Graphs Turn Relationships into Geometry

A graph represents paired values spatially. Horizontal position records the input; vertical position records the output. Geometry then reveals rate, intercepts, trends, and intersections.

Learning objectives

You will plot ordered pairs, interpret axes and scale, calculate slope, move among linear forms, and read solutions from intersections.

Coordinates require context

The point $(3,7)$ means input $3$, output $7$. Axis labels and units are part of the mathematics. A point at horizontal coordinate $3$ means something different when the axis is seconds rather than kilograms.

Slope is a ratio of changes

Between $(x_1,y_1)$ and $(x_2,y_2)$,

$$m=\frac{y_2-y_1}{x_2-x_1}=\frac{\Delta y}{\Delta x}.$$

Slope units are output units per input unit. Interchanging point order changes both signs and leaves the ratio unchanged.

Three useful forms

• $y=mx+b$ exposes slope and vertical intercept.
• $y-y_1=m(x-x_1)$ emphasizes a known point and slope.
• $Ax+By=C$ treats variables symmetrically and makes intercepts easy to find.

Choose the form that reveals the needed information rather than converting automatically.

Scale can mislead

Changing an axis scale changes visual steepness without changing the underlying rate. Truncated axes can exaggerate small differences. Always read numerical scales before interpreting shape.

Check your understanding

A line has slope $-3$ and passes through $(1,4)$. Find its equation and both intercepts.

Show the reasoning

$y-4=-3(x-1)$, so $y=-3x+7$. The $y$-intercept is $(0,7)$. Setting $y=0$ gives $x=7/3$, so the $x$-intercept is $(7/3,0)$.