key term - $b$

Definition

$b$ is a variable that represents an unknown or changing quantity in the context of modeling with linear functions. It is a key component of the general linear function equation, $y = mx + b$, where $b$ represents the y-intercept of the line.

5 Must Know Facts For Your Next Test

1. The y-intercept, $b$, represents the value of $y$ when $x = 0$ in the linear function equation $y = mx + b$.
2. The value of $b$ determines the vertical position of the line on the coordinate plane, while the slope, $m$, determines the direction and steepness of the line.
3. When modeling real-world situations with linear functions, the y-intercept, $b$, often represents an initial or starting value that is independent of the input variable, $x$.
4. The y-intercept, $b$, can be used to interpret the meaning of the linear function in the context of the problem being modeled.
5. Changing the value of $b$ in a linear function will shift the line vertically on the coordinate plane, while changing the value of $m$ will change the slope of the line.

Review Questions

• Explain the role of the y-intercept, $b$, in the general linear function equation $y = mx + b$.
  • The y-intercept, $b$, represents the value of $y$ when $x = 0$ in the linear function equation $y = mx + b$. It determines the vertical position of the line on the coordinate plane, independent of the input variable, $x$. The value of $b$ can be used to interpret the meaning of the linear function in the context of the problem being modeled, as it often represents an initial or starting value.
• Describe how changes in the value of $b$ affect the graph of a linear function.
  • Changing the value of $b$ in a linear function will shift the line vertically on the coordinate plane. If $b$ increases, the line will shift upward, and if $b$ decreases, the line will shift downward. The slope of the line, represented by $m$, will remain the same, but the y-intercept will change, affecting the overall position of the line on the graph.
• Analyze the relationship between the y-intercept, $b$, and the interpretation of a linear function in the context of a real-world problem.
  • In the context of modeling real-world situations with linear functions, the y-intercept, $b$, often represents an initial or starting value that is independent of the input variable, $x$. By understanding the meaning of the y-intercept in the problem context, you can gain valuable insights into the interpretation of the linear function and its practical applications. For example, if the linear function represents the cost of a product, the y-intercept may represent a fixed cost that is independent of the quantity purchased.

Related terms

Linear Function:

A linear function is a function that can be expressed in the form $y = mx + b$, where $m$ represents the slope of the line and $b$ represents the y-intercept.

Y-Intercept: The y-intercept is the point where a line crosses the y-axis, and is represented by the value of $b$ in the linear function equation $y = mx + b$.

Slope:

The slope, represented by $m$ in the linear function equation $y = mx + b$, is the rate of change or the steepness of the line.