Comparing the Steepness of Two Linear Functions

Oct 13, 2025 by miodeya.com 122 views

Introduction

In mathematics, the steepness of a line is often represented by its slope, which is a measure of how much the line rises (or falls) vertically over a given horizontal distance. In this article, we will compare the steepness of two linear functions represented by the equations $8x + y = 1$ and $-8x + y = 2$ . We will also explore the properties of an increasing linear function that passes through the point $(4,0)$ .

Comparing the Steepness of Two Linear Functions

To compare the steepness of two linear functions, we need to find their slopes. The slope of a line is given by the coefficient of the $x$ term in its equation. In the case of the two given equations, we can rewrite them in the slope-intercept form, which is $y = mx + b$ , where $m$ is the slope and $b$ is the y-intercept.

Equation 1: $8x + y = 1$

To rewrite the equation in slope-intercept form, we need to isolate $y$ on one side of the equation. We can do this by subtracting $8x$ from both sides of the equation:

8x + y = 1

y = -8x + 1

From this equation, we can see that the slope of the line is $-8$ .

Equation 2: $-8x + y = 2$

Similarly, we can rewrite the second equation in slope-intercept form by isolating $y$ on one side of the equation:

-8x + y = 2

y = 8x + 2

From this equation, we can see that the slope of the line is $8$ .

Conclusion

Based on the slopes of the two lines, we can conclude that the line represented by the equation $-8x + y = 2$ is steeper than the line represented by the equation $8x + y = 1$ . This is because the slope of the first line is $8$ , which is greater than the slope of the second line, which is $-8$ .

Properties of an Increasing Linear Function

An increasing linear function is a function that has a positive slope. In other words, as the input $x$ increases, the output $y$ also increases. We are given that the function $f(x)$ is an increasing linear function that passes through the point $(4,0)$ . We can use this information to find the equation of the function.

Finding the Equation of the Function

Since the function is increasing and passes through the point $(4,0)$ , we know that the slope of the function is positive. We can use the point-slope form of a linear equation to find the equation of the function. The point-slope form is given by:

y - y_1 = m(x - x_1)

where $(x_1, y_1)$ is a point on the line and $m$ is the slope.

In this case, we know that the point $(4,0)$ is on the line, so we can substitute $x_1 = 4$ and $y_1 = 0$ into the equation:

y - 0 = m(x - 4)

We also know that the slope of the function is positive, so we can write the equation as:

y = m(x - 4)

To find the value of $m$ , we can use the fact that the function passes through the point $(4,0)$ . We can substitute $x = 4$ and $y = 0$ into the equation:

0 = m(4 - 4)

Simplifying the equation, we get:

0 = 0

This equation is true for any value of $m$ , so we can conclude that the slope of the function is any positive value.

Finding the Equation of the Function

Since the slope of the function is any positive value, we can write the equation of the function as:

y = mx

where $m$ is a positive constant.

To find the value of $m$ , we can use the fact that the function passes through the point $(4,0)$ . We can substitute $x = 4$ and $y = 0$ into the equation:

0 = 4m

Simplifying the equation, we get:

m = 0

This is a contradiction, since we know that the slope of the function is positive. Therefore, we can conclude that the function $f(x)$ is not a linear function.

Conclusion

In this article, we compared the steepness of two linear functions represented by the equations $8x + y = 1$ and $-8x + y = 2$ . We found that the line represented by the equation $-8x + y = 2$ is steeper than the line represented by the equation $8x + y = 1$ . We also explored the properties of an increasing linear function that passes through the point $(4,0)$ and found that the function is not a linear function.

References

[1] Khan Academy. (n.d.). Slope-Intercept Form. Retrieved from https://www.khanacademy.org/math/algebra/x2f1f0/x2f1f1
[2] Math Open Reference. (n.d.). Linear Equations. Retrieved from https://www.mathopenref.com/linearequations.html

Keywords

Steepness of a line
Slope of a line
Linear function
Increasing linear function
Point-slope form
Slope-intercept form
Frequently Asked Questions (FAQs) =====================================

Q: What is the steepness of a line?

A: The steepness of a line is a measure of how much the line rises (or falls) vertically over a given horizontal distance. It is often represented by the slope of the line.

Q: How do you find the slope of a line?

A: To find the slope of a line, you can use the slope-intercept form of a linear equation, which is y = mx + b, where m is the slope and b is the y-intercept. You can also use the point-slope form, which is y - y1 = m(x - x1), where (x1, y1) is a point on the line.

Q: What is the difference between a linear function and a non-linear function?

A: A linear function is a function that has a constant slope, whereas a non-linear function is a function that has a variable slope. In other words, a linear function is a straight line, while a non-linear function is a curve.

Q: How do you determine if a function is increasing or decreasing?

A: To determine if a function is increasing or decreasing, you can look at the slope of the function. If the slope is positive, the function is increasing. If the slope is negative, the function is decreasing.

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

A: The point-slope form of a linear equation is y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope.

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

A: The slope-intercept form of a linear equation is y = mx + b, where m is the slope and b is the y-intercept.

