Working with Graphs

Functions, Patterns, and Rules

A function gives each permitted input from its domain exactly one output in its codomain; the range contains the outputs actually reached. The notations $f(x) = 2x + 1$ and $y = 2x + 1$ describe the same rule. For equally spaced inputs:

• Linear: constant first differences.
• Quadratic: constant non-zero second differences. For pattern terms $3, 6, 11, 18$ at positions $n = 1, 2, 3, 4$, the second difference is $2$. Half the second difference is $1$, the coefficient of $n^2$. Subtracting $n^2$ leaves $2$, giving $T_n = n^2 + 2$.
• Exponential: outputs multiply by a fixed ratio each step (such as doubling or tripling).

Plotting Functions and Domain Rules

Make a table of values by substituting each domain value into the function rule. Always use brackets when squaring negative inputs: $(-3)^2 = 9$.

If $x \in \mathbb{R}$, connect the points with a continuous line or smooth curve. If $x \in \mathbb{N}$ or $\mathbb{Z}$, plot each point as a separate dot and leave them unconnected.

Coordinate Geometry of the Line

For points $A(x_1, y_1)$ and $B(x_2, y_2)$:

• Distance: $|AB| = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.
• Midpoint: $\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$.
• Slope: $m = \frac{y_2 - y_1}{x_2 - x_1}$. Horizontal lines have slope $0$; vertical lines have undefined slope.
• Parallel and perpendicular: non-vertical parallel lines have equal slopes; vertical lines are parallel. Perpendicular lines with defined slopes satisfy $m_1 m_2 = -1$. Horizontal and vertical lines are perpendicular.
• Line equation: use $y - y_1 = m(x - x_1)$. A line with slope $3$ through $(2, 5)$ gives $y - 5 = 3(x - 2)$, simplifying to $y = 3x - 1$.
• Rearranging: to find the slope and y-intercept of $ax + by + c = 0$, isolate $y$. For $2x + 4y - 8 = 0$, $4y = -2x + 8$, giving $y = -\frac{1}{2}x + 2$, with slope $-\frac{1}{2}$ and intercept $(0, 2)$.

Reading Answers and Interpreting Graphs

For a straight segment, rate of change is vertical change divided by horizontal change. With distance travelled vertically and time horizontally, slope gives speed; a horizontal segment means stationary.

State $x$-values only when solving equations in $x$.

Intersections and Inequalities

When comparing two functions $f(x)$ and $g(x)$:

• Solve $f(x) = g(x)$: read the $x$-values of the intersection points.
• Point of intersection: state the full coordinate pair $(x, y)$. This solves simultaneous equations graphically.
• Solve $f(x) < g(x)$: find all $x$-values where the graph of $f$ lies below the graph of $g$. Give all permitted $x$-values, using one or more intervals or separate values as appropriate; strict inequalities exclude equality points.
• Number lines: for linear inequalities like $3x - 1 \le 8$, simplify to $x \le 3$. Draw a solid dot for $\le$ or $\ge$ and an open circle for $<$ or $>$. Shade a solid line with an arrow for $x \in \mathbb{R}$, but mark isolated dots for $x \in \mathbb{N}$ or $\mathbb{Z}$.

Key terms

Domain

The set of all permitted input values ($x$-values) for a function.

Range

The set of actual output values ($y$-values) generated by a function. Codomain: the stated target set containing all outputs.

Parabola

The symmetrical U-shaped or inverted U-shaped curve created by a quadratic function.

Turning point

The vertex where a curve changes from increasing to decreasing (maximum) or decreasing to increasing (minimum).

Root

An input value $x$ for which $f(x) = 0$, found where the graph meets the $x$-axis.

Rate of change

The change in the vertical quantity divided by the change in the horizontal quantity, represented by the slope of a line segment.

y-intercept

The coordinate point $(0, c)$ where a line or curve crosses the vertical axis.