Curve Sketching

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Kuta Software - Infinite Calculus Name___________________________________

Period____

Date________________

Curve Sketching

For each problem, find the: x and y intercepts, x-coordinates of the critical points, open intervals where the function is increasing and decreasing, x-coordinates of the inflection points, open intervals where the function is concave up and concave down, and relative minima and maxima. Using this information, sketch the graph of the function.

1) y =

− x³ 3

+ x²

x y

−8 −7 −6 −5 −4 −3 −2 −1 1 2 3 4 5 6 7 8

−8

−7

−6

−5

−4

−3

−2

−1 1 2 3 4 5 6 7 8

-1-

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2) y =

− x⁴ 4

+ x² − 1

x y

−8 −7 −6 −5 −4 −3 −2 −1 1 2 3 4 5 6 7 8

−8

−7

−6

−5

−4

−3

−2

−1 1 2 3 4 5 6 7 8

-2-

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3) y =

1

5

(

)

5

3 + 2

(

)

2 3

x y

−8 −7 −6 −5 −4 −3 −2 −1 1 2 3 4 5 6 7 8

−8

−7

−6

−5

−4

−3

−2

−1 1 2 3 4 5 6 7 8

-3-

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For each problem, find the: x and y intercepts, asymptotes, x-coordinates of the critical points, open intervals where the function is increasing and decreasing, x-coordinates of the inflection points, open intervals where the function is concave up and concave down, and relative minima and maxima. Using this information, sketch the graph of the function.

4) y =

7

x

2 − 7

x

3

x y

−8 −7 −6 −5 −4 −3 −2 −1 1 2 3 4 5 6 7 8

−8

−7

−6

−5

−4

−3

−2

−1 1 2 3 4 5 6 7 8

-4-

©a T2C0l1j3B HKIuatHat gSKobfytBwaa6rweu pLPLrCN.m 2 nA6lHld qrGilgChatxsC 8rQeGsGejrRvreXdJ.3 T MMQaqdHet Mw2istGhT UIXnzf7iKnji5tyeB hCeaXlRc4uClQuMsG.M Worksheet by Kuta Software LLC

Kuta Software - Infinite Calculus Name___________________________________

Period____

Date________________

Curve Sketching

For each problem, find the: x and y intercepts, x-coordinates of the critical points, open intervals where the function is increasing and decreasing, x-coordinates of the inflection points, open intervals where the function is concave up and concave down, and relative minima and maxima. Using this information, sketch the graph of the function.

1) y =

− x³ 3

+ x²

x y

−8 −7 −6 −5 −4 −3 −2 −1 1 2 3 4 5 6 7 8

−8

−7

−6

−5

−4

−3

−2

−1 1 2 3 4 5 6 7

8 x-intercepts at x = 0, 3

y-intercept at y = 0 Critical points at: x = 0, 2 Increasing:

(

)

Decreasing:

(

)

(

)

Inflection point at: x = 1 Concave up:

(

)

Concave down:

(

)

Relative minimum:

(

)

Relative maximum:

(

)

-1-

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2) y =

− x⁴ 4

+ x² − 1

x y

−8 −7 −6 −5 −4 −3 −2 −1 1 2 3 4 5 6 7 8

−8

−7

−6

−5

−4

−3

−2

−1 1 2 3 4 5 6 7

8 x-intercepts at x = − 2 , 2

y-intercept at y = −1

Critical points at: x = − 2 , 0, 2 Increasing:

(

)

(

)

Decreasing:

(

)

(

)

Inflection points at: x = − 6 3 , 6

3 Concave up:

(

)

Concave down:

(

)

(

)

Relative minimum:

(

)

Relative maxima:

(

)

(

)

-2-

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3) y =

1

5

(

)

5

3 + 2

(

)

2 3

x y

−8 −7 −6 −5 −4 −3 −2 −1 1 2 3 4 5 6 7 8

−8

−7

−6

−5

−4

−3

−2

−1 1 2 3 4 5 6 7

8 x-intercepts at x = −6, 4

y-intercept at y =

12 ³ 2 5 Critical points at: x = 0, 4 Increasing:

(

)

(

)

Decreasing:

(

)

Inflection point at: x = 6 Concave up:

(

)

Concave down:

(

)

(

)

Relative minimum:

(

)

Relative maximum:

(

12 ³2 5

)

-3-

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For each problem, find the: x and y intercepts, asymptotes, x-coordinates of the critical points, open intervals where the function is increasing and decreasing, x-coordinates of the inflection points, open intervals where the function is concave up and concave down, and relative minima and maxima. Using this information, sketch the graph of the function.

4) y =

7

x

2 − 7

x

3

x y

−8 −7 −6 −5 −4 −3 −2 −1 1 2 3 4 5 6 7 8

−8

−7

−6

−5

−4

−3

−2

−1 1 2 3 4 5 6 7

8 x-intercepts at x = −1, 1

No y-intercepts.

Vertical asymptote at: x = 0 Horizontal asymptote at: y = 0 Critical points at: x = − 3 , 3 Increasing:

(

)

(

)

Decreasing:

(

)

(

)

Inflection points at: x = − 6 , 6 Concave up:

(

)

(

)

Concave down:

(

)

(

)

Relative minimum:

(

)

Relative maximum:

(

14 3 9

)

-4-

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