A linear function models a relationship with a constant rate of change. SAT questions may present that relationship as an equation, table, graph, or verbal description and ask you to translate or interpret it.
Essential concepts
\[y=mx+b qquad m= rac{y_2-y_1}{x_2-x_1}\]
Use two distinct points to find the slope, then substitute either point to determine the intercept.
SAT strategy and application
Build a function from two points
A line passes through \((2,7)\) and \((5,16)\). Find its equation.
- Find the slope
\(m= rac{16-7}{5-2}=3\).
- Find the intercept
\(7=3(2)+b\), so \(b=1\).
Equation: \(y=3x+1\).
Mistakes and traps
Check your understanding
For \(f(x)=4x-3\), what is \(f(4)\)?
- 1
- 13
- 16
- 19
Show answer and explanation
Answer: \(13\)
\(f(4)=4(4)-3=16-3=13\).
Key takeaways
What to remember
- \(m\) is the constant rate of change and \(b\) is the value at input zero.
- Use units to interpret parameters in context.
- Check tables for constant first differences.
- Translate flexibly among equations, graphs, tables, and descriptions.
Continue learning
Put these notes into practice
Apply the ideas with SAT-style questions, then reinforce key details with flashcards.