Function & graph explorer

Drag a dot or a slider and watch the graph change, with the roots, vertex and working updating as you go. Plot lines, quadratics and polynomials, type any function, solve a pair of equations by graph, and try the distance and section formulas on points you place yourself.

Lessons

Pick your class, follow the steps in the lab, and check what you see against the working under the graph.

Class 9 — Coordinate geometry and the graph of a line

Concept: every point on the plane has coordinates (x, y). A linear equation $y = mx + c$ always draws a straight line: m is the slope (rise ÷ run) and c is where the line crosses the y-axis.

Open Line mode. Set m = 1 and c = 0: the line passes through the origin.
Move c to 3. The whole line slides up by 3 and now crosses the y-axis at (0, 3).
Change m to 2, then to 0.5. The line gets steeper, then flatter. Read the rise and the run in the little triangle.
Set m = −2. The line now falls as you move right: a negative slope.
Drag the two dots on the graph instead of using sliders, and watch the equation update.

Takeaway: slope = rise ÷ run, and c is the height at which the line meets the y-axis.

Class 10 — Pair of linear equations and zeros of a quadratic

Concept: the solution of two linear equations is the point where their lines meet. Two lines can meet once (one solution), never (parallel: no solution) or everywhere (the same line: infinitely many). For a quadratic $ax^2 + bx + c$ the zeros are where the parabola meets the x-axis, and the discriminant $D = b^2 - 4ac$ says how many there are.

Open Pair of lines and press one solution. Find the meeting point, then check the numbers under the graph.
Press no solution (parallel). Compare a₁/a₂, b₁/b₂ and c₁/c₂: the first two are equal but the third is not.
Press infinitely many (same line). Now all three ratios match.
Open Quadratic and press x² − 5x + 6. The zeros are 2 and 3, and D = 1.
Change c until D = 0 (the curve just touches the axis), then past it (D < 0, no real zeros).
Check the sum of zeros = −b/a and the product of zeros = c/a in the working.

Takeaway: D > 0 gives two real zeros, D = 0 gives one, D < 0 gives none.

Class 10 — Polynomials: zeros and the shape of the graph

Concept: a real zero of a polynomial is where its graph meets the x-axis. A cubic can have up to 3 real zeros and a quartic up to 4. If the zeros are $\alpha, \beta, \gamma$ then $\alpha + \beta + \gamma = -\tfrac{b}{a}$ .

Open Polynomial and choose Drag the zeros with a cubic.
Drag the three dots along the x-axis. The graph always passes through them.
Drag two dots together: the graph only touches the axis there (a repeated zero).
Check the sum and product of the zeros against the coefficients in the working.

Takeaway: the zeros decide where the curve crosses the axis, and they are tied to the coefficients.

Class 11 — Functions, symmetry and coordinate geometry

Concept: a function sends each x to one value f(x). It is even if $f(-x) = f(x)$ (symmetric about the y-axis) and odd if $f(-x) = -f(x)$ . For points A(x₁, y₁) and B(x₂, y₂), the distance is $AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ .

Open Any function and try x², then x³ − 3x. Read the Symmetry card: even, then odd.
Choose a·sin(bx)+c and move the sliders. a stretches it up and down, b squeezes it sideways, c lifts it.
Try 1/x and tan x: the graph breaks where the function is undefined.
Open Coordinate tools. Drag A and B, and check the distance against the working.
Set m : n = 2 : 3 and find the point P that divides AB internally in that ratio.

Takeaway: symmetry can be read from a graph, and the section formula gives the point that splits a segment in a given ratio.

Class 12 — One-one functions, inverses and a first look at slope

Concept: a function has an inverse only if it is one-one (no horizontal line meets the graph twice). The graph of the inverse is the reflection in the line $y = x$ . The slope of the tangent to $y = f(x)$ at a point is the derivative $f'(x)$ .

Open Any function, type x^3 and tick show the reflection in y = x. The reflection is again a function.
Type x^2: the reflection is a sideways parabola, which is not a function of x. So x² is not one-one on all reals.
Tick show the tangent and drag the moving point. Read f′ at each position.
Tick show the derivative f′(x). Where does f′ cross zero? That is where the tangent is flat.

Takeaway: reflecting in y = x reveals the inverse, and the derivative is the slope of the tangent.

The maths behind it

What each mode covers

Line —

y = mx + c

with two draggable dots. Slope as rise ÷ run, the angle with the x-axis, and both intercepts. Class 9 coordinate geometry and linear equations.

Quadratic —

y = ax^2 + bx + c

with a draggable vertex, the discriminant, real or complex zeros, standard, vertex and factored forms, and the sum and product of zeros. Class 10 quadratics and class 11 complex numbers.

Polynomial — cubics and quartics from coefficient sliders or by dragging their zeros, with turning points, end behaviour and the relations between zeros and coefficients. Class 9–10 polynomials.

Any function — type your own f(x) with sliders for a, b, c, then read its zeros, symmetry (even or odd), tangent and derivative, and reflect it in y = x to see the inverse. Class 11–12 functions.

