Understanding Y-Intercepts and Slopes in Linear Equations

When dealing with linear equations, it is crucial to understand the concepts of y-intercepts and slopes. These concepts are fundamental for many applications in mathematics, including graphing, optimization, and real-world problem-solving. In this article, we will explore how to find the y-intercept of different functions and compare them, as well as how to determine the slope of a line given a slope and a point.

Introduction to Linear Equations

A linear equation in two variables, such as ymx by mx b, is expressed in the slope-intercept form. Here, mm represents the slope of the line, and bb is the y-intercept.

Y-Intercept of Given Functions

We will analyze three different functions to understand how to find their y-intercepts:

1. Function f(x)3x

To find the y-intercept, we substitute 0 for xx in the function:

f(x)3xf(0)3?0f(0)0f(x) 3x f(0) 3 , 0 f(0) 0

The y-intercept is 0.

2. Equation 2x–3y122x - 3y 12

To find the y-intercept, we need to solve for yy when xx is 0:

2x–3y12-3y12-2?0-3y12y-12-3)y42x - 3y 12 -3y 12 - 2 , 0 -3y 12 y -frac{12}{-3) y 4

The y-intercept is 4.

3. Line with Slope 2 Passing Through Point (1, -4)

We are given a slope of 2 and a point (1, -4). Using the point-slope form of a line, which is y-y1m(x-x1)y-y_1 m(x-x_1), we can write the equation as:

y--42x-1y--4 2(x-1)

This simplifies to:

y--42x-1y-42x-1y--4 2(x-1) y-4 2(x-1)

Now, we find the y-intercept by setting xx to 0:

y-420-1y-4-2y-2 4y2y-4 2(0-1) y-4 -2 y -2 4 y 2

The y-intercept is 2.

Comparison and Conclusion

Comparing the y-intercepts of the given functions, we can see that the second function (2x - 3y 12) has the greatest y-intercept, which is 4. The first function (f(x) 3x) and the line with slope 2 passing through the point (1, -4) have y-intercepts of 0 and 2, respectively.

Understanding y-intercepts and slopes is essential for various applications in science, engineering, and economics. These concepts help in modeling real-world phenomena and solving practical problems.