Practice: one over a function and transformations

Calculus, grade 11 (4 units) Lesson 3 of 20Practice

Practice: one over a function and transformations

$\displaystyle {f(x)}={x^2-6x+8}$ , and $\displaystyle {g(x)}={\frac{1}{f(x)}}$ .

Solve the equation $\displaystyle {f(x)}={0}$ , and write the domain of $\displaystyle g$ .

Write the equations of the asymptotes of $\displaystyle g$ on the board, and mark its extremum point.

\displaystyle f(x)=x^{2}-6x+8

\displaystyle g(x)=\frac{1}{x^{2}-6x+8}

On which intervals is $\displaystyle g$ negative? On which intervals does it rise?

$\displaystyle {f(x)}={x^2+4x+5}$ , and $\displaystyle {g(x)}={\frac{1}{f(x)}}$ .

Mark the extremum point of $\displaystyle g$ .

\displaystyle g(x)=\frac{1}{x^{2}+4x+5}

For which values of $\displaystyle x$ is $\displaystyle {g(x)}={\frac{1}{2}}$ ?

What is the domain of $\displaystyle g$ ? How many asymptotes does it have, and what are they? On which intervals does it rise, and on which does it fall?

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

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

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

By which operations is $\displaystyle g$ made from the function $\displaystyle \frac{1}{x}$ ?

Write the equations of the asymptotes of $\displaystyle g$ on the board, and mark the point where the graph crosses the $\displaystyle x$ -axis.

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

On the board – the graph of $\displaystyle {g(x)}={\frac{1}{x^2-4}}$ , and beside it the graph of $\displaystyle h(x)$ , made from $\displaystyle g$ by one transformation.
Write $\displaystyle h(x)$ on the board.

\displaystyle g(x)=\frac{1}{x^{2}-4}

\displaystyle h(x)=\,?

$\displaystyle {g(x)}={\frac{1}{x^2-1}}$ .
Solve the equation $\displaystyle {\left|g(x)\right|}={1}$ .

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 $\displaystyle {x}={b}$
$\displaystyle S$ $\displaystyle =$ $\displaystyle \int_{a}^{b}\left(f(x)-g(x)\right)dx$ $\displaystyle f(x)\geq g(x)$