Find The Linear Function From Point-Slope Equation A Detailed Explanation
Hey there, math enthusiasts! Today, we're diving into the world of linear functions and point-slope equations. We've got a fun little puzzle to solve: figuring out which linear function matches the line defined by the point-slope equation y + 7 = -2/3(x + 6). So, let's put on our detective hats and get started!
Understanding the Point-Slope Form
Okay, before we jump into solving, let's quickly recap the point-slope form of a linear equation. This form is super handy because it lets us write an equation if we know a point on the line and the slope of the line. The general form looks like this: y - y1 = m(x - x1), where (x1, y1) is the point and m is the slope.
In our case, we're given the equation y + 7 = -2/3(x + 6). Notice how it closely resembles the point-slope form? We can see that the slope, m, is -2/3. Also, we can identify a point on the line by looking at the values being subtracted from y and x. Remember that the formula has subtractions, so we have to consider the signs carefully. Here, we have y + 7, which can be rewritten as y - (-7), and x + 6, which is the same as x - (-6). This tells us that the point on the line is (-6, -7).
To further solidify our understanding, let's break down why this form is so useful. Imagine you're given a graph with a line on it, and you can easily spot a point and calculate the slope. With the point-slope form, you can immediately write down the equation of the line without needing to calculate the y-intercept first. It's a real time-saver and a fundamental concept in linear equations. Mastering the point-slope form allows us to swiftly transition between different representations of linear equations and understand their properties more intuitively.
Converting to Slope-Intercept Form
Now, our goal is to find the linear function in the form f(x) = mx + b, which is the familiar slope-intercept form. This form is great because it directly shows us the slope (m) and the y-intercept (b) of the line. To get there, we need to convert our point-slope equation into slope-intercept form. This involves a little bit of algebraic manipulation, but nothing we can't handle!
The key is to isolate y on one side of the equation. Let's start with our equation: y + 7 = -2/3(x + 6). The first step is to distribute the -2/3 on the right side of the equation. This means multiplying -2/3 by both x and 6: y + 7 = (-2/3)x + (-2/3)6. Simplifying the multiplication, we get y + 7 = -2/3x* - 4. See? We're getting closer!
Next, we need to get y by itself. To do this, we subtract 7 from both sides of the equation: y + 7 - 7 = -2/3x - 4 - 7. This simplifies to y = -2/3x - 11. And there we have it! We've successfully converted our point-slope equation into slope-intercept form. The equation y = -2/3x - 11 tells us that the slope of the line is -2/3 and the y-intercept is -11. Now, we can easily identify the correct linear function.
Understanding this conversion process is crucial because it highlights the relationship between different forms of linear equations. Being able to switch between point-slope and slope-intercept forms gives us flexibility in solving problems and interpreting linear relationships. It's like having different tools in our mathematical toolbox, each useful in its own way.
Identifying the Correct Linear Function
Alright, we've done the heavy lifting and transformed our equation into the slope-intercept form: y = -2/3x - 11. Now comes the fun part: matching this equation to the given options. Remember, we're looking for the linear function that represents the same line.
The options are given in function notation, f(x) = mx + b. This is just a fancy way of saying y = mx + b, where f(x) represents the value of y for a given x. So, we're essentially looking for the option that has the same slope and y-intercept as our equation.
Let's quickly recap our equation: y = -2/3x - 11. The slope (m) is -2/3, and the y-intercept (b) is -11. Now, let's look at the options:
- A. f(x) = -2/3 x - 11
- B. f(x) = -2/3 x - 1
- C. f(x) = -2/3 x + 3
- D. f(x) = -2/3 x + 13
By comparing the options to our equation, we can see that option A, f(x) = -2/3 x - 11, perfectly matches our slope and y-intercept. It has a slope of -2/3 and a y-intercept of -11, just like our equation. Therefore, option A is the correct answer!
This process of comparing equations highlights the importance of understanding the different forms of linear equations and how they relate to each other. By converting to slope-intercept form, we made it easy to visually identify the matching function. It's a powerful technique that can be applied to many similar problems.
Why Other Options Are Incorrect
It's always a good idea to understand why the other options are incorrect. This helps solidify our understanding of the concepts and prevents us from making similar mistakes in the future. So, let's take a quick look at why options B, C, and D are not the correct linear functions.
- Option B: f(x) = -2/3 x - 1. This option has the correct slope (-2/3), but the y-intercept is -1, which is different from our calculated y-intercept of -11. So, this line would have the same steepness as our line, but it would cross the y-axis at a different point.
- Option C: f(x) = -2/3 x + 3. Again, the slope is correct (-2/3), but the y-intercept is +3. This line would also have the same steepness, but it would cross the y-axis at a different point, higher up than our line.
- Option D: f(x) = -2/3 x + 13. This option has the correct slope (-2/3), but the y-intercept is +13. This line, like the others, would have the same steepness but a different y-intercept.
The key takeaway here is that the y-intercept is crucial in defining a linear function. Even if the slope is the same, different y-intercepts will result in different lines. This understanding is fundamental to working with linear equations and graphing lines accurately.
Conclusion
Great job, everyone! We successfully navigated through the point-slope form, converted it to slope-intercept form, and identified the correct linear function. Our journey started with the equation y + 7 = -2/3(x + 6) and ended with us confidently choosing option A, f(x) = -2/3 x - 11, as the matching linear function.
Remember, the key to success with these types of problems is understanding the different forms of linear equations and how to convert between them. The point-slope form is great for writing equations when you know a point and the slope, while the slope-intercept form is perfect for quickly identifying the slope and y-intercept. By mastering these forms and practicing conversions, you'll be well-equipped to tackle any linear function challenge that comes your way.
So, keep practicing, keep exploring, and most importantly, keep having fun with math! You've got this!
