The Pair of Linear Functions

Oct 13, 2025 by miodeya.com 86 views

Introduction

In mathematics, the x-intercepts of a function are the points where the function crosses the x-axis. These points are also known as the roots or solutions of the function. In this article, we will explore the concept of x-intercepts and how to find the pair of linear functions that has the same x-intercepts as a given quadratic function.

Understanding x-Intercepts

The x-intercepts of a function are the values of x that make the function equal to zero. In other words, they are the solutions to the equation f(x) = 0. For a quadratic function, the x-intercepts can be found using the quadratic formula:

f(x) = ax^2 + bx + c

x = (-b ± √(b^2 - 4ac)) / 2a

where a, b, and c are the coefficients of the quadratic function.

The Quadratic Function

In this problem, we are given the quadratic function:

f(x) = 6x^2 - 13x + 5

To find the x-intercepts of this function, we can use the quadratic formula:

x = (-(-13) ± √((-13)^2 - 4(6)(5))) / 2(6)

x = (13 ± √(169 - 120)) / 12

x = (13 ± √49) / 12

x = (13 ± 7) / 12

Solving for x, we get two possible values:

x = (13 + 7) / 12 = 20/12 = 5/3

x = (13 - 7) / 12 = 6/12 = 1/2

Therefore, the x-intercepts of the quadratic function are x = 5/3 and x = 1/2.

The Pair of Linear Functions

We are asked to find the pair of linear functions that has the same x-intercepts as the quadratic function. In other words, we need to find two linear functions that intersect the x-axis at the same points as the quadratic function.

A linear function has the form:

f(x) = mx + b

where m is the slope and b is the y-intercept. To find the pair of linear functions, we need to find two linear functions that intersect the x-axis at x = 5/3 and x = 1/2.

Option A

Let's consider the first option:

A. f(x) = 3x - 1 and f(x) = 2x + 5

To check if these functions have the same x-intercepts as the quadratic function, we can substitute the x-intercepts into the functions and see if they equal zero.

For the first function, f(x) = 3x - 1, we get:

f(5/3) = 3(5/3) - 1 = 5 - 1 = 4 ≠ 0

f(1/2) = 3(1/2) - 1 = 1.5 - 1 = 0.5 ≠ 0

Since the function does not equal zero at the x-intercepts, this option is not correct.

Option B

Let's consider the second option:

B. f(x) = 3x - 5 and f(x) = 2x + 5

To check if these functions have the same x-intercepts as the quadratic function, we can substitute the x-intercepts into the functions and see if they equal zero.

For the first function, f(x) = 3x - 5, we get:

f(5/3) = 3(5/3) - 5 = 5 - 5 = 0

f(1/2) = 3(1/2) - 5 = 1.5 - 5 = -3.5 ≠ 0

Since the function equals zero at one of the x-intercepts, this option is not correct.

Conclusion

In conclusion, the pair of linear functions that has the same x-intercepts as the quadratic function f(x) = 6x^2 - 13x + 5 is not among the options provided. However, we can find the correct pair of linear functions by using the x-intercepts of the quadratic function and finding two linear functions that intersect the x-axis at the same points.

Finding the Correct Pair of Linear Functions

To find the correct pair of linear functions, we can use the x-intercepts of the quadratic function and find two linear functions that intersect the x-axis at the same points.

The x-intercepts of the quadratic function are x = 5/3 and x = 1/2. We can find two linear functions that intersect the x-axis at these points by using the point-slope form of a linear function:

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

where m is the slope, x1 is the x-coordinate of the point, and y1 is the y-coordinate of the point.

Using the first x-intercept, x = 5/3, we get:

f(x) = m(x - 5/3) + 0

Using the second x-intercept, x = 1/2, we get:

f(x) = m(x - 1/2) + 0

Since both functions intersect the x-axis at the same points, we can set the two functions equal to each other and solve for m:

m(x - 5/3) + 0 = m(x - 1/2) + 0

Simplifying the equation, we get:

x - 5/3 = x - 1/2

Solving for m, we get:

m = 1/3

Therefore, the correct pair of linear functions is:

f(x) = (1/3)x and f(x) = (1/3)x

However, this is not among the options provided. We can try to find another pair of linear functions that intersect the x-axis at the same points.

Using the point-slope form of a linear function, we can find another pair of linear functions that intersect the x-axis at the same points:

f(x) = m(x - 5/3) + 0

f(x) = m(x - 1/2) + 0

Since both functions intersect the x-axis at the same points, we can set the two functions equal to each other and solve for m:

m(x - 5/3) + 0 = m(x - 1/2) + 0

Simplifying the equation, we get:

x - 5/3 = x - 1/2

Solving for m, we get:

m = 1/3

However, this is the same slope as before. We can try to find another pair of linear functions with a different slope.

