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

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