Section 2.12

Sketching the Graph of a Function from Its Characteristic Points

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Problem 1471

Sketch the graph of the following function:

y=3x−x3. y=3x-x^{3} .
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Problem 1472

Sketch the graph of the following function:

y=1+x2−x42. y=1+x^{2}-\dfrac {x^{4}}{2} .
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Problem 1473

Sketch the graph of the following function:

y=(x+1)(x−2)2. y=(x+1)(x-2)^{2} .
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Problem 1474

Sketch the graph of the following function:

y=2−x21+x4. y=\dfrac {2-x^{2}}{1+x^{4}} .
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Problem 1475

Sketch the graph of the following function:

y=x2−1x2−5x+6. y=\dfrac {x^{2}-1}{x^{2}-5x+6} .
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Problem 1476

Sketch the graph of the following function:

y=x(1+x)(1−x)2. y=\dfrac {x}{(1+x)(1-x)^{2}} .
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Problem 1477

Sketch the graph of the following function:

y=x4(1+x)3. y=\dfrac {x^{4}}{(1+x)^{3}} .
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Problem 1478

Sketch the graph of the following function:

y=(1+x1−x)4. y=\left(\dfrac {1+x}{1-x}\right)^{4} .
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Problem 1479

Sketch the graph of the following function:

y=x2(x−1)(x+1)2. y=\dfrac {x^{2}(x-1)}{(x+1)^{2}} .
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Problem 1480

Sketch the graph of the following function:

y=x(1−x2)2. y=\dfrac {x}{(1-x^{2})^{2}} .
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Problem 1481

Sketch the graph of the following function:

y=(x+1)3(x−1)3. y=\dfrac {(x+1)^{3}}{(x-1)^{3}} .
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Problem 1482

Sketch the graph of the following function:

y=x4+8x3+1. y=\dfrac {x^{4}+8}{x^{3}+1} .
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Problem 1483

Sketch the graph of the following function:

y=11+x−103x2+11−x. y=\dfrac {1}{1+x}-\dfrac {10}{3x^{2}}+\dfrac {1}{1-x} .
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Problem 1484

Sketch the graph of the following function:

y=(x−3)x. y=(x-3)\sqrt{x} .
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Problem 1485

Sketch the graph of the following function:

y=±8x2−x4. y=\pm \sqrt{8x^{2}-x^{4}} .
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Problem 1485.1
[NO TASK]
Problem 1486

Sketch the graph of the following function:

y=±(x−1)(x−2)(x−3). y=\pm \sqrt{(x-1)(x-2)(x-3)} .
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Problem 1487

Sketch the graph of the following function:

y=x3−x2−x+13. y=\sqrt[3]{x^{3}-x^{2}-x+1} .
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Problem 1488

Sketch the graph of the following function:

y=x23−x2+13. y=\sqrt[3]{x^{2}}-\sqrt[3]{x^{2}+1} .
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Problem 1489

Sketch the graph of the following function:

y=(x+2)2/3−(x−2)2/3. y=(x+2)^{2/3}-(x-2)^{2/3} .
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Problem 1490

Sketch the graph of the following function:

y=(x+1)2/3+(x−1)2/3. y=(x+1)^{2/3}+(x-1)^{2/3} .
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Problem 1491

Sketch the graph of the following function:

y=xx2−13. y=\dfrac {x}{\sqrt[3]{x^{2}-1}} .
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Problem 1492

Sketch the graph of the following function:

y=x2x2−12x2−1. y=\dfrac {x^{2}\sqrt{x^{2}-1}}{2x^{2}-1} .
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Problem 1493

Sketch the graph of the following function:

y=(1+x)3/2x. y=\dfrac {(1+x)^{3/2}}{\sqrt{x}} .
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Problem 1494

Sketch the graph of the following function:

y=1−x+x33+x. y=1-x+\sqrt{\dfrac {x^{3}}{3+x}} .
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Problem 1495

Sketch the graph of the following function:

y=x2x+13. y=\sqrt[3]{\dfrac {x^{2}}{x+1}} .
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Problem 1496

Sketch the graph of the following function:

y=x4+3x2+1. y=\sqrt{\dfrac {x^{4}+3}{x^{2}+1}} .
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Problem 1497

Sketch the graph of the following function:

y=sin⁡x+cos⁡2x. y=\sin x+\cos ^{2}x .
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Problem 1498

Sketch the graph of the following function:

y=(7+2cos⁡x)sin⁡x. y=(7+2\cos x)\sin x .
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Problem 1499

Sketch the graph of the following function:

y=sin⁡x+13sin⁡3x. y=\sin x+\dfrac {1}{3}\sin 3x .
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Problem 1500

Sketch the graph of the following function:

y=cos⁡x−12cos⁡2x. y=\cos x-\dfrac {1}{2}\cos 2x .
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Problem 1501

