Graphs of linear, quadratic and power functions

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Problem 237 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 1.4 "Graphical Representation of Functions"

Draw the graphs of the linear homogeneous function

y=ax y=ax

for a=0,12,1,2,−1a=0,\dfrac {1}{2},1,2,-1.

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Problem 238 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 1.4 "Graphical Representation of Functions"

Draw the graphs of the linear function

y=x+b y=x+b

for b=0,1,2,−1b=0,1,2,-1.

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Problem 239 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 1.4 "Graphical Representation of Functions"

Draw the graphs of the linear functions:

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(a)

y=2x+3y=2x+3;

(b)

y=2−0.1xy=2-0.1x;

(c)

y=−x2−1y=-\dfrac {x}{2}-1.

Problem 242 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 1.4 "Graphical Representation of Functions"

Plot the graphs of the quadratic rational integral function (parabola):

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(a)

y=ax2y=ax^{2}, for a=1,12,2,−1a=1, \dfrac {1}{2}, 2, -1;

(b)

y=(x−x0)2y=(x-x_{0})^{2}, for x0=0,1,2,−1x_{0}=0, 1, 2, -1;

(c)

y=x2+cy=x^{2}+c, for c=0,1,2,−1c=0, 1, 2, -1.

Problem 243 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 1.4 "Graphical Representation of Functions"

Convert the quadratic trinomial

y=ax2+bx+c y=ax^{2}+bx+c

into the form

y=y0+a(x−x0)2, y=y_{0}+a(x-x_{0})^{2},

plot its graph, and study the examples:

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(a)

y=8x−2x2y=8x-2x^{2};

(b)

y=x2+3x+2y=x^{2}+3x+2;

(c)

y=−x2+2x−1y=-x^{2}+2x-1;

(d)

y=12x2+x+1y=\dfrac {1}{2}x^{2}+x+1.

Problem 249 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 1.4 "Graphical Representation of Functions"

Plot the graph of the linear fractional function (hyperbola): y=1xy=\dfrac {1}{x}.

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Problem 274 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 1.4 "Graphical Representation of Functions"

Graph the power function

y=xn y=x^{n}

for: (a) n=1,3,5n=1,3,5; (b) n=2,4,6n=2,4,6.

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(a)
(b)
Problem 275 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 1.4 "Graphical Representation of Functions"

Graph the power function

y=xn y=x^{n}

for: (a) n=−1,−3n=-1,-3; (b) n=−2,−4n=-2,-4.

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(a)
(b)
Problem 276 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 1.4 "Graphical Representation of Functions"

Graph the radical

y=xm y=\sqrt[m]{x}

for: (a) m=2,4m=2,4; (b) m=3,5m=3,5.

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(a)
(b)
Problem 277 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 1.4 "Graphical Representation of Functions"

Let:

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(a)

m=2,k=1m=2, k=1;

(b)

m=2,k=3m=2, k=3;

(c)

m=3,k=1m=3, k=1;

(d)

m=3,k=2m=3, k=2;

(e)

m=3,k=4m=3, k=4;

(f)

m=4,k=2m=4, k=2;

(g)

m=4,k=3m=4, k=3.

Plot the graph of the radical

y=xkm. y=\sqrt[m]{x^{k}} .
Problem 885 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 3.3 "The Concept of a Function. Continuity. Graphical Representation of Functions"

Plot point by point the graph of the entire function of the 2nd degree (parabola) y=x24y = \frac{x^{2}}{4}.

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Problem 886 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 3.3 "The Concept of a Function. Continuity. Graphical Representation of Functions"

Plot point by point the graph of the entire function of the 2nd degree (parabola) y=−x22y = -\frac{x^{2}}{2}.

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Problem 887 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 3.3 "The Concept of a Function. Continuity. Graphical Representation of Functions"

Plot point by point the graph of the entire function of the 2nd degree (parabola) y=x2−3x4y = \frac{x^{2}-3x}{4}.

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Problem 888 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 3.3 "The Concept of a Function. Continuity. Graphical Representation of Functions"

Plot point by point the graph of the entire function of the 2nd degree (parabola) y=4x−x23y = \frac{4x-x^{2}}{3}.

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Problem 889 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 3.3 "The Concept of a Function. Continuity. Graphical Representation of Functions"

Plot point by point the graphs of the functions y=ax2y = ax^{2} for a=4,2,1,12,14a = 4, 2, 1, \frac{1}{2}, \frac{1}{4}.

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Problem 890 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 3.3 "The Concept of a Function. Continuity. Graphical Representation of Functions"

Plot point by point the graphs of the functions y=ax2+x−1y = ax^{2}+x-1 for a=1,14,120a = 1, \frac{1}{4}, \frac{1}{20}, and the line x=y+1x = y+1.

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Problem 891 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 3.3 "The Concept of a Function. Continuity. Graphical Representation of Functions"

Using the identity

ax2+bx+c=a(x+b2a)2+4ac−b24a(a≠0), ax^{2}+bx+c = a\left(x+\frac{b}{2a}\right)^{2}+\frac{4ac-b^{2}}{4a} \quad (a \neq 0),

plot the graph of the function y=ax2+bx+cy = ax^{2}+bx+c.

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Problem 900 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 3.3 "The Concept of a Function. Continuity. Graphical Representation of Functions"

Plot the hyperbola that is the graph of the fractional function y=1xy = \frac{1}{x}.

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Problem 1.7.21 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Предел. Непрерывность. Дифференцируемость." [ККЧ+03]More problems in Subsection 1.7.2 "Problems"

In one coordinate system, plot the graphs of the functions:

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(1)

y=x,y=x2,y=x3,y=x4y = x, y = x^{2}, y = x^{3}, y = x^{4};

(2)

y=1x,y=1x2,y=1x3y = \frac{1}{x}, y = \frac{1}{x^{2}}, y = \frac{1}{x^{3}};

(3)

y=x,y=x1/2,y=x1/3,y=x1/4y = x, y = x^{1 / 2}, y = x^{1 / 3}, y = x^{1 / 4};

(4)

y=1x,y=1x,y=1x3y = \frac{1}{x}, y = \frac{1}{\sqrt{x}}, y = \frac{1}{\sqrt[3]{x}}.

References

[Dem97] Demidovich B.P. Сборник задач и упражнений по математическому анализу. 13-е изд., Издательство Московского университета / Издательство "ЧеРо", 1997.
[ГюКу03] Гюнтер Н.М., Кузьмин Р.О. Сборник задач по высшей математике. 13-е изд., Издательство «Лань», 2003.
[ККЧ+03] Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. Предел. Непрерывность. Дифференцируемость. 2-е изд., ФИЗМАТЛИТ, 2003.