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Linear Functions

Linear functions of the form

f(x) = a x + b

and the properties of their graphs are explored interactively using an applet. Another tutorial on
graphing linear functions is included in this website.

The exploration is carried out by changing the parameters a and b defining the linear function. Answers to the questions in the tutorial are at the bottom of the page.

a =

b =


Similar tutorials on
quadratic and rational functions are also included in this site.


1 - Use the sliders to set parameter a to 1 and change parameter b. How does the y intercept change as b changes? Give a quantitative answer and explain it analytically.
2 - Set parameter b to any value and change parameter a. For what values of parameter a is the graph of function f increasing? For what values of the parameter a is the graph of f decreasing? For what value of a if f constant?
3 - The graph of f is a line. Set a to a value and use two points on the graph to find the slope of the line. Compare the value of the slope found to the value of parameter a. Do this for many values of a. What does a represent?
4 - What is the domain of the linear function f?
5 - What is the range of function f when parameter a is not equal to 0? What is the range of f when parameter a is equal to 0?

Answers to the Above Questions

2 - If we set x = 0 in f(x) = a x + b, we obtain f(0) = b. The y intercept is the point with coordinates (0 , b).
3 - If a is positive, f is an increasing function on (-∞ , +∞).
If a is negative, f is a decreasing function on (-∞ , +∞).
If a is equal to 0, f is a constant function on (-∞ , +∞).
4 - The graph of function f is a line, hence the name linear function. Parameter a represents the slope of this line.
5 - The domain of all linear functions is the set of all real numbers.
6 - If a is not equal to 0, the range of any linear function is the set of all real numbers. If a is equal to 0, f(x) = b a constant function and the range is {b}.

More References and Links

Solver to Analyze and Graph a Linear Function