Linear Function
A linear function is a type of mathematical function that can be represented by a straight line on a graph
A linear function is a type of mathematical function that can be represented by a straight line on a graph. It is a function that follows the form of:
f(x) = mx + b
Where:
– f(x) represents the output or dependent variable.
– x represents the input or independent variable.
– m represents the slope of the line.
– b represents the y-intercept, which is the point where the line crosses the y-axis.
The slope (m) of a linear function determines how steep or flat the line is. It represents the rate of change or the amount by which the dependent variable changes for each unit change in the independent variable. A positive slope indicates that the line is increasing from left to right, while a negative slope indicates a decreasing line.
The y-intercept (b) is the value of the dependent variable (y) when the independent variable (x) is equal to zero. It represents the initial value or starting point of the function.
To graph a linear function, you can plot the y-intercept on the y-axis and then use the slope to find additional points. You can do this by using the formula:
y = mx + b
For example, let’s say we have the equation y = 2x + 3.
The slope is 2, so for every unit increase in x, the y-value will increase by 2. The y-intercept is 3, so the line crosses the y-axis at (0, 3).
To find additional points, you can choose x-values and substitute them into the equation to solve for y. For example, if we substitute x = 1, we get y = 2(1) + 3 = 2 + 3 = 5. So the point (1, 5) is on the line.
You can continue this process to find more points, or you can plot two points and draw a straight line through them to represent the linear function.
Linear functions have many applications in real-world situations, such as representing the relationship between distance and time, cost and quantity, or temperature and time. They are also fundamental in algebraic concepts such as slope-intercept form, point-slope form, and finding equations of lines.
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