Transformations of a graph

Calculus, grade 11 (4 units) Lesson 2 of 20Learning

Transformations of a graph

A reminder from grade 10

In grade 10 you shifted and stretched graphs of polynomials, for example of $\displaystyle {y}={x^2}$ :

$\displaystyle {y}={(x-3)^2}$ – the graph moves 3 units to the right;
$\displaystyle {y}={x^2+2}$ – the graph moves 2 units up;
$\displaystyle {y}={-x^2}$ – the graph is reflected in the $\displaystyle x$ -axis;
$\displaystyle {y}={3x^2}$ – the graph is stretched by 3 along the $\displaystyle y$ -axis.

The same operations work on any function, and on $\displaystyle {g(x)}={\frac{1}{f(x)}}$ too.

The transformations of

\displaystyle \frac{1}{f(x)}

$\displaystyle {g(x)}={\frac{1}{f(x)}}$ , and each operation gives a new function:

A horizontal shift: $\displaystyle {y}={\frac{1}{f(x-p)}}$ – the graph moves $\displaystyle p$ units to the right ( $\displaystyle p>0$ ) or to the left ( $\displaystyle p<0$ ).
A vertical shift: $\displaystyle {y}={\frac{1}{f(x)}+k}$ – the graph moves $\displaystyle k$ units up ( $\displaystyle k>0$ ) or down ( $\displaystyle k<0$ ).
A reflection in the $\displaystyle x$ -axis: $\displaystyle {y}={-\frac{1}{f(x)}}$ – every value changes its sign.
A reflection in the $\displaystyle y$ -axis: $\displaystyle {y}={\frac{1}{f(-x)}}$ – what was at $\displaystyle {x}={3}$ is now at $\displaystyle {x}={-3}$ .
A stretch or a squeeze: $\displaystyle {y}={\frac{a}{f(x)}}$ – every value is multiplied by $\displaystyle a$ : a stretch when $\displaystyle a>1$ , a squeeze when $\displaystyle 0<a<1$ .

The operations can be combined:

\displaystyle y=\frac{a}{f(x-p)}+k

The asymptotes move with the graph

An asymptote is part of the graph’s picture, so it goes through the same operation:

A shift to the right by $\displaystyle p$ : a vertical asymptote $\displaystyle {x}={c}$ moves to $\displaystyle {x}={c+p}$ .
A shift up by $\displaystyle k$ : the horizontal asymptote $\displaystyle {y}={0}$ moves to $\displaystyle {y}={k}$ .
A reflection in the $\displaystyle y$ -axis: $\displaystyle {x}={c}$ moves to $\displaystyle {x}={-c}$ .
A reflection in the $\displaystyle x$ -axis, a stretch and a squeeze: the asymptotes stay – $\displaystyle \frac{a}{0}$ is still undefined, and $\displaystyle a$ times a number close to 0 is still close to 0.

Example

$\displaystyle {f(x)}={\frac{1}{x}}$ has two asymptotes: $\displaystyle {x}={0}$ and $\displaystyle {y}={0}$ .
In the example below we move the graph 2 units to the right and 1 unit up:

$\displaystyle {g(x)}={\frac{1}{x-2}+1}$ ;
the vertical asymptote moves from $\displaystyle {x}={0}$ to $\displaystyle {x}={2}$ ;
the horizontal asymptote moves from $\displaystyle {y}={0}$ to $\displaystyle {y}={1}$ ;
the domain of $\displaystyle g$ : $\displaystyle x\neq2$ .

\displaystyle f(x)=\frac{1}{x}

\displaystyle g(x)=\frac{1}{x-2}+1

\displaystyle x=2

\displaystyle y=1

Every point of $\displaystyle f$ ’s graph moves in the same way: $\displaystyle (1,1)$ moves to $\displaystyle (3,2)$ , and $\displaystyle (-1,-1)$ moves to $\displaystyle (1,0)$ – the point where $\displaystyle g$ crosses the $\displaystyle x$ -axis.

Absolute value

$\displaystyle {y}={\left|\frac{1}{f(x)}\right|}$ : where the value is positive it stays, and where it is negative it changes its sign.
The part of the graph below the $\displaystyle x$ -axis is reflected above it, and the part above it does not change. The asymptotes stay.
On the board: $\displaystyle {g(x)}={\frac{1}{x-1}}$ (dashed) and $\displaystyle {h(x)}={\left|\frac{1}{x-1}\right|}$ .

\displaystyle g(x)=\frac{1}{x-1}

\displaystyle h(x)=\left|\frac{1}{x-1}\right|

Exercise 1

On the board – the graph of $\displaystyle {f(x)}={\frac{1}{x}}$ .
It is moved 3 units to the right and 2 units down.
Write the new function on the board, and the equations of its two asymptotes.

\displaystyle f(x)=\frac{1}{x}

Exercise 2

On the board – the graph of $\displaystyle {g(x)}={\frac{1}{x^2+1}}$ , and beside it the graph of $\displaystyle h(x)$ .
$\displaystyle h$ was made from $\displaystyle g$ by a reflection in the $\displaystyle x$ -axis and a stretch by 2.
Write $\displaystyle h(x)$ on the board.

