Find The Function With A Rate Of Change Of -3

Hey there, math whizzes! Today, we're diving deep into the fascinating world of functions and their rates of change. You know, that slope thingy that tells us how much our y-value is dancing around as our x-value takes a little step. We've got a specific mission, guys: to pinpoint which function among the options provided has a constant rate of change that's precisely -3. That means for every step up in x, our y should be taking a dip of 3, and this pattern needs to hold true all the time for the function to qualify. Let's break down each option and see which one fits the bill. Get ready, because we're about to become rate-of-change detectives!

Analyzing Option A: The Table of Values

First up, let's get our hands dirty with Option A, which is presented in a neat little table. This is often the easiest way to spot a constant rate of change, especially if the x-values are increasing by a consistent amount. In this table, our x-values are going up by 1 each time (0, 1, 2, 3). This is perfect for our investigation! Now, let's look at how the y-values are behaving. We start at y=2 when x=0. When x jumps to 1, y leaps to 5. What's the change in y? It's 5 - 2 = 3. Okay, so the rate of change from x=0 to x=1 is +3. Now, let's check the next step. From x=1 to x=2, y goes from 5 to 8. The change in y here is 8 - 5 = 3. Still +3! And finally, from x=2 to x=3, y moves from 8 to 11. The change in y is 11 - 8 = 3. So, for every increase of 1 in x, the y-value consistently increases by 3. This means the constant rate of change for the function represented by Table A is +3, not -3. So, unfortunately, this option doesn't meet our -3 requirement. Keep those detective hats on, though; we've got more clues to follow!

Investigating Option B: The Set of Ordered Pairs

Alright, moving on to Option B, which is given as a set of ordered pairs: {(1,5), (2,2), (3,-5), (4,4)}. Here, the x-values are increasing by 1 each time (1, 2, 3, 4), which is super helpful. We can now calculate the change in y for each step and see if it's a constant -3. Let's start with the jump from (1,5) to (2,2). The change in x is 2 - 1 = 1. The change in y is 2 - 5 = -3. Awesome! So far, the rate of change is -3. Now, let's look at the next pair, from (2,2) to (3,-5). The change in x is 3 - 2 = 1. The change in y is -5 - 2 = -7. Uh oh. The rate of change here is -7. Since the rate of change isn't consistent between these steps (-3 then -7), this function does not have a constant rate of change. Therefore, Option B is also not our target function. We're getting closer, guys! Don't give up on this mathematical mystery.

Cracking Option C: The Equation

Finally, we arrive at Option C, presented as an equation: $2y = -6x + 12$. This one requires a little bit of algebraic manipulation to get it into a more recognizable form, specifically the slope-intercept form, $y = mx + b$. In this form, 'm' directly represents the constant rate of change (the slope), and 'b' is the y-intercept. Our goal is to isolate 'y' on one side of the equation. So, we have $2y = -6x + 12$. To get 'y' by itself, we need to divide every term on both sides of the equation by 2. Let's do it:

$\frac{2y}{2} = \frac{-6x}{2} + \frac{12}{2}$

This simplifies to:

$y = -3x + 6$

Now, look at this! It's in the $y = mx + b$ format. The coefficient of 'x' is -3, which is our 'm', and the constant term is 6, which is our 'b'. This means the constant rate of change for the function represented by the equation $2y = -6x + 12$ is indeed -3. This is exactly what we were looking for! So, Option C is our winner, guys. This function consistently decreases by 3 units in its y-value for every 1-unit increase in its x-value. It's a perfect match for our requirement of a constant rate of change equal to -3. High fives all around!

Wrapping It Up: Understanding Rate of Change

So, what did we learn from this mathematical expedition? The constant rate of change, often referred to as the slope in linear functions, is a fundamental characteristic that tells us how one variable changes in relation to another. When this rate is constant, the function is linear, meaning its graph is a straight line. A positive rate of change signifies that as x increases, y also increases (an upward trend). A negative rate of change, like the -3 we found in Option C, indicates that as x increases, y decreases (a downward trend). Zero rate of change means y stays constant regardless of x (a horizontal line). A situation with an undefined rate of change typically corresponds to a vertical line where x is constant. Understanding how to calculate and identify the rate of change from different representations – tables, sets of points, or equations – is a crucial skill in algebra and beyond. It helps us model real-world phenomena, predict future values, and understand the relationships between different quantities. Keep practicing, and you'll become rate-of-change ninjas in no time!

Final Answer Recap

To recap our detective work:

• Option A showed a constant rate of change of +3.
• Option B did not have a constant rate of change; it varied.
• Option C, after rearranging the equation to $y = -3x + 6$, clearly demonstrates a constant rate of change of -3.

Therefore, the function that has a constant rate of change equal to -3 is Option C. Well done for following along, everyone! Keep exploring the awesome world of math!