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Aim: What concepts have we available to aide us in sketching functions?

Aim: What concepts have we available to aide us in sketching functions?. Do Now:. Find the domain of. Concepts used in Sketching. x - and y -intercepts. symmetry. domain & range. continuity. vertical asymptotes. differentiability. relative extrema. concavity.

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Aim: What concepts have we available to aide us in sketching functions?

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  1. Aim: What concepts have we available to aide us in sketching functions? Do Now: Find the domain of

  2. Concepts used in Sketching • x- and y-intercepts • symmetry • domain & range • continuity • vertical asymptotes • differentiability • relative extrema • concavity • points of inflection • horizontal asymptotes Use them all? If not all, which are best?

  3. Guidelines for Analyzing Graph 1. Determine the domain and range of the function. 2. Determine the intercepts and asymptotes of the graph. 3. Locate the x-values for which f’(x) and f’’(x) are either zero or undefined. Use the results to determine relative extrema and points of inflection. Also helpful: symmetry; end behavior

  4. Abridged Guidelines – the 4 Tees T1 Test the function T2 Test the 1st Derivative T3 Test the 2nd Derivative T4 Test End Behavior

  5. Model Problem 1 Analyze the graph of 1. find domain & range exclusions at zeros of denominator domain: all reals except ±2

  6. Model Problem 1 Analyze the graph of 2. find intercepts & asymptotes y-intercept x-intercept

  7. Model Problem 1 Analyze the graph of 2. find intercepts & asymptotes verticals asymptotes found at zeros of denominator x = ±2 horizontal asymptote If degree of p = degree of q, then the line y = an/bm is a horizontal asymptote. y = 2

  8. Model Problem 1 Analyze the graph of 3. find f’(x) = 0 and f’’(x) = 0 or undefined x = 0 (x2 – 4)2 = 0 undefined at zeros of denominator x = ±2

  9. Model Problem 1 Analyze the graph of 3. find f’(x) = 0 and f’’(x) = 0 or undefined no real solution no possible points of inflection

  10. vertical asymptote vertical asymptote Model Problem 1 3. test intervals decreasing, concave down decreasing, concave up + + 0 relative minimum increasing, concave up + + increasing, concave down +

  11. decreasing, concave up -2 < x < 0 increasing, concave up 0 < x < 2 increasing, concave down - < x < -2 Model Problem 1 (0, 9/2) relative minimum increasing, concave down 2 < x < 

  12. Model Problem 2 – What the cusp!! Analyze the graph of T1 Find Domain all reals Find intercepts & asymptotes no vertical or horizontal asymptotes

  13. Model Problem 2 – What the cusp!! Analyze the graph of T2 1st Derivative Test x at 0 is undefined BUT . . . f’ > 0 inc f’ < 0 dec x = 0 is defined for original function a cusp!!!

  14. cusp Model Problem 2 – What the cusp!! Analyze the graph of T3 2nd Derivative Test x at 0 is undefined f’’ > 0 con up f’’ > 0 con up

  15. f’ f’’ (-2,2) relative max. inflection points: (-1.4,1.2), (0,0), (1.4,-1.2) relative min. (2,-2) Model Problem 3 < 0 dec < 0 dec > 0 inc > 0 inc > 0 c.u. < 0 c.d. > 0 c.u. < 0 c.d.

  16. Model Problem 4 Analyze the graph of 1. find Domain 2. find intercepts & asymptotes verticals asymptotes found a zeros of denominator x = ±2 1 + sin x = 0; sin x = -1

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