Comparing Linear Functions: A Detailed Guide

Hey guys! Let's dive into the fascinating world of linear functions. We're going to compare two of them, focusing on their key features like x-intercepts and slopes. Our mission? To understand how they stack up against each other. Buckle up, it's going to be a fun ride!

Understanding Linear Functions and Their Components

First things first, what even is a linear function? Well, in simple terms, it's a function that, when graphed, produces a straight line. The equation representing a linear function typically takes the form of y = mx + b, where:

• y represents the output or dependent variable.
• x represents the input or independent variable.
• m represents the slope of the line, which tells us how much y changes for every unit change in x.
• b represents the y-intercept, which is the point where the line crosses the y-axis (when x = 0).

Now, let's break down the crucial elements we'll be dealing with in this comparison: the x-intercept and the slope. The x-intercept is the point where the line intersects the x-axis. At this point, the y-value is always zero. Finding the x-intercept is super important for understanding where the line "starts" on the horizontal axis. The slope, as mentioned earlier, tells us the direction and steepness of the line. A positive slope means the line goes upwards from left to right, a negative slope means it goes downwards, a slope of zero is a horizontal line, and undefined is a vertical line. The steeper the line, the greater the absolute value of the slope. A larger absolute value means a steeper line. These two components give us a complete picture of the linear function's behavior.

Knowing the x-intercept and the slope is like having two vital pieces of a puzzle. Knowing these is crucial to easily plot the graph. With this information, we can graph the function and understand how it behaves across different values of x. Remember, the equation y = mx + b is the cornerstone for understanding and working with linear functions. Let's clarify by working through the first function, where we have the $x$-intercept and slope, and then we will see how that applies to the table provided to compare both linear functions. To reiterate, the x-intercept is the point where the function crosses the x-axis. At this point, the y value is always equal to zero. Knowing that, we can get our first function's equation, which we will compare with our second function from the table.

To recap, we have the first linear function with the x-intercept of 12 and a slope of 3/8. This means that the line crosses the x-axis at the point (12, 0). To compare this, we will need to find the equation of the line. Since we know the x-intercept (12,0) and the slope (3/8), we can utilize the slope-intercept form of a linear equation, which is y = mx + b, where m is the slope and b is the y-intercept. But we are not given b, so we need to find the y-intercept by using the point-slope form of a linear equation: y - y1 = m(x - x1). We can input our data to find the equation. Using the slope and the x-intercept, we can find the equation and compare it to the table, which we will do later on. Let's see, in this context, m = 3/8, x1 = 12 and y1 = 0. So, plugging in the data we get y - 0 = (3/8)(x - 12). Simplify it and we get y = (3/8)x - (3/8)(12). Multiplying it, we get y = (3/8)x - (36/8). Then we get y = (3/8)x - 4.5. Therefore, the equation for the first linear function is y = (3/8)x - 4.5. Knowing the equation, now we can easily find and compare the second linear function. Pretty easy, right?

Analyzing the First Linear Function: X-Intercept and Slope

Now, let's take a closer look at our first linear function. We are given that it has an x-intercept of 12 and a slope of 3/8. The x-intercept of 12 tells us that the line crosses the x-axis at the point (12, 0). This single point is where the y-coordinate is zero. The slope of 3/8 means that for every 8 units we move to the right along the x-axis, the line goes up 3 units on the y-axis. This also means that the line goes up. This positive slope indicates an increasing linear function. We can utilize this information to draw a graph of our first function. But let's get into the equation. We previously calculated the equation as y = (3/8)x - 4.5. This equation is in slope-intercept form, making it super easy to visualize and work with the graph. The slope is 3/8 (the coefficient of x), and the y-intercept is -4.5 (the constant term). The y-intercept indicates that the line crosses the y-axis at the point (0, -4.5). So we have two points, the x-intercept at (12, 0) and the y-intercept at (0, -4.5). We can plot those two points, and using the slope of 3/8, we can find the rest of the line, so that we can fully understand how our function looks like and how it behaves. This first linear function's properties are pretty straightforward, we have the slope and the intercept, and it gives us a good baseline for comparing it with our second function derived from the table.

