Graphing Functions lessons

The function machine

Plug in different values and watch the function produce outputs. This builds intuition for how inputs and outputs relate.

Interactive function machine: slide the input and cycle functions to see the output computed.

Practice: Evaluate a function

Type the output of a randomised linear or quadratic function at a given input.

The coordinate plane

The coordinate plane is a flat surface formed by two number lines crossing at right angles. The horizontal axis is called x and the vertical axis is called y.

The point where they cross is the origin, written (0, 0). Every point on the plane has an address written as (x, y).

Coordinate-plane explorer: click to place points and read off their coordinates.

Practice: Plot the point

Click the grid to plot a target coordinate across five scored rounds.

Linear functions

A linear function has the form y = mx + b. Its graph is always a straight line.

The number m is the slope. It tells you how steep the line is and whether it rises or falls. The number b is the y-intercept, where the line crosses the y-axis.

Slope-intercept form: y = mx + b

Slope and intercept explorer

The slope m measures rise over run. A positive slope goes uphill, a negative slope goes downhill, and zero slope gives a flat line.

1. m = \frac{\textcolor{#f871b9}{rise}}{\textcolor{#8b5cf6}{run}} = \frac{\textcolor{#f871b9}{y_2 - y_1}}{\textcolor{#8b5cf6}{x_2 - x_1}} (slope formula)

Slope and intercept sliders morph y = mx + b live with rise/run arrows.

Practice: Find the slope

Compute the slope through two plotted points, typed as a fraction or decimal.

Transformations: Shifting graphs

You can move any graph without changing its shape. Adding a number outside the function shifts it vertically. Adding a number inside the function shifts it horizontally.

Vertical shifts are intuitive: f(x) + k moves the graph up by k units. Horizontal shifts feel backwards: f(x - h) moves the graph right by h units.

1. f(x) + k (shift up by k)
2. f(x) - k (shift down by k)
3. f(x - h) (shift right by h)
4. f(x + h) (shift left by h)

Transformations: Stretches and reflections

Multiplying the output by a constant stretches or compresses the graph vertically. af(x) stretches by a factor of a when a > 1 and compresses when 0 < a < 1.

Negating the output, -f(x), reflects the graph across the x-axis. Negating the input, f(-x), reflects it across the y-axis.

Stretch, shift and reflect a parabola while the grey original stays fixed.

Practice: Identify the transformation

Pick which transformation maps the dashed base graph onto the pink one.

Quadratic functions

A quadratic function has the form y = ax^2 + bx + c. Its graph is a U-shaped curve called a parabola.

When a > 0 the parabola opens upward. When a < 0 it opens downward. The turning point is called the vertex.

Standard form: y = ax^2 + bx + c

Anatomy of a parabola

Every parabola has a vertex, an axis of symmetry, and zero, one, or two x-intercepts (roots). The vertex sits at the peak or trough of the curve.

1. \textcolor{#f871b9}{x_{vertex}} = \frac{-b}{2a} (x-coordinate of vertex)
2. \textcolor{#8b5cf6}{y_{vertex}} = f\left(\frac{-b}{2a}\right) (substitute back for y)

a, b, c sliders reshape a parabola with live vertex, axis of symmetry and roots.

Practice: Match the quadratic

Click the equation that matches a plotted parabola from four options.

Cubic and polynomial functions

A polynomial is a sum of terms where each term is a constant times a power of x. The highest power is called the degree.

Linear functions have degree 1, quadratics have degree 2, and cubics have degree 3. Higher degree polynomials have more bends in their graphs.

A cubic function: y = x^3 - 3x

Cubic graphs have an S-shape. They can cross the x-axis up to three times and always extend to both positive and negative infinity.

Polynomial shape explorer

The degree of a polynomial controls how many times the graph can change direction. A degree-n polynomial has at most n - 1 turning points.

Pick a degree and scale the leading coefficient to see roots and turning points.

Practice: Identify the degree

Read a polynomial graph and press its degree from 1 to 4.

Reciprocal and rational functions

A reciprocal function like y = \frac{1}{x} produces a curve that never touches the axes. The axes act as invisible barriers called asymptotes.

An asymptote is a line the graph approaches but never reaches. For y = \frac{1}{x}, both the x-axis and y-axis are asymptotes.

1. y = \frac{1}{x} (basic reciprocal)
2. x \to \infty,\ \ y \to 0 (approaches x-axis)
3. x \to 0,\ \ y \to \pm\infty (approaches y-axis)

Asymptotes in action

When we shift or scale a reciprocal function, the asymptotes move too. The general form y = \frac{a}{x - h} + k has a vertical asymptote at x = h and a horizontal asymptote at y = k.

Move a reciprocal curve while its vertical and horizontal asymptotes track along.

Practice: Find the asymptotes

Type the vertical and horizontal asymptotes of a randomised rational function.

Exponential functions

An exponential function has the variable in the exponent: y = a^x. These functions grow (or decay) much faster than polynomials.

When a > 1, the function grows rapidly as x increases. When 0 < a < 1, it decays toward zero. The graph never touches the x-axis.

Exponential function: y = a^x,\ \ a > 0

Growth and decay curves

Exponential growth appears in population models, compound interest, and radioactive decay. The base a controls how fast the curve rises or falls.

1. y = \textcolor{#f871b9}{2}^x (doubles each step)
2. y = \textcolor{#8b5cf6}{\left(\frac{1}{2}\right)}^x (halves each step)

Change the base a to compare exponential growth against decay with a live value table.

Practice: Exponential or not?

Classify a table of values as linear, quadratic or exponential from its differences.

Square root functions

The square root function y = \sqrt{x} is defined only for x \geq 0. Its graph starts at the origin and curves gently upward, growing slower and slower.

It is the inverse of y = x^2 (restricted to x \geq 0). Transformations work the same way: y = \sqrt{x - h} + k shifts the starting point to (h, k).

Shift the square-root curve and watch its starting point move against x² and y = x.

Practice: Sketch the function type

Click the graph thumbnail whose shape matches a given equation across five rounds.

Domain and range

The domain of a function is the set of all valid inputs (x-values). The range is the set of all possible outputs (y-values).

Some functions accept any input. Others have restrictions, like y = \frac{1}{x} which cannot take x = 0, or y = \sqrt{x} which requires x \geq 0.

Finding domain and range visually

On a graph, the domain is the horizontal extent (how far left and right the curve stretches) and the range is the vertical extent (how far up and down it goes).

1. y = x^2 (domain: all real numbers)
2. (range: y \geq 0)
3. y = \sqrt{x} (domain: x \geq 0)
4. (range: y \geq 0)

Reveal domain and range strips on the graph to see a function's horizontal and vertical extent.

Practice: State the domain and range

Type the domain and range of a randomised function in interval notation.

Reading graphs: Intercepts and zeros

The y-intercept is where the graph crosses the y-axis (set x = 0). The x-intercepts (or zeros) are where the graph crosses the x-axis (set y = 0).

These key points tell you a lot about a function at a glance. Every function type has its own pattern of intercepts.

Reading graphs: Increasing and decreasing

A function is increasing where the graph goes uphill from left to right, and decreasing where it goes downhill. At turning points, the function switches direction.

Hover across a curve to see where it is increasing (green) or decreasing (pink).

Practice: Read the graph

Answer a random question about a plotted curve: intercepts, direction or maximum.