Graphical Representation of Functions

Draw the graphs of the linear homogeneous function

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

Draw the graphs of the linear function

for $b=0,1,2,-1$.

Draw the graphs of the linear functions:

The coefficient of linear expansion of iron is $a=1.2 \times 10^{-6}$. On a suitable scale, plot the graph of the function

where $T$ denotes the temperature (in degrees) and $l$ denotes the length of the iron rod at temperature $T$. Assume that when $T=0^{\circ }$, $l=100$ cm.

Two particles move on a number axis. The first particle, at time $t=0$, is at a point 20 meters to the left of the origin, with velocity $v_{1}=10$ m/s; the second particle, at $t=0$, is at a point 30 meters to the right of the origin $O$, with velocity $v_{2}=-20$ m/s. Plot the graphs of the equations of motion of the two points and find the time and position at which they meet.

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

Convert the quadratic trinomial

into the form

plot its graph, and study the examples:

A particle is launched with initial velocity $v_{0}=600$ meters per second in a direction making angle $\alpha =45^{\circ }$ with the horizontal plane. Plot the graph of the trajectory, and find the maximum height and the range of the flight (assume $g \approx 10$ m/s${}^{2}$, air resistance neglected).

Plot the graph of the rational integral function of degree higher than two: $y=x^{3}+1$.

Plot the graph of the rational integral function of degree higher than two: $y=(1-x^{2})(2+x)$.

Plot the graph of the rational integral function of degree higher than two: $y=x^{2}-x^{4}$.

Plot the graph of the rational integral function of degree higher than two: $y=x(a-x)^{2}(a+x)^{3} \quad (a>0)$.

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

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

Convert the linear fractional function

into the form

Then plot its graph. Study the example

A gas at pressure $p_{0}=1$ atmosphere occupies volume $v_{0}=12$ cubic meters. Assuming the temperature of the gas remains constant, plot the graph of the gas volume $v$ as a function of the pressure (Boyle--Mariotte law).

Plot the graph of the following rational fractional function: $y=x+\dfrac {1}{x}$ (hyperbola).

Plot the graph of the following rational fractional function: $y=x^{2}+\dfrac {1}{x}$ (Newton's cubic curve).

Plot the graph of the following rational fractional function: $y=x+\dfrac {1}{x^{2}}$.

Graph the following rational fractional function: $y=\dfrac {1}{1+x^{2}}$ (the witch of Agnesi).

Graph the following rational fractional function: $y=\dfrac {2x}{1+x^{2}}$ (the serpentine of Newton).

Graph the following rational fractional function: $y=\dfrac {1}{1-x^{2}}$.

Graph the following rational fractional function: $y=\dfrac {x}{1-x^{2}}$.

Graph the following rational fractional function: $y=\dfrac {1}{1+x}-\dfrac {2}{x}+\dfrac {1}{1-x}$.

Graph the following rational fractional function: $y=\dfrac {1}{1+x}-\dfrac {2}{x^{2}}+\dfrac {1}{1-x}$.

Graph the following rational fractional function: $y=\dfrac {(x+1)(x-2)}{(x-1)(x+2)}$.

Reduce the function

to the form

and then sketch its graph. Study the example

A particle is at distance $x$ from a center of gravitational attraction. Given that the force of attraction is $F=10$ kg when $x=1$ meter, graph the absolute value of the force of attraction $F$ on the particle (Newton's law).

According to the van der Waals law (Закон Ван-дер-Ваальса), at constant temperature the volume $v$ of a real gas and its pressure $p$ are related by

Given $a=2, b=0.1$ and $c=10$, graph the function $p=p(v)$.

Graph the following irrational function: $y=\pm \sqrt{-x-2}$ (a parabola).

Graph the following irrational function: $y=\pm x\sqrt{x}$ (a semicubical parabola).

Graph the following irrational function: $y=\pm \dfrac {1}{2}\sqrt{100-x^{2}}$ (an ellipse).

Graph the following irrational function: $y=\pm \sqrt{x^{2}-1}$ (a hyperbola).

Graph the following irrational function: $y=\pm \sqrt{\dfrac {1-x}{1+x}}$.

Graph the following irrational function: $y=\pm x\sqrt{100-x^{2}}$.

