Function plotter with custom terms

What is it about?

A function plotter draws the graph of an expression. Type up to three expressions $f(x)$ , $g(x)$ and $h(x)$ . The simulation typesets each one, so you can see how your input was understood, and draws the graph. If there is a typo, the spot is marked and explained.

Input	means
`2x`, `3(x+1)`	$2x$ , $3(x+1)$ – the multiplication sign may be left out
`x^2`, `2^x`, `x^(1/2)`	$x^2$ , $2^x$ , $x^{\frac{1}{2}}$
`1/(x-1)`	$\dfrac{1}{x-1}$ – put the denominator in brackets
`sqrt(x)`, `abs(x)`	$\sqrt{x}$ , $\lvert x \rvert$
`sin(x)`, `e^x`, `ln(x)`, `log_2(x)`	angles in radians
`pi`, `e`, `a`, `b`	constants and parameters (sliders appear)

The usual order of operations applies, so 1/2x means $\frac{1}{2} \cdot x$ ; for $\frac{1}{2x}$ type 1/(2x).

The simulation can mark zeros, intersections (solutions of $f(x) = g(x)$ ), maxima and minima, poles (dashed asymptotes) and holes in the visible window. These are approximations rounded to two decimal places ( $\approx$ ). Unlike a naive plotter, it never joins the two sides of a pole with a vertical line – that line would not belong to the graph.

Try it

Load parabola and sine and drag the traced point. Roughly which numbers solve $x^2 - 2 = 2 \sin x$ ?
Type 1/x for $h(x)$ , pan and zoom.
Make deliberate mistakes such as 2x+ or sinus(x).
Change $a$ in the family of parabolas.
Compare x² and 2ˣ: how many intersections are there?
Look at (x² − 1)/(x − 1) near $x = 1$ .

Tasks

Task 1: Zeros

Load task Use the trace to find the zeros of $f(x) = x^3 - 2x - 1$ to two decimal places.

Show solution

$x_1 = -1$ exactly, $x_2 \approx -0.62$ , $x_3 \approx 1.62$ (exactly $\frac{1 \pm \sqrt{5}}{2}$ ). Show

Task 2: Solving graphically

Load task Solve $2^x = 3 - x$ graphically. Why is there only one solution?

Show solution

$x = 1$ , since $2^1 = 2 = 3 - 1$ . $2^x$ always increases and $3 - x$ always decreases, so the graphs meet only once.

Task 3: Pole or hole?

Load task Both functions are undefined at $x = 2$ . How do the graphs differ there?

Show solution

$f(x) = x + 2$ for $x \neq 2$ : a line with a hole at $(2, 4)$ . $g$ cannot be simplified; its values grow without bound: a pole with asymptote $x = 2$ . Show

Ideas for the classroom

Use: from grade 8 (functions and their graphs) up to calculus and university; tangents and families of functions for upper grades.
Expressions are evaluated by a custom parser (no eval). In task links, hide the result lists with _hide=1 and switch off the marks in the picture (nst=0, sct=0, ext=0, asy=0).
All special points are numerical approximations within the window; very flat turning points may be missed.
Common stumbling blocks: the vertical line at a pole is part of the graph; the computer gives exact values; 1/2x instead of 1/(2x); expecting degrees in sin(x).