Transformations: F(x) = -6x - 6 To G(x) = -3x - 3
Hey guys! Let's dive into a cool math problem today that involves graph transformations. We're going to figure out what kind of transformation turns the graph of the function f(x) = -6x - 6 into the graph of g(x) = -3x - 3. This is a classic question that tests our understanding of how different operations on a function affect its visual representation on a coordinate plane. So, buckle up, and let's get started!
Understanding the Problem
To understand the problem fully, let's break down the key components. We have two linear functions, f(x) and g(x), and we need to identify the transformation that maps one onto the other. The options given are horizontal stretch, vertical shrink, horizontal shrink, and vertical stretch. Each of these transformations has a distinct effect on the graph of a function, so let's explore them one by one to determine the correct answer. When dealing with these types of questions, it's crucial to first identify the parent function, which in this case is a linear function. Then, we need to analyze how the coefficients and constants in the functions change and how these changes affect the graph. This involves understanding the relationship between algebraic manipulations and their geometric interpretations.
Analyzing the Functions f(x) and g(x)
First, let's take a closer look at our functions:
- f(x) = -6x - 6
- g(x) = -3x - 3
Notice that both functions are in the slope-intercept form, which is y = mx + b, where m represents the slope and b represents the y-intercept. For f(x), the slope is -6 and the y-intercept is -6. For g(x), the slope is -3 and the y-intercept is -3. By comparing these values, we can see that both the slope and the y-intercept have changed. The slope has gone from -6 to -3, and the y-intercept has gone from -6 to -3. This means that the graph has been altered in some way, but how? That's what we need to figure out! Understanding the slope and y-intercept is crucial here. The slope tells us how steep the line is and in what direction it's going (positive or negative). The y-intercept tells us where the line crosses the y-axis. Changes in these values will directly affect the orientation and position of the line on the graph.
Exploring Transformation Options
Now, let's explore the transformation options given to us and see which one fits the changes we observe between f(x) and g(x). This is where we put on our detective hats and start eliminating possibilities until we arrive at the correct solution. It's like a process of elimination, but with mathematical reasoning!
A. Horizontal Stretch
A horizontal stretch makes the graph wider along the x-axis. This happens when we replace x with x/k, where k > 1. In this case, a horizontal stretch would affect the slope and y-intercept in a way that's not immediately obvious from our equations. So, let's hold off on this one for a moment and explore the other options. Horizontal transformations are a bit tricky because they affect the x-values, which can sometimes be counterintuitive. A stretch makes the graph wider, while a shrink makes it narrower.
B. Vertical Shrink
A vertical shrink compresses the graph towards the x-axis. This occurs when we multiply the entire function by a constant between 0 and 1. For example, if we multiply f(x) by 1/2, we'd get a vertical shrink. This sounds promising because the coefficients in g(x) are smaller than those in f(x). Vertical transformations are often more straightforward to visualize because they directly affect the y-values. A shrink makes the graph flatter, while a stretch makes it taller.
C. Horizontal Shrink
A horizontal shrink compresses the graph along the x-axis. This happens when we replace x with kx, where k > 1. Similar to a horizontal stretch, this would affect the slope and y-intercept in a less direct way. So, let's keep this in mind but explore the remaining option first. Remember, horizontal shrinks work in the opposite way to what you might expect. Multiplying x by a number greater than 1 actually shrinks the graph horizontally.
D. Vertical Stretch
A vertical stretch pulls the graph away from the x-axis. This occurs when we multiply the entire function by a constant greater than 1. In our case, since the coefficients in g(x) are smaller than those in f(x), a vertical stretch wouldn't make sense. This is because a stretch would make the graph taller, but we need the graph to be compressed. Vertical stretches are essentially the opposite of vertical shrinks. They make the graph taller and steeper.
Finding the Transformation Factor
Okay, guys, so far, it looks like a vertical shrink is the most likely candidate. But let's confirm this by finding the transformation factor. To do this, we can compare the coefficients of x and the constant terms in both functions. This will help us quantify the amount of shrink that has occurred. It's like measuring the distance between two points to understand the scale of the transformation.
Comparing Coefficients
- The coefficient of x in f(x) is -6, and in g(x), it's -3.
- The constant term in f(x) is -6, and in g(x), it's -3.
Notice that both the coefficient of x and the constant term in g(x) are half of those in f(x). This suggests that we're multiplying the entire function f(x) by 1/2 to get g(x). This is the key to confirming our suspicion about a vertical shrink! When the corresponding parts of the function have a consistent ratio, it's a strong indication of a linear transformation like a stretch or shrink.
Verifying the Transformation
Let's verify that multiplying f(x) by 1/2 indeed gives us g(x). This is the crucial step to ensure we haven't made any mistakes and that our reasoning is sound. It's like double-checking your work after solving a complex problem.
Multiplying f(x) by 1/2
1/2 * f(x) = 1/2 * (-6x - 6) = -3x - 3 = g(x)
Woo-hoo! It works! This confirms that the transformation is indeed a vertical shrink by a factor of 1/2. We've successfully connected the dots between the two functions and identified the transformation that links them.
Conclusion
Therefore, the correct answer is B. vertical shrink. The graph of f(x) = -6x - 6 is transformed into the graph of g(x) = -3x - 3 by a vertical shrink by a factor of 1/2. Understanding transformations like these is super important in mathematics as they help us visualize and manipulate functions in various ways. Keep practicing, guys, and you'll master these concepts in no time!
So, next time you encounter a similar problem, remember to analyze the functions, compare the coefficients, and consider the effects of different transformations. You've got this!