Question

For the graph of the linear function \(f(x)=m x+b, m\) is the ____ and \(b\) is the _____.

Short answer

slope, y-intercept

Step-by-step solution

- Identify the Linear Function

Recognize the given linear function. The general form of a linear function is given as \(f(x) = mx + b\).

- Understand the Slope

In the equation \(f(x) = mx + b\), the coefficient \(m\) is the slope of the line. The slope \(m\) represents the steepness or incline of the line, indicating how much the value of \(y\) changes for a unit change in \(x\).

- Understand the Y-intercept

In the same equation, the term \(b\) is the y-intercept. The y-intercept \(b\) is the point where the line crosses the y-axis. It represents the value of \(f(x)\) when \(x = 0\).

- Conclusion

Summarize the information: \(m\) is the slope and \(b\) is the y-intercept in the linear equation \(f(x) = mx + b\).

Key concepts

Understand the Linear Equation

A linear equation is one of the fundamental concepts in algebra. It is often written in the form: \(f(x) = mx + b\). This representation makes it easy to identify both the slope and the y-intercept.
Linear equations graph as straight lines in a coordinate plane. They model relationships where the rate of change is consistent. This makes them quite useful in various real-world applications such as predicting trends or comparing rates.
It's important to recognize the variables in a linear equation: \(x\) is the independent variable, while \(f(x)\) (or sometimes written as \(y\)) is the dependent variable. The values of \(x\) and \(f(x)\) change based on the equation and depict a linear relationship.

Understanding the Slope

In the equation \(f(x) = mx + b\), the value \(m\) is known as the slope. The slope of a line measures how steep the line is. It tells us the rate at which \(f(x)\) (or \(y\)) changes as \(x\) changes.
The slope is calculated as the ratio of the vertical change to the horizontal change between two points on the line. This means: \[ m = \frac{\Delta y}{\Delta x} \] If \(m\) is positive, the line inclines upwards. If \(m\) is negative, the line inclines downwards. A higher value of \(m\) means a steeper line.
Understanding slope helps in various practical scenarios, such as figuring out the speed of a moving object, or the incline of a hill.

Understanding the Y-intercept

The y-intercept \(b\) in the linear equation \(f(x) = mx + b\) tells us where the line crosses the y-axis. This occurs when \(x = 0\).
The y-intercept represents the value of the function (or \(f(x)\)) when the independent variable \(x\) is zero. Thus, \(b\) provides a starting point for plotting the linear equation on the graph.
For example, in the equation \(f(x) = 2x + 3\), the y-intercept is 3. This means the line will intersect the y-axis at the point \((0, 3)\). They intersect the y-axis at different points, but the slope tells us how steeply they have inclined or declined from that point.
Understanding the y-intercept is crucial for graphing a linear function quickly and accurately.