📖Definition
A linear function is a function whose graph is a straight line. It can be written in the form f(x) = mx + b, where m is the slope and b is the y-intercept. Linear functions have a constant rate of change.
📐Formula
In f(x) = mx + b: m is the slope (rate of change), b is the y-intercept (where the line crosses the y-axis), x is the input, and f(x) is the output.
📝Step-by-Step Guide
Identify the Form
Write the function in slope-intercept form f(x) = mx + b.
Find Slope and Y-Intercept
The coefficient of x is the slope (m). The constant is the y-intercept (b).
Graph the Function
Plot the y-intercept, then use the slope to find more points. Connect with a straight line.
Evaluate the Function
To find f(a), substitute a for x and calculate.
⚠️Common Mistakes to Avoid
- Confusing f(x) with f times x
- Misidentifying slope and y-intercept
- Plotting points incorrectly
- Forgetting that linear means constant rate of change
- Not recognizing equivalent forms of linear functions
✏️Practice Problems
For f(x) = 2x + 3, find f(4)
Answer: 11
Find the slope and y-intercept of f(x) = -3x + 7
Answer: Slope = -3, y-intercept = 7
A linear function has f(2) = 5 and f(5) = 14. Find the function.
Answer: f(x) = 3x - 1
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