Geometric transformations of graphs

Translating graphs along the coordinate axes. Suppose the function $y=f(x)$ and the numbers $a,b>0$. Then:

1. the graph of the function $y=f(x-a)$ is obtained by translating the graph of $y=f(x)$ by $a$ units to the right;

2. the graph of the function $y=f(x+a)$ is obtained by translating the graph of $y=f(x)$ by $a$ units to the right;

3. the graph of the function $y-b=f(x)$ or $y=f(x)+b$ is obtained by translating the graph of $y=f(x)$ by $b$ units in the positive direction of the $Oy$ axis (upwards);

4. the graph of the function $y+b=f(x)$ or $y=f(x)-b$ is obtained by translating the graph of $y=f(x)$ by $b$ units in the negative direction of the $Oy$ axis (downwards).

The graph of $y-3={{(x-2)}^{2}}$ is obtained from the graph of $y={{x}^{2}}$ by two translations.

Compressing and stretching graphs. Suppose the function $y=f(x)$ and the number $a>1$. Then:

1. the graph of the function $y=f(ax)$ is obtained by compressing the graph of $y=f(x)$ horizontally by a factor of $a$;

The graph of $g(x)=\sin (2\cdot x)$ is obtained from the graph of $f(x)=\sin (x)$ by a contraction along the x-axis.

2. the graph of the function $y=f(\frac{x}{a})$ is obtained by stretching the graph of $y=f(x)$ horizontally by a factor of $a$;

3. the graph of the function $y=af(x)$ is obtained by stretching the graph of $y=f(x)$ vertically by a factor of $a$;

The graphs of the functions $f(x)=\sin (x)$ and $g(x)=3\cdot \sin (x)$

The graph of the function $f(x)=a{{x}^{2}}$ for the following values of a = 1, 2, 3.

4. the graph of the function $y=\frac{f(x)}{a}$ is obtained by compressing the graph of $y=f(x)$ vertically by a factor of $a$;

Reflecting graphs in the coordinate axes. Suppose the function $y=f(x)$. Then:

1. the graph of the function $y=f(-x)$ is obtained by reflecting the graph of $y=f(x)$ in the $Oy$ axis;

2. the graph of the function $y=-f(x)$ is obtained by reflecting the graph of $y=f(x)$ in the $Ox$ axis.

In physics one meets the function: $f:\mathbb{R}\to \mathbb{R},\text{ }f(x)=A\cdot \cos (\omega x+b), \omega \ne 0$

$A=amplitudine,$ $\omega =pulsatie,$ $b=faza iniial\breve{a}.$ The fundamental period of the function is $T=\frac{2\pi }{\left| \omega \right|}.$

The graph of the function $f(x)=\sin (x-b)$ for $b=0, \frac{\pi }{4}, \pi , \frac{3\pi }{2}$

Try it

Every transformation in this chapter, on one curve.

A stretches it vertically (the amplitude), ω squeezes it horizontally (the angular frequency), φ slides it sideways (the phase), and d lifts it. This is the physics function from the end of the chapter, made movable.

Notice that changing φ alone moves the curve left or right without changing its shape — a translation along Ox. Changing d alone moves it up or down. Those are the first two rules of the chapter, and you can see them separately.

Take a through zero: the parabola flattens, then opens the other way. That is the reflection in the Ox axis the chapter describes.