Graph the following irrational function: $y=\pm x\sqrt{\dfrac {x}{10-x}}$ (the cissoid).

Graph the following irrational function: $y=\pm \sqrt{(x^{2}-1)(9-x^{2})}$.

Graph the power function

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

Graph the power function

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

Graph the radical

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

Let:

Plot the exponential function

for $a=\dfrac {1}{2}, 1, 2, e, 10$.

Plot the graph of the composite exponential function

where:

Plot the graph of the logarithmic function $y=\log_{a}x$ for $a=\dfrac {1}{2}, 2, e, 10$.

Plot the graphs of the following functions:

Let:

Plot the graph of the function

Plot the graph of the function

for $A=1, 10, -2$.

Plot the graph of the function

for $x_{0}=0, \dfrac {\pi }{4}, \dfrac {\pi }{2}, \dfrac {3\pi }{4}, \pi$.

Plot the graph of the function

Let $n=1, 2, 3, \dfrac {1}{2}, \dfrac {1}{3}$.

Convert the function

into the form

and then plot its graph.

Study the example

Plot the graph of the following trigonometric function: $y=\cos x$.

Plot the graph of the following trigonometric function: $y=\tg x$.

Plot the graph of the following trigonometric function: $y=\ctg x$.

Plot the graph of the following trigonometric function: $y=\sec x$.

Plot the graph of the following trigonometric function: $y=\csc x$.

Plot the graph of the following trigonometric function: $y=\sin^{2}x$.

Plot the graph of the following trigonometric function: $y=\sin^{3}x$.

Plot the graph of the following trigonometric function: $y=\ctg^{2}x$.

Plot the graph of the function: $y=\sin x \cdot \sin 3x$.

Plot the graph of the function: $y=\pm \sqrt{\cos x}$.

Plot the graph of the following function: $y=\sin x^{2}$.

Plot the graph of the following function: $y=\sin \dfrac {1}{x}$.

Plot the graph of the following function: $y=x\cos \dfrac {\pi }{x}$.

Plot the graph of the following function: $y=\tg \dfrac {\pi }{x}$.

Plot the graph of the following function: $y=x\left(2+\sin \dfrac {1}{x}\right)$.

Plot the graph of the following function: $y=\pm \sqrt{1-x^{2}}\sin \dfrac {\pi }{x}$.

Plot the graph of the following function: $y=\dfrac {\sin x}{x}$.

Plot the graph of the following function: $y=e^{x}\cos x$.

Plot the graph of the following function: $y=\pm 2^{-x}\sqrt{\sin \pi x}$.

Graph the following function: $y=\dfrac {\cos x}{1+x^{2}}$.

Graph the following function: $y=\ln (\cos x)$.

Graph the following function: $y=\cos (\ln x)$.

Graph the following function: $y=e^{\frac{1}{\sin x}}$.

Graph the following inverse trigonometric function: $y=\arcsin x$.

Graph the following inverse trigonometric function: $y=\arccos x$.

Graph the following inverse trigonometric function: $y=\arctg x$.

Graph the following inverse trigonometric function: $y=\arcctg x$.

Graph the following inverse trigonometric function: $y=\arcsin \dfrac {1}{x}$.

Graph the following inverse trigonometric function: $y=\arccos \dfrac {1}{x}$.

Graph the following inverse trigonometric function: $y=\arctg \dfrac {1}{x}$.

Graph the following inverse trigonometric function: $y=\arcsin (\sin x)$.

Graph the following inverse trigonometric function: $y=\arcsin (\cos x)$.

Graph the following inverse trigonometric function: $y=\arccos (\cos x)$.

Graph the following inverse trigonometric function: $y=\arctg (\tg x)$.

Graph the following inverse trigonometric function: $y=\arcsin (2\sin x)$.

Let

Let

$y_{1}=x^{2}$;

Given the graph of the function $y=f(x)$, construct the graphs of the following functions

Given the graph of the function $y=f(x)$, construct the graphs of the following functions:

$y=f(x-x_{0})$;

Construct the graphs of the functions

Given the graph of the function $y=f(x)$, construct the graphs of the following functions:

Given the graph of the function $y=f(x)$, construct the graphs of the following functions:

$y=f^{2}(x)$;

$y=\sqrt{f(x)}$;

Given the graphs of the functions $y=f(x)$ and $y=g(x)$, construct the graphs of the following functions:

Using graphical addition, construct the graph of the function: $y=1+x+e^{x}$.

