The Equation of a Linear Function

Oct 13, 2025 by miodeya.com 137 views

Introduction

In mathematics, a linear function is a polynomial function of degree one, which means it has the form f(x) = ax + b, where 'a' and 'b' are constants. The graph of a linear function is a straight line. In this article, we will learn how to find the equation of a linear function that passes through two given points.

The Equation of a Linear Function

The equation of a linear function is given by f(x) = ax + b, where 'a' is the slope of the line and 'b' is the y-intercept. The slope of a line is a measure of how steep it is, and it is calculated as the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line.

Finding the Equation of a Linear Function Passing Through Two Points

To find the equation of a linear function passing through two points, we can use the following steps:

Calculate the slope (a): The slope of a line passing through two points (x1, y1) and (x2, y2) is given by the formula:

a = (y2 - y1) / (x2 - x1)

Calculate the y-intercept (b): Once we have the slope, we can use one of the points to calculate the y-intercept. We can use the point-slope form of a linear equation, which is given by:

f(x) = a(x - x1) + y1

Example 1: Finding the Equation of a Linear Function Passing Through (-8, -1) and (8, 7)

Let's find the equation of a linear function passing through the points (-8, -1) and (8, 7).

Step 1: Calculate the slope (a)

Using the formula for the slope, we get:

a = (7 - (-1)) / (8 - (-8)) = (7 + 1) / (8 + 8) = 8 / 16 = 1/2

Step 2: Calculate the y-intercept (b)

Using the point-slope form of a linear equation, we get:

f(x) = (1/2)(x - (-8)) + (-1) = (1/2)(x + 8) - 1

Simplifying the Equation

To simplify the equation, we can multiply both sides by 2 to get rid of the fraction:

2f(x) = x + 8 - 2 2f(x) = x + 6

Dividing both sides by 2, we get:

f(x) = (1/2)x + 3

Example 2: Finding the Equation of a Linear Function Passing Through (9, -7) and (-5, 3)

Let's find the equation of a linear function passing through the points (9, -7) and (-5, 3).

Step 1: Calculate the slope (a)

Using the formula for the slope, we get:

a = (3 - (-7)) / (-5 - 9) = (3 + 7) / (-5 - 9) = 10 / (-14) = -5/7

Step 2: Calculate the y-intercept (b)

Using the point-slope form of a linear equation, we get:

f(x) = (-5/7)(x - 9) + (-7) = (-5/7)(x - 9) - 7

Simplifying the Equation

To simplify the equation, we can multiply both sides by 7 to get rid of the fraction:

7f(x) = -5(x - 9) - 49 7f(x) = -5x + 45 - 49 7f(x) = -5x - 4

Dividing both sides by 7, we get:

f(x) = (-5/7)x - 4/7

Conclusion

In this article, we learned how to find the equation of a linear function passing through two given points. We used the formula for the slope and the point-slope form of a linear equation to find the equation of a linear function passing through the points (-8, -1) and (8, 7), and the points (9, -7) and (-5, 3). We also simplified the equations to get the final answers.

Key Takeaways

The equation of a linear function is given by f(x) = ax + b, where 'a' is the slope and 'b' is the y-intercept.
The slope of a line passing through two points (x1, y1) and (x2, y2) is given by the formula a = (y2 - y1) / (x2 - x1).
The point-slope form of a linear equation is given by f(x) = a(x - x1) + y1.
To find the equation of a linear function passing through two points, we can use the formula for the slope and the point-slope form of a linear equation.

Practice Problems

Find the equation of a linear function passing through the points (2, 4) and (6, 10).
Find the equation of a linear function passing through the points (-3, 2) and (1, -4).
Find the equation of a linear function passing through the points (0, 5) and (4, 15).

Solutions

f(x) = (1/2)x + 3
f(x) = (-3/2)x + 1
f(x) = (5/4)x + 5/4

Introduction

In our previous article, we learned how to find the equation of a linear function passing through two given points. In this article, we will answer some frequently asked questions about linear function equations.

Q: What is the equation of a linear function?

A: The equation of a linear function is given by f(x) = ax + b, where 'a' is the slope and 'b' is the y-intercept.

Q: How do I find the slope of a line passing through two points?

A: To find the slope of a line passing through two points (x1, y1) and (x2, y2), you can use the formula a = (y2 - y1) / (x2 - x1).

Q: What is the point-slope form of a linear equation?

A: The point-slope form of a linear equation is given by f(x) = a(x - x1) + y1.

Q: How do I find the equation of a linear function passing through two points?

A: To find the equation of a linear function passing through two points, you can use the formula for the slope and the point-slope form of a linear equation.

Q: Can I use any two points to find the equation of a linear function?

A: Yes, you can use any two points to find the equation of a linear function. However, you should make sure that the two points are not the same point, as this would result in a vertical line.

Q: How do I simplify the equation of a linear function?

A: To simplify the equation of a linear function, you can multiply both sides of the equation by a common factor to get rid of any fractions.

Q: What is the y-intercept of a linear function?

A: The y-intercept of a linear function is the point where the line intersects the y-axis. It is given by the value of 'b' in the equation f(x) = ax + b.

Q: Can I find the equation of a linear function if I only know one point and the slope?

A: Yes, you can find the equation of a linear function if you only know one point and the slope. You can use the point-slope form of a linear equation to find the equation.

Q: How do I find the equation of a linear function if I only know two points and the slope is not given?

A: To find the equation of a linear function if you only know two points and the slope is not given, you can use the formula for the slope and the point-slope form of a linear equation.

Q: Can I use a graphing calculator to find the equation of a linear function?

A: Yes, you can use a graphing calculator to find the equation of a linear function. You can enter the two points and the calculator will give you the equation of the line.

Q: How do I check if a given equation is a linear function?

A: To check if a given equation is a linear function, you can look at the equation and see if it is in the form f(x) = ax + b. If it is, then the equation is a linear function.

Conclusion

In this article, we answered some frequently asked questions about linear function equations. We hope that this article has been helpful in clarifying any doubts you may have had about linear function equations.

Key Takeaways

The equation of a linear function is given by f(x) = ax + b, where 'a' is the slope and 'b' is the y-intercept.
The slope of a line passing through two points (x1, y1) and (x2, y2) is given by the formula a = (y2 - y1) / (x2 - x1).
The point-slope form of a linear equation is given by f(x) = a(x - x1) + y1.
To find the equation of a linear function passing through two points, you can use the formula for the slope and the point-slope form of a linear equation.

Practice Problems

Find the equation of a linear function passing through the points (2, 4) and (6, 10).
Find the equation of a linear function passing through the points (-3, 2) and (1, -4).
Find the equation of a linear function passing through the points (0, 5) and (4, 15).

Solutions

f(x) = (1/2)x + 3
f(x) = (-3/2)x + 1
f(x) = (5/4)x + 5/4