Math Curve sketching Intersection and Zero of a function

# Intersection and Zero of a function

## Intersections with the x-axis

At the intersection with the x-axis, $y=0$. The general form is:

$S_x(x_N|0)$

$x_N$ is called the zero of a function.

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### Method

1. Set function equal to zero: $x_N\Leftrightarrow f(x_N)=0$
2. Solve the equation
3. Specify intersection(s)

### Example

$f(x)=x^2-9$

1. #### Set function equal to zero

$x^2-9=0$
2. #### Solve equation for $x$

$x^2-9=0\quad|+9$
$x^2=9\quad|\pm\sqrt{}$
$x_{ N1 }=+\sqrt{ 9 }=3$
$x_{ N2 }=-\sqrt{ 9 }=-3$
3. #### Specify intersections

$S_{x1}(3|0)$ und $S_{x2}(-3|0)$

## Intersection with the y-axis

At the intersection with the y-axis, $x=0$. The general form is:

$S_y(0|f(0))$
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### Remember

A function can have at most one intersection with the y-axis.
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### Method

1. Calculate $f(0)$
2. Specify intersection

### Example

$f(x)=x^2-9$

1. #### Calculate $f(0)$

$f(0)=0^2-9=-9$
2. #### Specify intersection

$S_{y}(0|-9)$