Now that we know the equation and the behavior of the first function, we can now move to the second function, the one provided in the table. We will need to calculate the slope and intercepts. Let's compare both to have a better understanding.

Decoding the Second Linear Function from the Table

Alright, time to crack the code of the second linear function! We're given a table of values:

From this table, we need to extract enough info to compare this second function with our first function. What we need to do is to calculate the slope and the y-intercept. First, let's get the slope. We can do this by picking any two points from the table and using the slope formula: m = (y2 - y1) / (x2 - x1). Let's use the first two points in the table (-2/3, -3/4) and (0, 1/2). Now we just insert the points into the equation to get our slope. m = (1/2 - (-3/4)) / (0 - (-2/3)). Simplify to get m = (5/4) / (2/3). Finally, m = 15/8. The slope is positive, so the function increases. Now we have the slope, which is 15/8. And we can easily get the y-intercept. Notice that when x=0, y = 1/2. So, the y-intercept is 1/2. Now we can get the equation of the linear function from the table, using the slope-intercept form which is y = mx + b. In this case, m = 15/8 and b = 1/2. Therefore, the equation is y = (15/8)x + 1/2. Having found the second function's equation, we can now compare the properties and behavior of both linear functions.

To do this comparison, we need to identify the slope and intercepts of the second function. We've already done it, but let's review! We used two points, (-2/3, -3/4) and (0, 1/2). Then we computed the slope to find m = 15/8. From the table, we directly identified the y-intercept as 1/2, since that is the value of y when x is 0. So, the y-intercept is (0, 1/2). With the slope and y-intercept, we were able to define the equation of the second function. Now we have all the needed information to do a proper and in-depth comparison.

Comparing the Functions: Slope, Intercepts, and Behavior

Okay, guys, time for the grand finale! We've analyzed each function and found their equations, and now we can compare them side by side. Let's put all the information together.

• Function 1: The first function has the equation y = (3/8)x - 4.5. It has a slope of 3/8 and an x-intercept of 12. The y-intercept is -4.5.
• Function 2: From the table, the equation is y = (15/8)x + 1/2. The slope is 15/8, and the y-intercept is 1/2.

Now, let's compare them:

• Slope: Function 2 (15/8) has a much steeper slope than Function 1 (3/8). This means that the line for Function 2 rises more rapidly as x increases. The line for Function 2 is steeper.
• Y-Intercept: Function 1 crosses the y-axis at -4.5, while Function 2 crosses at 1/2. This means that Function 1 starts below the x-axis, and Function 2 starts above it.
• X-Intercept: Function 1 crosses the x-axis at 12. We would need to compute the x-intercept of Function 2 to get a full comparison. We know that the x-intercept occurs when y=0. So, we can set y to zero in the equation, and then solve for x. For Function 2, its equation is y = (15/8)x + 1/2. If y = 0, then 0 = (15/8)x + 1/2. Solve for x: -1/2 = (15/8)x. Multiply both sides by 8/15 to get -4/15 = x. The x-intercept for Function 2 is -4/15 (or -0.267). This means that Function 2 crosses the x-axis to the left of the origin, while Function 1 crosses it to the right.
• Overall behavior: Function 2 increases more rapidly than Function 1. Function 2's line will rise faster than Function 1. Function 1 starts below the x-axis and increases. Function 2 starts above the x-axis and also increases.

So, there you have it! We've successfully compared two linear functions, looking at their key features to understand their behavior. I hope you found this exploration helpful and insightful. Keep practicing, and you'll become a linear function master in no time!

Remember guys, understanding the properties of linear functions helps us to understand lots of real-world scenarios. See you next time, and happy calculating!