Pair of lines — two linear equations drawn together. One solution, none or infinitely many, with the ratio test and Cramer's rule shown step by step. Class 10 pair of linear equations.

Coordinate tools — drag points A, B and C for the distance, midpoint, slope, section formula, area of a triangle and centroid. Class 9–11 coordinate geometry and straight lines.

Part of Shunya. See the syllabus map for the chapters these modes map to. Every number on this page is computed live from what you drag.

Formulas used in this lab

Coordinate geometry and the graph of a line Class 9

Equation of a line: $y = mx + c$
Slope (rise ÷ run): $m = \dfrac{y_2 - y_1}{x_2 - x_1}$
y-intercept (put x = 0): $(0,\ c)$
x-intercept (put y = 0): $x = -\dfrac{c}{m}$
Angle the line makes with the x-axis: $\theta = \tan^{-1} m$

Pair of linear equations and zeros of a quadratic Class 10

Pair of linear equations: $a_1x + b_1y = c_1,\qquad a_2x + b_2y = c_2$
Determinants (Cramer's rule): $D = a_1b_2 - a_2b_1,\quad D_x = c_1b_2 - c_2b_1,\quad D_y = a_1c_2 - a_2c_1$
The one solution when D ≠ 0: $x = \dfrac{D_x}{D},\qquad y = \dfrac{D_y}{D}$
One solution (lines cross): $\dfrac{a_1}{a_2} \ne \dfrac{b_1}{b_2}$
No solution (parallel lines): $\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} \ne \dfrac{c_1}{c_2}$
Infinitely many (same line): $\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} = \dfrac{c_1}{c_2}$
Quadratic: $y = ax^2 + bx + c$
Discriminant: $D = b^2 - 4ac$
Zeros (D > 0): $x = \dfrac{-b \pm \sqrt{D}}{2a}$
Repeated zero (D = 0): $x = -\dfrac{b}{2a}$
Complex roots (D < 0): $x = \dfrac{-b \pm i\sqrt{-D}}{2a}$
Sum and product of the zeros: $\alpha + \beta = -\dfrac{b}{a},\qquad \alpha\beta = \dfrac{c}{a}$
Vertex form: $y = a(x - h)^2 + k,\quad h = -\dfrac{b}{2a},\ k = c - \dfrac{b^2}{4a}$
Factored form: $y = a(x - \alpha)(x - \beta)$

Polynomials: zeros and the shape of the graph Class 10

Cubic: $y = ax^3 + bx^2 + cx + d$
Sum of the zeros: $\alpha + \beta + \gamma = -\dfrac{b}{a}$
Sum of products of pairs: $\alpha\beta + \beta\gamma + \gamma\alpha = \dfrac{c}{a}$
Product of the zeros: $\alpha\beta\gamma = -\dfrac{d}{a}$
Polynomial built from its zeros: $y = a(x - \alpha)(x - \beta)(x - \gamma)\cdots$
Number of real zeros of a degree-n polynomial: $\le n$

Functions, symmetry and coordinate geometry Class 11

Even function (symmetric about the y-axis): $f(-x) = f(x)$
Odd function (half-turn symmetry about the origin): $f(-x) = -f(x)$
Stretched sine wave: $y = a\sin(bx) + c$
Amplitude, period, vertical lift: $|a|,\qquad \dfrac{2\pi}{|b|},\qquad c$
Where a function is undefined: $\dfrac{1}{x}:\ x = 0, \qquad \tan x:\ x = \dfrac{\pi}{2} + n\pi$
Distance between two points: $AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$
Midpoint: $M = \left(\dfrac{x_1 + x_2}{2},\ \dfrac{y_1 + y_2}{2}\right)$
Section formula (internal, m : n): $P = \left(\dfrac{mx_2 + nx_1}{m + n},\ \dfrac{my_2 + ny_1}{m + n}\right)$
Area of a triangle: $\tfrac{1}{2}\,\bigl|\,x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)\,\bigr|$

One-one functions, inverses and a first look at slope Class 12

Inverse function: $f^{-1}\bigl(f(x)\bigr) = x$
Reflection in y = x: $(a,\ b) \ \longleftrightarrow\ (b,\ a)$
One-one test: $f(x_1) = f(x_2) \Rightarrow x_1 = x_2$
Derivative (slope of the tangent): $f'(x) = \lim_{h \to 0} \dfrac{f(x + h) - f(x)}{h}$
Tangent line at x₀: $y - f(x_0) = f'(x_0)\,(x - x_0)$
Power rule: $\dfrac{d}{dx}\,x^n = n x^{n-1}$
Sum, product and quotient rules: $(u \pm v)' = u' \pm v',\quad (uv)' = u'v + uv',\quad \left(\dfrac{u}{v}\right)' = \dfrac{u'v - uv'}{v^2}$
Chain rule: $\dfrac{d}{dx}\,f\bigl(g(x)\bigr) = f'\bigl(g(x)\bigr)\cdot g'(x)$
Standard derivatives: $(\sin x)' = \cos x,\ (\cos x)' = -\sin x,\ (\tan x)' = \sec^2 x,\ (e^x)' = e^x,\ (\ln x)' = \dfrac{1}{x}$