Understanding y5x-9: A Linear Function Explained

Is y5x-9 a linear function?

Yes, the equation y5x-9 represents a linear function.

The General Form of a Linear Function

A linear function can be expressed in the general form:

(y mx b) where m is the slope of the line, and b is the y-intercept. In the given example, m 5 and b -9.

The Slope and Y-Intercept

The slope of the line is 5, indicating that for every unit increase in x, y increases by 5 units. The y-intercept is -9, which means the line crosses the y-axis at the point (0, -9).

Visualizing the Linear Function

When graphed on a coordinate plane, y5x-9 forms a straight line. You can plot several points to visualize this:

When x 0, y -9, so the line intersects the y-axis at (-9, 0). When x 1, y 5(1) - 9 -4, so the line passes through the point (1, -4). When x 2, y 5(2) - 9 1, so the line passes through the point (2, 1).

You can plot these points and draw a straight line through them, confirming that the equation is indeed a linear function.

Is y5x-9 a Function?

In the context of how functions are defined, a linear function is typically written as f(x) 5x - 9. This notation indicates that f(x) depends on x, which is the independent variable, and is a linear relationship.

The term 'relationship' used for y5x-9 can be a bit ambiguous. In a broader sense, y5x-9 describes a set of points that lie on a line. However, strictly speaking, for it to be a function, every x value must map to exactly one y value, which y5x-9 satisfies.

Distinguishing between Linear Relationships and Functions

It's important to understand that there are two distinct meanings of 'linear function' in mathematics:

Linear in the First Sense

In one sense, a linear function is a polynomial function of degree 1, meaning the highest power of the independent variable (usually x) is 1. In this sense, y5x-9 is a linear function because the variable x is raised to the power of 1.

Linear in the Second Sense

A linear function can also be understood in the context of differential equations, where a function y is considered linear if it satisfies a linear differential equation. In this sense, y5x-9 is a linear function, as its derivative with respect to y is a constant (5).

Differentiating y5x-9

Let's differentiate (y 5x - 9) with respect to x: [frac{dy}{dx} 5] This constant derivative signifies that the rate of change of y with respect to x is constant, which is a key characteristic of a linear function.

Conclusion

In summary, y5x-9 is indeed a linear function in both senses of the term. It describes a straight line on a coordinate plane and is represented by the general form (y mx b) with a slope of 5 and a y-intercept of -9.