Linear functions

Definition

Linear functions are mathematical expressions that create a straight line when graphed on a coordinate plane. They can be represented in the form of an equation, typically written as $$y = mx + b$$, where $$m$$ is the slope and $$b$$ is the y-intercept. This structure makes linear functions fundamental in understanding relationships between variables, as they demonstrate constant rates of change.

5 Must Know Facts For Your Next Test

1. Linear functions graph as straight lines, indicating that for every increase in $$x$$, there is a consistent increase or decrease in $$y$$ based on the slope.
2. The slope $$m$$ in the linear function equation $$y = mx + b$$ indicates the direction and steepness of the line, with positive values rising and negative values falling.
3. If two linear functions have the same slope but different y-intercepts, they will be parallel and never intersect.
4. The x-intercept can be found by setting $$y = 0$$ in the linear equation and solving for $$x$$.
5. Transformations such as shifts, stretches, and reflections can change the appearance of a linear function's graph while retaining its linearity.

Review Questions

• How does the slope of a linear function affect its graph, and what does it indicate about the relationship between the variables?
  • The slope of a linear function determines how steep the line is and the direction it moves on a graph. A positive slope means that as one variable increases, the other also increases, indicating a direct relationship. Conversely, a negative slope shows that as one variable increases, the other decreases, demonstrating an inverse relationship. Thus, understanding slope is crucial for interpreting how changes in one variable impact another.
• In what ways can transformations alter the graph of a linear function without changing its fundamental nature?
  • Transformations such as vertical shifts, horizontal shifts, and reflections can change the position or orientation of a linear function's graph. For instance, adding a constant to the y-value shifts the graph upward while subtracting shifts it downward. Similarly, multiplying by -1 reflects the graph across the x-axis. These transformations maintain the straight-line characteristic of linear functions but modify their appearance on a coordinate plane.
• Evaluate how understanding linear functions can impact real-world applications and problem-solving.
  • Understanding linear functions is essential for modeling real-world situations where relationships between quantities are consistent. For instance, they can be used in fields like economics to represent costs and revenues or in physics to depict motion with constant speed. Recognizing how to interpret and manipulate linear equations allows for effective decision-making and predictions based on trends. By applying this knowledge creatively, one can tackle complex problems across various disciplines.

Related terms

slope: The slope is a measure of the steepness or incline of a line, calculated as the ratio of the vertical change to the horizontal change between two points on the line.

y-intercept: The y-intercept is the point where a line crosses the y-axis, representing the value of $$y$$ when $$x$$ is zero in a linear function.

horizontal line: A horizontal line has a slope of zero and represents a linear function where the value of $$y$$ remains constant regardless of the value of $$x$$.