Using graphical addition, construct the graph of the function: $y=(x+1)^{-2}+(x-1)^{-2}$.

Using graphical addition, sketch the graph of the following function:$y=x+\sin x$.

Using graphical addition, sketch the graph of the following function:$y=x+\operatorname {arctg}x$.

Using graphical addition, sketch the graph of the following function:$y=\cos x+\dfrac {1}{2}\cos 2x+\dfrac {1}{3}\cos 3x$.

Using graphical addition, sketch the graphs of the following functions: $y=\sin x-\dfrac {1}{3}\sin 3x+\dfrac {1}{5}\sin 5x$.

Using graphical addition, sketch the graph of the following function:$y=\sin^{4}x+\cos^{4}x$.

Using graphical addition, sketch the graph of the following function:$y=\left|1-x\right|+\left|1+x\right|$.

Using graphical addition, sketch the graph of the following function:$y=\left|1-x\right|-\left|1+x\right|$.

Sketch the graphs of the hyperbolic functions:

Using graphical multiplication, sketch the graphs of the following functions: $y=x\sin x$.

Using graphical multiplication, sketch the graphs of the following functions: $y=x\cos x$.

Using graphical multiplication, sketch the graphs of the following functions: $y=x^{2}\sin^{2}x$.

Using graphical multiplication, sketch the graph of the following function:$y=\dfrac {\sin x}{1+x^{2}}$.

Using graphical multiplication, sketch the graph of the following function:$y=e^{-x^{2}}\cos 2x$.

Using graphical multiplication, sketch the graph of the following function:$y=x\operatorname {sgn}(\sin x)$.

Using graphical multiplication, sketch the graph of the following function:$y=[x] \cdot \left|\sin \pi x\right|$.

Using graphical multiplication, sketch the graph of the following function:$y=\cos x \cdot \operatorname {sgn}(\sin x)$.

Let

Sketch the graph of the function $y=f(x)f(a-x)$ when (a) $a=0$, (б) $a=1$, (в) $a=2$.

Sketch the graph of the function $y=x+\sqrt{x}\operatorname {sgn}(\sin \pi x)$.

Sketch the graph of the function $y=\dfrac {1}{f(x)}$, where $f(x)=x^{2}(1-x^{2})$.

Sketch the graph of the function $y=\dfrac {1}{f(x)}$, where $f(x)=x(1-x)^{2}$.

Sketch the graph of the function $y=\dfrac {1}{f(x)}$, where $f(x)=\sin^{2}x$.

Sketch the graph of the function $y=\dfrac {1}{f(x)}$, where $f(x)=\ln x$.

Sketch the graph of the function $y=\dfrac {1}{f(x)}$, where $f(x)=e^{x}\sin x$.

Let

Sketch the graph of the composite function $y=f(u)$, where $u=2\sin x$.

Let $\varphi (x)=\dfrac {1}{2}(x+\left|x\right|)$ and

Sketch the graphs of the following functions:

Let

and

Sketch the graphs of the functions:

A function $f(x)$ is defined on the domain of positive numbers $x>0$. Extend $f(x)$ to the domain of negative numbers $x<0$ so that the resulting function is: (1) even; (2) odd. Let

Determine about which vertical axis the graph of each of the following functions is symmetric:

Determine the center of symmetry of the graph of each of the following functions:

Sketch the graphs of the periodic functions:

Prove: if the graph of the function $y=f(x)$ ($-\infty <x<+\infty$) is symmetric with respect to the two vertical lines $x=a$ and $x=b$ ($b>a$), then the function $f(x)$ is periodic.

Prove: if the graph of the function $y=f(x)$ ($-\infty <x<+\infty$) is symmetric with respect to the two points $A(a,y_{0})$ and $B(b,y_{1})$ ($b>a$), then the function $f(x)$ is the sum of a linear function and a periodic function. In particular, if $y_{0}=y_{1}$, then $f(x)$ is a periodic function.