Points on the axes, positive and negative

Calculus, grade 11 (4 units) Lesson 5 of 20Learning

Points on the axes, positive and negative

The points on the axes

On the $\displaystyle x$ -axis: a fraction is 0 exactly when its numerator is 0 (and its denominator is not). Solve the equation “the numerator $\displaystyle {}={0}$ ”, and keep only the solutions in the domain.
On the $\displaystyle y$ -axis: substitute $\displaystyle {x}={0}$ , if 0 is in the domain. The point: $\displaystyle (0,f(0))$ . If 0 is not in the domain, the graph does not meet the $\displaystyle y$ -axis.

Example

$\displaystyle {f(x)}={\frac{x^2-4}{x-1}}$ , and the domain is $\displaystyle x\neq1$ .
In the example below we find the points on the axes:

The numerator: $\displaystyle {x^2-4}={0}$ , so $\displaystyle {x}={2}$ or $\displaystyle {x}={-2}$ . The points $\displaystyle (-2,0)$ and $\displaystyle (2,0)$ .
$\displaystyle {f(0)}={\frac{-4}{-1}}={4}$ . The point $\displaystyle (0,4)$ .

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

\displaystyle x=1

\displaystyle A=(-2,0)

\displaystyle B=(2,0)

\displaystyle C=(0,4)

Where the sign changes

A rational function can go from positive values to negative ones only at two kinds of points:

at a zero of the numerator – there the graph crosses the $\displaystyle x$ -axis;
at a zero of the denominator – there the graph jumps from one side of a vertical asymptote to the other.

Between two such points the sign does not change. So it is enough to check the sign at one number in each interval.

A sign table

In $\displaystyle {f(x)}={\frac{x^2-4}{x-1}}$ the points are $\displaystyle -2$ , $\displaystyle 1$ , $\displaystyle 2$ , and they split the line into four intervals.
In each interval check the sign of the numerator and of the denominator, and from them the sign of the fraction:

$\displaystyle x$	$\displaystyle x<-2$	$\displaystyle -2<x<1$	$\displaystyle 1<x<2$	$\displaystyle x>2$
$\displaystyle x^2-4$	$\displaystyle +$	$\displaystyle -$	$\displaystyle -$	$\displaystyle +$
$\displaystyle x-1$	$\displaystyle -$	$\displaystyle -$	$\displaystyle +$	$\displaystyle +$
$\displaystyle f(x)$	$\displaystyle -$	$\displaystyle +$	$\displaystyle -$	$\displaystyle +$

$\displaystyle f(x)>0$ when $\displaystyle -2<x<1$ or $\displaystyle x>2$ .
$\displaystyle f(x)\geq0$ adds the zeros of the numerator: $\displaystyle -2\leq x<1$ or $\displaystyle x\geq2$ .
$\displaystyle {x}={1}$ is never part of the solution: $\displaystyle f$ is not defined there.

Exercise 1

$\displaystyle {f(x)}={\frac{x+3}{x-1}}$ .
Mark on the board the points where the graph meets the axes.

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

Exercise 2

Solve the inequalities.

Exercise 3

Solve the inequality $\displaystyle x-\frac{4}{x}>0$ .
First write the left side as one fraction, over a common denominator.

Exercise 4

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

Mark on the board the points where the graph meets the axes.

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

On which intervals is $\displaystyle f(x)\geq0$ ?

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