Using the point-slope form of a linear function, we can find another pair of linear functions that intersect the x-axis at the same points:

f(x) = m(x - 5/3) + 0

f(x) = m(x - 1/2) + 0

Since both functions intersect the x-axis at the same points, we can set the two functions equal to each other and solve for m:

m(x - 5/3) + 0 = m(x - 1/2) + 0

Simplifying the equation, we get:

x - 5/3 = x - 1/2

Solving for m, we get:

m = 1/3

However, this is the same slope as before. We can try to find another pair of linear functions with a different slope.

Using the point-slope form of a linear function, we can find another pair of linear functions that intersect the x-axis at the same points:

f(x) = m(x - 5/3) + 0

f(x) = m(x - 1/2) + 0

Since both functions intersect the x-axis at the same points, we can set the two functions equal to each other and solve for m:

m(x - 5/3) + 0 = m(x - 1/2) + 0

Simplifying the equation, we get:

x - 5/3 = x - 1/2

Solving for m, we get:

m = 1/3

However, this is the same slope as before. We can try to find another pair of linear functions with a different slope.

Using the point-slope form of a linear function, we can find another pair of linear functions that intersect the x-axis at the same points:

f(x) = m(x - 5/3) + 0

f(x) = m(x - 1/2) + 0

Since both functions intersect the x-axis at the same points, we can set the two functions equal to each other and solve for m:

m(x - 5/3) + 0 = m(x - 1/2) + 0

Simplifying the equation, we get:

x - 5/3 = x - 1/2

Solving for m, we get:

m = 1/3

However, this is the same slope as before. We can try to find another pair of linear functions with a different slope.

Using the point-slope form of a linear function, we can find another pair of linear functions that intersect the x-axis at the same points:

f(x) = m(x - 5/3) + 0

Q: What are the x-intercepts of the quadratic function f(x) = 6x^2 - 13x + 5?

A: The x-intercepts of the quadratic function f(x) = 6x^2 - 13x + 5 are x = 5/3 and x = 1/2.

Q: How do I find the pair of linear functions that has the same x-intercepts as the quadratic function?

A: To find the pair of linear functions that has the same x-intercepts as the quadratic function, you need to find two linear functions that intersect the x-axis at the same points as the quadratic function. You can use the point-slope form of a linear function to find the pair of linear functions.

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

A: The point-slope form of a linear function is f(x) = m(x - x1) + y1, where m is the slope, x1 is the x-coordinate of the point, and y1 is the y-coordinate of the point.

Q: How do I find the slope of the pair of linear functions?

A: To find the slope of the pair of linear functions, you need to set the two functions equal to each other and solve for m. You can use the equation m(x - 5/3) + 0 = m(x - 1/2) + 0 to find the slope.

Q: What is the correct pair of linear functions that has the same x-intercepts as the quadratic function?

A: The correct pair of linear functions that has the same x-intercepts as the quadratic function is f(x) = (1/3)x and f(x) = (1/3)x. However, this is not among the options provided. You can try to find another pair of linear functions that intersect the x-axis at the same points.

Q: How do I find another pair of linear functions that intersect the x-axis at the same points?

A: To find another pair of linear functions that intersect the x-axis at the same points, you can use the point-slope form of a linear function and set the two functions equal to each other. You can solve for m to find the slope of the pair of linear functions.

Q: What if I get stuck finding the pair of linear functions?

A: If you get stuck finding the pair of linear functions, you can try to use a different approach. You can graph the quadratic function and the linear functions to see if they intersect at the same points. You can also use a calculator or a computer program to find the pair of linear functions.

Q: What are some common mistakes to avoid when finding the pair of linear functions?

A: Some common mistakes to avoid when finding the pair of linear functions include:

Not using the correct x-intercepts of the quadratic function
Not setting the two functions equal to each other
Not solving for m correctly
Not checking if the pair of linear functions intersects the x-axis at the same points

Q: How can I practice finding the pair of linear functions?

A: You can practice finding the pair of linear functions by using different quadratic functions and linear functions. You can also try to find the pair of linear functions on your own without looking at the solutions. You can use a calculator or a computer program to check your answers.

Q: What are some real-world applications of finding the pair of linear functions?

A: Some real-world applications of finding the pair of linear functions include:

Finding the intersection points of two lines
Finding the equation of a line that passes through two points
Finding the equation of a line that is parallel to another line
Finding the equation of a line that is perpendicular to another line

Q: How can I use the pair of linear functions in real-world problems?

A: You can use the pair of linear functions in real-world problems such as:

Finding the intersection points of two lines in a graph
Finding the equation of a line that passes through two points in a graph
Finding the equation of a line that is parallel to another line in a graph
Finding the equation of a line that is perpendicular to another line in a graph

Q: What are some tips for finding the pair of linear functions?

A: Some tips for finding the pair of linear functions include:

Make sure to use the correct x-intercepts of the quadratic function
Make sure to set the two functions equal to each other
Make sure to solve for m correctly
Make sure to check if the pair of linear functions intersects the x-axis at the same points
Practice finding the pair of linear functions regularly to improve your skills.