1. Lesson Video 6.8 “What are we trying to accomplish?” Algebra 2 2013-14 Example by Mrs. G Give It a Try

2. Lesson 6.8 • Determine behavior of the graph at the x-intercepts. • Determine end behavior of the graph. • Sketch a reasonable graph of a polynomial function. Objective Video Example by Mrs. G Give It a Try

3. Video YouTube mathispower4u Objective Video Example by Mrs. G Give It a Try

4. Example by Mrs. G • We already know how the graphs of some simple polynomials look… Linear y = x Quadratic y = x2 Let’s look at another simple graph, this time with degree 3. Click to continue. Objective Video Example by Mrs. G Give It a Try

5. Example by Mrs. G • Cubic y = x3 Click to continue. Objective Video Example by Mrs. G Give It a Try

6. Example by Mrs. G • Cubic y = x3 • • • • • Click to continue. Objective Video Example by Mrs. G Give It a Try

7. Example by Mrs. G • The simplest polynomial functions look like this… Linear y = x Quadratic y = x2 Cubic y = x3 In General: y = x(even)y = x(odd) Click to continue. Objective Video Example by Mrs. G Give It a Try

8. Example by Mrs. G • “End Behavior” is what happens on the graph as x gets REALLY large (+∞) and REALLY small (-∞). It is what is happening on the “ends”. • The graph of has this end behavior. EVEN DEGREE • For an > 0 and n even, as x  -∞, f(x)  +∞ and as x  +∞, f(x)  +∞ . • If leading coefficient is positive & degree is even, both ”ends” go up. • For an < 0 and n even, as x  -∞, f(x)  -∞ and as x  +∞, f(x)  -∞ . • If leading coefficient is negative & degree is even, both “ends” go down. • Negative a reflects graph and makes ends go down. Click to continue. Objective Video Example by Mrs. G Give It a Try

9. Example by Mrs. G • “End Behavior” is what happens on the graph as x gets REALLY large (+∞) and REALLY small (-∞). It is what is happening on the “ends”. • The graph of has this end behavior. ODD DEGREE • For an > 0 and n odd, as x  -∞, f(x)  -∞ and as x  +∞, f(x)  +∞ . • If leading coefficient is positive & degree is odd, left ”end” goes down, right “end” up. • For an < 0 and n odd, as x  -∞, f(x)  -∞ and as x  +∞, f(x)  -∞ . • If leading coefficient is negative & degree is odd, left “end” goes up, right “end” down. • Negative a reflects graph and makes ends go opposite directions. Click to continue. Objective Video Example by Mrs. G Give It a Try

10. Example by Mrs. G • Describe the end behavior of the graph of the polynomial function by completing the statement as x  -∞, f(x)  ____ and as x  +∞, f(x)  _____ . • f(x) = x3 – 5x • Identify Leading coefficient: 1 (positive) and Degree: 3 (odd) • as x  -∞, f(x)  ____ and as x  +∞, f(x)  _____ • left “end” goes down, right “end” goes up • f(x) = -x8 + 9x5 – 2x4 • Identify Leading coefficient: -1 (negative) and Degree: 8 (even) • as x  -∞, f(x)  ____ and as x  +∞, f(x)  _____ • both “ends” go down -∞ +∞ Now fill in statement. -∞ -∞ Now fill in statement. Click to continue. Objective Video Example by Mrs. G Give It a Try

11. Give It a Try • Describe the end behavior of the graph of the polynomial function by completing the statement as x  -∞, f(x)  ____ and as x  +∞, f(x)  _____ . • f(x) = -x5 – 3x3 + 2 • f(x) = x4 – 4x2 + x Bring your work and answer to our next class! Objective Video Example by Mrs. G Give It a Try