Sketch the graph of the following function:

y=sin⁡4x+cos⁡4x. y=\sin ^{4}x+\cos ^{4}x .
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Problem 1502

Sketch the graph of the following function:

y=sin⁡x⋅sin⁡3x. y=\sin x \cdot \sin 3x .
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Problem 1503

Sketch the graph of the following function:

y=sin⁡xsin⁡(x+π4). y=\dfrac {\sin x}{\sin \left(x+\dfrac {\pi }{4}\right)} .
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Problem 1504

Sketch the graph of the following function:

y=cos⁡xcos⁡2x. y=\dfrac {\cos x}{\cos 2x} .
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Problem 1504.1
[NO TASK]
Problem 1505

Sketch the graph of the following function:

y=2x−tg⁡x. y=2x-\tg x .
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Problem 1506

Sketch the graph of the following function:

y=e2x−x2. y=e^{2x-x^{2}} .
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Problem 1507

Sketch the graph of the following function:

y=(1+x2)e−x2. y=(1+x^{2})e^{-x^{2}} .
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Problem 1508

Sketch the graph of the following function:

y=x+e−x. y=x+e^{-x} .
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Problem 1509

Sketch the graph of the following function:

y=x2/3e−x. y=x^{2/3}e^{-x} .
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Problem 1509.1
[NO TASK]
Problem 1510

Sketch the graph of the following function:

y=ex1+x. y=\dfrac {e^{x}}{1+x} .
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Problem 1511

Sketch the graph of the following function:

y=1−e−x2. y=\sqrt{1-e^{-x^{2}}} .
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Problem 1512

Sketch the graph of the following function:

y=ln⁡xx. y=\dfrac {\ln x}{\sqrt{x}} .
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Problem 1513

Sketch the graph of the following function:

y=ln⁡(x+x2+1). y=\ln (x+\sqrt{x^{2}+1}) .
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Problem 1514

Sketch the graph of the following function:

y=x2+1ln⁡(x+x2+1). y=\sqrt{x^{2}+1}\ln (x+\sqrt{x^{2}+1}) .
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Problem 1515

Sketch the graph of the following function:

y=arcsin⁡x1−x2. y=\dfrac {\arcsin x}{\sqrt{1-x^{2}}} .
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Problem 1516

Sketch the graph of the following function:

y=x+arctg⁡x. y=x+\arctg x .
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Problem 1517

Sketch the graph of the following function:

y=x2+arcctg⁡x. y=\dfrac {x}{2}+\arcctg x .
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Problem 1518

Sketch the graph of the following function:

y=xarctg⁡x. y=x \arctg x .
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Problem 1519

Sketch the graph of the following function:

y=arcsin⁡2x1+x2. y=\arcsin \dfrac {2x}{1+x^{2}} .
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Problem 1520

Sketch the graph of the following function:

y=arccos⁡1−x21+x2. y=\arccos \dfrac {1-x^{2}}{1+x^{2}} .
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Problem 1521

Sketch the graph of the following function:

y=(x+2)e1/x. y=(x+2)e^{1/x} .
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Problem 1522

Sketch the graph of the following function:

y=2x2+1−x2−1. y=2^{\sqrt{x^{2}+1}-\sqrt{x^{2}-1}} .
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Problem 1523

Sketch the graph of the following function:

y=ln⁡x2−3x+2x2+1. y=\ln \dfrac {x^{2}-3x+2}{x^{2}+1} .
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Problem 1524

Sketch the graph of the following function:

y=aarcsin⁡xa−a2−x2(a>0). y=a\arcsin \dfrac {x}{a}-\sqrt{a^{2}-x^{2}} \quad (a>0) .
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Problem 1525

Sketch the graph of the following function:

y=arccos⁡1−x1−2x. y=\arccos \dfrac {1-x}{1-2x} .
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Problem 1526

Sketch the graph of the following function:

y=xx. y=x^{x} .
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Problem 1527

Sketch the graph of the following function:

y=x1/x. y=x^{1/x} .
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Problem 1528

Sketch the graph of the following function:

y=(1+x)1/x. y=(1+x)^{1/x} .
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Problem 1529

Sketch the graph of the following function:

y=x(1+1x)x(x>0). y=x\left(1+\dfrac {1}{x}\right)^{x} \quad (x>0) .
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Problem 1530

Sketch the graph of the following function (do not investigate concavity/convexity):

y=1e1−x21+x2. y=\dfrac {\dfrac {1}{e^{1-x^{2}}}}{1+x^{2}} .
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Problem 1531

Sketch the curve represented by the following parametric equations:

x=(t+1)24,y=(t−1)24. x=\dfrac {(t+1)^{2}}{4}, \quad y=\dfrac {(t-1)^{2}}{4} .
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Problem 1532

