Linear Functions

Definition

Linear functions are mathematical expressions that create straight lines when graphed on a coordinate plane. They can be represented in the slope-intercept form, which is written as $$y = mx + b$$, where $$m$$ represents the slope of the line and $$b$$ is the y-intercept. These functions maintain a constant rate of change, meaning for every unit increase in the input (x-value), there is a consistent increase or decrease in the output (y-value).

5 Must Know Facts For Your Next Test

1. A linear function can be represented in various forms, including standard form $$Ax + By = C$$ and slope-intercept form $$y = mx + b$$.
2. The slope of a line can be positive, negative, zero, or undefined, affecting the direction and nature of the linear function.
3. Linear functions have a domain and range that can be all real numbers unless restricted by context or conditions.
4. The graph of a linear function is always a straight line, which illustrates the concept of constant rate of change.
5. Transformations such as translations and reflections can affect the position and orientation of linear functions on a graph.

Review Questions

• How does changing the slope and y-intercept affect the graph of a linear function?
  • Changing the slope of a linear function alters its steepness and direction; a larger slope value results in a steeper line, while a negative slope indicates a downward trend. Adjusting the y-intercept shifts the graph vertically without changing its slope. For example, increasing the y-intercept raises the entire line upward on the graph, while decreasing it moves it downward. Together, these changes can lead to different linear equations that represent distinct real-world scenarios.
• Discuss how transformations such as vertical shifts or horizontal shifts impact linear functions.
  • Vertical shifts occur when you add or subtract a constant from the output of a linear function. This results in moving the entire line up or down without affecting its slope. Horizontal shifts happen when you adjust the input (x-value) by adding or subtracting a constant within the function. This effectively moves the line left or right on the graph. Both transformations maintain the linearity of the function but change its position on the coordinate plane.
• Evaluate how understanding linear functions can apply to solving real-world problems, particularly in modeling relationships between variables.
  • Understanding linear functions allows us to model relationships between two variables effectively, making it easier to predict outcomes based on given inputs. For example, in economics, one might use a linear function to illustrate how supply changes with price or how demand might decrease with increasing costs. By analyzing these relationships through their slopes and intercepts, one can derive valuable insights that influence decision-making in various fields such as business, engineering, and environmental science.

Related terms

Slope: The slope of a linear function indicates its steepness and direction, calculated as the ratio of the change in the y-values to the change in the x-values.

Y-Intercept: The y-intercept is the point where the linear function crosses the y-axis, representing the output value when the input is zero.

Graphing: Graphing is the process of visually representing linear functions on a coordinate plane, allowing for easier interpretation of their behavior and characteristics.