Write linear functions and equations of lines using slope-intercept and point-slope form, from a graph, two points, or a real-world description (TN A1.F.LE.A.2, A1.A.CED.A.2).

What this topic is asking

This page covers writing a linear function from various starting points: a slope and a point, two points, a graph, or a context (A1.F.LE.A.2 and A1.A.CED.A.2). The tools are the slope formula and the slope-intercept and point-slope forms. The reference sheet gives the slope formula, but the line forms are not on it, so memorize them.

The slope formula and the line forms

Slope-intercept is best when you know the slope and the $y$-intercept; point-slope is best when you know the slope and any point.

Writing a line from two points

The reliable two-step routine: find the slope, then substitute a point into point-slope.

Writing a line from a context

In a word problem, the constant rate is the slope and the fixed/starting value is the $y$-intercept. "$15$ per month with a $40$ setup fee" is $y = 15x + 40$. If you are given a rate and a non-starting point, use point-slope and simplify, the pool example does exactly this with a negative slope for draining.

How TNReady examines this topic

• Equation response. Write a line in slope-intercept or point-slope form, scored by exact match.
• Numeric response. Find the slope, the $y$-intercept, or a starting value from a context.
• Multiple choice. Choose the equation matching a graph or table, with slope-sign and intercept distractors.

A clarifying idea is that the slope is the average rate of change of a linear function, constant on every interval, and in a data setting it is the slope of the linear model fitted to points. The same number appears across the Functions and Statistics categories.

Why two forms, and when to convert

Slope-intercept and point-slope describe the same line, so why keep both? Because each is fastest for a different starting point. If you already know where the line crosses the $y$-axis, slope-intercept is immediate. If you know the slope and some other point, point-slope lets you write the equation in one step without solving for $b$. The test often supplies a slope and a point that is not the $y$-intercept, which is exactly when point-slope shines, you plug in and simplify. Converting between the forms is just algebra: distribute and isolate $y$ to go from point-slope to slope-intercept. Knowing both, and converting fluently, means you are never stuck regardless of which information the item hands you. A negative slope simply means the line falls (a decreasing function), as in a draining or depreciation context, and the method is identical.

Try this

Q1. Write the line with slope $3$ through $(0, -2)$. [1 point]

• Cue. Slope-intercept directly: $y = 3x - 2$.

Q2. Write the line through $(1, 5)$ and $(4, 14)$. [2 points]

• Cue. $m = \frac{14 - 5}{4 - 1} = 3$; $y - 5 = 3(x - 1) \Rightarrow y = 3x + 2$.