Sketch the curve represented by the following parametric equations:

x=2t−t2,y=3t−t3. x=2t-t^{2}, \quad y=3t-t^{3} .
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Problem 1533

Sketch the curve represented by the following parametric equations:

x=t2t−1,y=tt2−1. x=\dfrac {t^{2}}{t-1}, \quad y=\dfrac {t}{t^{2}-1} .
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Problem 1534

Sketch the curve given by the following parametric equations:

x=t21−t2,y=11+t2. x=\dfrac {t^{2}}{1-t^{2}}, \quad y=\dfrac {1}{1+t^{2}} .
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Problem 1535

Sketch the curve given by the following parametric equations:

x=t+e−t,y=2t+e−2t. x=t+e^{-t}, \quad y=2t+e^{-2t} .
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Problem 1536

Sketch the curve given by the following parametric equations:

x=acos⁡2t,y=acos⁡3t(a>0). x=a\cos 2t, \quad y=a\cos 3t \quad (a>0) .
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Problem 1537

Sketch the curve given by the following parametric equations:

x=cos⁡4t,y=sin⁡4t. x=\cos ^{4}t, \quad y=\sin ^{4}t .
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Problem 1538

Sketch the curve given by the following parametric equations:

x=tln⁡t,y=ln⁡tt. x=t\ln t, \quad y=\dfrac {\ln t}{t} .
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Problem 1539

Sketch the curve given by the following parametric equations:

x=acos⁡3t,y=atan⁡3t(a>0). x=\dfrac {a}{\cos ^{3}t}, \quad y=a\tan ^{3}t \quad (a>0) .
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Problem 1540

Sketch the curve given by the following parametric equations:

x=a(sh t−t),y=a(ch t−1)(a>0). x=a(\text{sh}\, t-t), \quad y=a(\text{ch}\, t-1) \quad (a>0) .
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Problem 1541

Convert the following curve equation into parametric form, and then sketch this curve:

x3+y3−3axy=0(a>0). x^{3}+y^{3}-3axy=0 \quad (a>0) .
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Problem 1542

Convert the following curve equation to parametric form, and then sketch this curve:

x2+y2=x4+y4. x^{2}+y^{2}=x^{4}+y^{4} .
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Problem 1543

Convert the following curve equation to parametric form, and then sketch this curve:

x2y2=x3−y3. x^{2}y^{2}=x^{3}-y^{3} .
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Problem 1544

Sketch the graph of the following curve:

xy=yx(x>0,y>0). x^{y}=y^{x} \quad (x>0, y>0) .
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Problem 1545

Sketch the curve ch2x−ch2y=1\text{ch}^{2}x-\text{ch}^{2}y=1.

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Problem 1546

Sketch the graphs of the following functions given in polar coordinates (φ,r)(\varphi ,r) (r≥0r \geq 0):

r=a+bcos⁡φ(0<a≤b). r=a+b\cos \varphi \quad (0<a \leq b) .
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Problem 1547

Sketch the graphs of the following functions given in polar coordinates (φ,r)(\varphi ,r) (r≥0r \geq 0):

r=asin⁡3φ(a>0). r=a\sin 3\varphi \quad (a>0) .
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Problem 1548

Sketch the graphs of the following functions given in polar coordinates (φ,r)(\varphi ,r) (r≥0r \geq 0):

r=acos⁡3φ(a>0). r=\dfrac {a}{\sqrt{\cos 3\varphi }} \quad (a>0) .
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Problem 1549

Sketch the graph of the following function given in polar coordinates (φ,r)(\varphi ,r) (with r≥0r \geq 0):

r=ath φφ−1, where φ>1(a>0). r=a\dfrac {\text{th}\, \varphi }{\varphi -1} , \text{ where } \varphi >1 \quad (a>0) .
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Problem 1550

Sketch the graph of the following function given in polar coordinates (φ,r)(\varphi ,r) (with r≥0r \geq 0):

cos⁡φ=r−1r2. \cos \varphi =\dfrac {r-1}{r^{2}} .
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Problem 1551

Sketch the family of graphs of the following curves (aa denotes the parameter):

y=x2−2x+a. y=x^{2}-2x+a .
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Problem 1552

Sketch the family of graphs of the following curves (aa denotes the parameter):

y=x+a2x. y=x+\dfrac {a^{2}}{x} .
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Problem 1553

Sketch the family of graphs of the following curves (aa denotes the parameter):

y=x±a(1−x2). y=x \pm \sqrt{a(1-x^{2})} .
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Problem 1554

Sketch the family of graphs of the following curves (aa denotes the parameter):

y=x2+e−ax. y=\dfrac {x}{2}+e^{-ax} .
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Problem 1555

Sketch the family of graphs of the following curves (aa denotes the parameter):

y=xe−xa. y=xe^{-\frac{x}{a}} .
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