Linear function

Definition

A linear function is a type of function that creates a straight line when graphed on a coordinate plane. This function can be expressed in the form $$f(x) = mx + b$$, where $$m$$ represents the slope and $$b$$ represents the y-intercept. The characteristics of a linear function include constant rates of change, and its graph showcases key properties like symmetry, slope, and intercepts.

5 Must Know Facts For Your Next Test

1. A linear function has a constant slope, meaning that for any two points on its graph, the ratio of the change in y to the change in x remains the same.
2. The graph of a linear function is always a straight line, which can be upward sloping, downward sloping, or horizontal based on the slope's value.
3. Linear functions can be represented in various forms, including slope-intercept form $$y = mx + b$$ and standard form $$Ax + By = C$$.
4. The slope-intercept form makes it easy to identify both the slope and y-intercept directly from the equation.
5. Linear functions are used extensively in real-life applications, such as economics for modeling cost and revenue relationships, physics for speed and distance calculations, and many other fields.

Review Questions

• How can you determine if a given function is linear based on its equation?
  • To determine if a function is linear from its equation, check if it can be expressed in the form $$y = mx + b$$, where $$m$$ and $$b$$ are constants. If the equation involves higher powers of x (like squares or cubes), or variables multiplied together, it is not linear. An equation representing a straight line will maintain that consistent form without additional complexities.
• Compare and contrast the different forms of linear functions and their uses in graphing.
  • Linear functions can be expressed in several forms, most notably slope-intercept form $$y = mx + b$$ and standard form $$Ax + By = C$$. The slope-intercept form allows for quick identification of slope and y-intercept, making it easier for graphing. On the other hand, standard form can sometimes make it simpler to find intercepts and perform operations like adding or subtracting equations. Both forms lead to straight-line graphs but offer different advantages depending on the context.
• Evaluate how changes in the parameters of a linear function affect its graph.
  • When evaluating how changes in parameters affect the graph of a linear function, consider how modifications to slope (m) or y-intercept (b) impact its position and angle. Increasing m will make the line steeper, while decreasing m will flatten it. Adjusting b shifts the line up or down without altering its slope. By analyzing these changes, one can predict how variations in real-world situations (like cost or distance) impact outcomes represented by linear relationships.

Related terms

slope: The slope is a measure of the steepness of a line, calculated as the ratio of the rise (change in y) to the run (change in x) between two points on the line.

y-intercept: The y-intercept is the point where the graph of a function intersects the y-axis, represented by the value of $$b$$ in the linear function equation.

function: A function is a relationship between a set of inputs and a set of possible outputs, where each input is related to exactly one output.