Q: How do you find the equation of a line given two points?

A: To find the equation of a line given two points, you can use the point-slope form of a linear equation. You can substitute the coordinates of the two points into the equation and solve for the slope and y-intercept.

Q: What is the y-intercept of a line?

A: The y-intercept of a line is the point where the line intersects the y-axis. It is represented by the constant term in the slope-intercept form of a linear equation.

Q: How do you determine if a line is parallel or perpendicular to another line?

A: To determine if a line is parallel or perpendicular to another line, you can look at the slopes of the two lines. If the slopes are equal, the lines are parallel. If the slopes are negative reciprocals of each other, the lines are perpendicular.

Q: What is the slope of a horizontal line?

A: The slope of a horizontal line is 0, since the line does not rise or fall vertically.

Q: What is the slope of a vertical line?

A: The slope of a vertical line is undefined, since the line does not rise or fall vertically.

Q: How do you find the equation of a line given its slope and a point?

A: To find the equation of a line given its slope and a point, you can use the point-slope form of a linear equation. You can substitute the slope and the coordinates of the point into the equation and solve for the y-intercept.

Q: What is the equation of a line that passes through the origin?

A: The equation of a line that passes through the origin is y = mx, where m is the slope of the line.

Q: How do you determine if a line is tangent to a circle?

A: To determine if a line is tangent to a circle, you can look at the distance from the center of the circle to the line. If the distance is equal to the radius of the circle, the line is tangent to the circle.

Q: What is the equation of a line that is tangent to a circle?

A: The equation of a line that is tangent to a circle is given by the equation of the circle and the equation of the line. You can use the point-slope form of a linear equation to find the equation of the line.

Q: How do you find the equation of a line that is perpendicular to another line?

A: To find the equation of a line that is perpendicular to another line, you can use the slope of the other line and the point-slope form of a linear equation. You can substitute the slope and the coordinates of a point on the line into the equation and solve for the y-intercept.

Q: What is the equation of a line that is parallel to another line?

A: The equation of a line that is parallel to another line is given by the equation of the other line and the slope of the line. You can use the slope-intercept form of a linear equation to find the equation of the line.

Q: How do you determine if a line is a diameter of a circle?

A: To determine if a line is a diameter of a circle, you can look at the distance from the center of the circle to the line. If the distance is equal to the radius of the circle, the line is a diameter of the circle.

Q: What is the equation of a line that is a diameter of a circle?

A: The equation of a line that is a diameter of a circle is given by the equation of the circle and the equation of the line. You can use the point-slope form of a linear equation to find the equation of the line.

Q: How do you find the equation of a line that passes through two points?

A: To find the equation of a line that passes through two points, you can use the point-slope form of a linear equation. You can substitute the coordinates of the two points into the equation and solve for the slope and y-intercept.

Q: What is the equation of a line that passes through the midpoint of two points?

A: The equation of a line that passes through the midpoint of two points is given by the equation of the line that passes through the two points. You can use the point-slope form of a linear equation to find the equation of the line.

Q: How do you determine if a line is a chord of a circle?

A: To determine if a line is a chord of a circle, you can look at the distance from the center of the circle to the line. If the distance is less than the radius of the circle, the line is a chord of the circle.

Q: What is the equation of a line that is a chord of a circle?

A: The equation of a line that is a chord of a circle is given by the equation of the circle and the equation of the line. You can use the point-slope form of a linear equation to find the equation of the line.

Q: How do you find the equation of a line that passes through the center of a circle?

A: To find the equation of a line that passes through the center of a circle, you can use the equation of the circle and the point-slope form of a linear equation. You can substitute the coordinates of the center of the circle into the equation and solve for the slope and y-intercept.

Q: What is the equation of a line that passes through the center of a circle and is perpendicular to another line?

A: The equation of a line that passes through the center of a circle and is perpendicular to another line is given by the equation of the circle and the equation of the line. You can use the point-slope form of a linear equation to find the equation of the line.

Q: How do you determine if a line is a tangent to a circle and a chord of another circle?

A: To determine if a line is a tangent to a circle and a chord of another circle, you can look at the distance from the center of each circle to the line. If the distance is equal to the radius of one circle and less than the radius of the other circle, the line is a tangent to the first circle and a chord of the second circle.

Q: What is the equation of a line that is a tangent to a circle and a chord of another circle?

A: The equation of a line that is a tangent to a circle and a chord of another circle is given by the equation of the two circles and the equation of the line. You can use the point-slope form of a linear equation to find the equation of the line.

Q: How do you find the equation of a line that passes through the center of a circle and is parallel to another line?

A: To find the equation of a line that passes through the center of a circle and is parallel to another line, you can use the equation of the circle and the equation of the line. You can substitute the coordinates of the center of the circle into the equation and solve for the slope and y-intercept.

Q: What is the equation of a line that passes through the center of a circle and is parallel to another line?

A: The equation of a line that passes through the center of a circle and is parallel to another line is given by the equation of the circle and the equation of the line. You can use the point-slope form of a linear equation to find the equation of the line.

**Q: How do you determine if a line is a