Finding Matching Slopes: A Linear Function Guide

Hey everyone! Let's dive into the world of linear functions and slopes. We're going to figure out which linear function shares the same slope as the one represented by the table you provided. This is super useful for understanding how lines behave and how they relate to each other. So, grab your pencils, and let's get started!

Understanding Linear Functions and Slopes

First things first, what exactly is a linear function, and what's all the fuss about the slope? Well, a linear function is just a fancy way of saying a function that, when graphed, makes a straight line. Pretty straightforward, right? These lines can go up, down, or stay perfectly horizontal. The slope is the key to understanding how a line behaves. Think of the slope as the "steepness" of the line. It tells us how much the y-value changes for every one-unit change in the x-value. Mathematically, the slope (often represented by the letter 'm') is calculated as "rise over run" or the change in y divided by the change in x.

So, why is the slope so important? Well, it tells us a lot about the relationship between the x and y variables. A positive slope means the line goes uphill from left to right, while a negative slope means it goes downhill. A slope of zero means the line is horizontal (no change in y), and an undefined slope means the line is vertical (no change in x). Recognizing the slope helps us predict where a line will be on a graph and how the y-value will change as x-value increases or decreases. If two lines have the same slope, they are parallel, meaning they will never intersect. This concept is fundamental to understanding linear equations and their behavior in various contexts, from simple graphs to complex systems. Getting the hang of slopes opens the door to understanding a vast range of mathematical concepts. Remember, the slope is a crucial piece of the puzzle in linear functions.

Now, how do you find the slope? When given a table of x and y values, like the one we have, you can calculate the slope by picking any two points from the table and using the slope formula. The formula is: m = (y2 - y1) / (x2 - x1), where (x1, y1) and (x2, y2) are the coordinates of your chosen points. Let's practice that. For example, if you know that one point is (1, 2) and another one is (3, 6), then your slope is: (6 - 2) / (3 - 1) = 4 / 2 = 2. So the slope is 2. Easy peasy, right?

Calculating the Slope from the Table

Alright, let's get down to the nitty-gritty and calculate the slope from the table you provided. Remember, the table gives us a set of (x, y) coordinates. To calculate the slope, we'll use the slope formula: m = (y2 - y1) / (x2 - x1). Here is the table again:

Let's choose two points from the table to work with. How about (-1/2, 1/5) and (-1/5, 7/50)? Let's plug those values into our slope formula. First, let's label our points:

• (x1, y1) = (-1/2, 1/5)
• (x2, y2) = (-1/5, 7/50)

Now, let's substitute these values into the slope formula:

m = (7/50 - 1/5) / (-1/5 - (-1/2))

First, we need to simplify the numerator:

• 7/50 - 1/5 = 7/50 - 10/50 = -3/50

Next, simplify the denominator:

• -1/5 - (-1/2) = -1/5 + 1/2 = -2/10 + 5/10 = 3/10

Now, let's divide the numerator by the denominator:

• m = (-3/50) / (3/10)

To divide fractions, we multiply by the reciprocal of the second fraction:

• m = (-3/50) * (10/3)

Multiply the numerators and the denominators:

• m = -30/150

Finally, simplify the fraction:

• m = -1/5

So, the slope of the linear function represented by the table is -1/5. Keep that value in mind, because this is our key to unlocking the answer to our original question.

Finding Functions with the Same Slope

Now that we know the slope of our original linear function, which is -1/5, the next step is to find other linear functions that have the same slope. To do this, we need to examine other linear functions (usually provided in the form of equations or tables) and calculate their slopes. The goal is to identify which one has a slope of -1/5. Remember, parallel lines have the same slope, and we're essentially looking for lines that will be parallel to the one represented by your table. Let's consider a few possible formats for linear functions:

• Linear Equations: These are usually written in slope-intercept form (y = mx + b) or point-slope form. In the slope-intercept form, 'm' is the slope, so we can directly identify it. In the point-slope form, we'd need to convert it or use the slope formula.
• Tables of Values: Similar to the one we started with, we can calculate the slope using the slope formula with any two points from the table.
• Graphs: We can find the slope from a graph by identifying two points on the line and calculating the rise over run.

The process involves calculating or identifying the slope for each given function and comparing it to our target slope of -1/5. If the slopes match, we have found a linear function with the same slope. This process might seem tedious, but it's a fundamental skill in algebra, enabling us to compare and relate different linear relationships. By practicing and understanding different function formats, you can quickly assess slopes and identify relationships between linear functions.

Examples and Practice

Let's get some practice by looking at some example linear functions and figuring out if they have the same slope as the original table (-1/5). Remember, the key is to calculate the slope for each example and compare it to -1/5. Let's start with a few scenarios:

• Example 1: Linear Equation in Slope-Intercept Form: Suppose we have the equation y = (-1/5)x + 3. In this case, the slope is explicitly given in the equation. The slope is -1/5. Bingo! This linear function has the same slope as the original table.
• Example 2: Linear Equation in Point-Slope Form: Suppose we have the equation y - 2 = -1/5 (x - 5). We can see that the slope is -1/5. This function has the same slope as the original table.
• Example 3: Table of Values: Let's create a new table. In this case we need to calculate the slope using the formula.

Let's calculate the slope using the points (0, 4) and (5, 3):

m = (3 - 4) / (5 - 0) = -1/5

This function also has the same slope as the original table. Cool!

• Example 4: A Different Table of Values: Let's try another table.

Let's calculate the slope using the points (0, 1) and (1, 3):

m = (3 - 1) / (1 - 0) = 2

This function does not have the same slope as the original table.

These examples illustrate how to identify and compare slopes. You may need to convert the linear functions into the same format, calculate the slope, and compare them to determine which one matches the original table's slope.

Conclusion

So, there you have it! Finding linear functions with the same slope is all about calculating or identifying the slope and comparing it to the reference slope. Whether you're working with equations, tables, or graphs, the process remains the same. The slope tells us everything. Remember, parallel lines have the same slope, and that's the key concept to keep in mind. We've gone through the process step-by-step, providing examples and practice. Mastering this skill will make you a pro at dealing with linear functions. Keep practicing, and you'll be identifying matching slopes in no time. If you have any questions, feel free to ask. Cheers!