\displaystyle g(x)=\frac{1}{x^{2}+1}

\displaystyle h(x)=\,?

Exercise 3

$\displaystyle {g(x)}={\frac{3}{x+1}-2}$ .

Which function $\displaystyle \frac{1}{f(x)}$ was $\displaystyle g$ made from, and by which operations?

Write the equations of the asymptotes of $\displaystyle g$ on the board.

\displaystyle g(x)=\frac{3}{x+1}-2

Exercise 4

On the board – the graph of $\displaystyle {g(x)}={\frac{1}{x-2}}$ .
Write on the board the function made by reflecting $\displaystyle g$ in the $\displaystyle y$ -axis, and the equation of its vertical asymptote.

\displaystyle g(x)=\frac{1}{x-2}

Exercise 5

On the board – the graph of $\displaystyle {g(x)}={\frac{1}{x^2-1}}$ .
Write $\displaystyle {y}={\left|g(x)\right|}$ on the board.

\displaystyle g(x)=\frac{1}{x^{2}-1}

On which intervals is the graph of $\displaystyle \left|g(x)\right|$ different from the graph of $\displaystyle g(x)$ ? What is its extremum point on the interval $\displaystyle -1<x<1$ ?

Formulas: calculus, grade 11 (4 units)

Open as a page

Transformations of the graph of $f$

A horizontal shift: $\displaystyle p$ units to the right ( $\displaystyle p>0$ ) or to the left ( $\displaystyle p<0$ )
$\displaystyle y=f(x-p)$
A vertical shift: $\displaystyle k$ units up ( $\displaystyle k>0$ ) or down ( $\displaystyle k<0$ )
$\displaystyle y=f(x)+k$
A reflection in the $\displaystyle x$ -axis
$\displaystyle y=-f(x)$
A reflection in the $\displaystyle y$ -axis
$\displaystyle y=f(-x)$
A vertical stretch ( $\displaystyle a>1$ ) or squeeze ( $\displaystyle 0<a<1$ )
$\displaystyle y=a\cdot f(x)$
Absolute value: the part below the $\displaystyle x$ -axis is reflected above it
$\displaystyle y=|f(x)|$

The rational function

The horizontal asymptote when the numerator and the denominator have the same degree – $\displaystyle a$ and $\displaystyle b$ the coefficients of the highest power
$\displaystyle y=\frac{a}{b}$
The horizontal asymptote when the numerator’s degree is lower than the denominator’s
$\displaystyle y=0$
An even function – the graph is symmetric about the $\displaystyle y$ -axis
$\displaystyle f(-x)$ $\displaystyle =$ $\displaystyle f(x)$
An odd function – the graph is symmetric about the origin
$\displaystyle f(-x)$ $\displaystyle =$ $\displaystyle -f(x)$

Derivatives and the tangent

The derivative of a power
$\displaystyle \left(x^{n}\right)'=nx^{n-1}$
The derivative of a product
$\displaystyle \left(u\cdot v\right)'$ $\displaystyle =$ $\displaystyle u'\cdot v+u\cdot v'$
The derivative of a quotient
$\displaystyle \left(\frac{u}{v}\right)'$ $\displaystyle =$ $\displaystyle \frac{u'\cdot v-u\cdot v'}{v^{2}}$
The derivative of $\displaystyle \frac{c}{x}$
$\displaystyle \left(\frac{c}{x}\right)'$ $\displaystyle =$ $\displaystyle -\frac{c}{x^{2}}$
The derivative of a root – $\displaystyle u$ a function of $\displaystyle x$
$\displaystyle \left(\sqrt{u}\right)'$ $\displaystyle =$ $\displaystyle \frac{u'}{2\sqrt{u}}$
The equation of the tangent at the point $\displaystyle (x_1,y_1)$
$\displaystyle y-y_1$ $\displaystyle =$ $\displaystyle m(x-x_1)$
The slope of the tangent
$\displaystyle m$ $\displaystyle =$ $\displaystyle f'(x_1)$

Investigating a function with its derivative

The function rises
$\displaystyle f'(x)>0$
The function falls
$\displaystyle f'(x)<0$
A possible extremum point – check the sign of $\displaystyle f'$ on each side of it
$\displaystyle f'(x)=0$

Integrals and areas

The integral of a power
$\displaystyle \int x^{n}\,dx$ $\displaystyle =$ $\displaystyle \frac{x^{n+1}}{n+1}+C$ $\displaystyle n\neq-1$
The integral of $\displaystyle \frac{c}{(ax+b)^{2}}$
$\displaystyle \int\frac{c}{(ax+b)^{2}}\,dx$ $\displaystyle =$ $\displaystyle -\frac{c}{a(ax+b)}+C$
The definite integral – $\displaystyle F$ an antiderivative of $\displaystyle f$
$\displaystyle \int_{a}^{b}f(x)\,dx$ $\displaystyle =$ $\displaystyle F(b)-F(a)$
The area between two graphs, from $\displaystyle {x}={a}$ to $\displaystyle {x}={b}$
$\displaystyle S$ $\displaystyle =$ $\displaystyle \int_{a}^{b}\left(f(x)-g(x)\right)dx$ $\displaystyle f(x)\geq g(x)$