Transforming F(x) = -10x - 5 Into G(x) = -2x - 1: How?

by TextBrain Team 55 views

Hey guys! Ever wondered how one graph can morph into another? Let's dive into the fascinating world of graph transformations! Today, we're tackling a specific question: What kind of transformation turns the graph of f(x) = -10x - 5 into the graph of g(x) = -2x - 1? This is a classic problem in algebra, and understanding it will give you serious superpowers when it comes to visualizing and manipulating functions. So, buckle up, and let's get started!

Understanding the Functions

Before we jump into the transformations, let's make sure we're all on the same page about what these functions, f(x) and g(x), actually look like. Both f(x) = -10x - 5 and g(x) = -2x - 1 are linear functions. This means they represent straight lines when graphed on a coordinate plane. The general form of a linear function is y = mx + b, where m is the slope (which tells us how steep the line is) and b is the y-intercept (the point where the line crosses the vertical y-axis).

  • For f(x) = -10x - 5, the slope (m) is -10 and the y-intercept (b) is -5. This tells us that the line is quite steep and slopes downwards from left to right (because the slope is negative). It also crosses the y-axis at the point (0, -5).
  • For g(x) = -2x - 1, the slope (m) is -2 and the y-intercept (b) is -1. This line is also sloping downwards, but it's less steep than f(x) (because the absolute value of -2 is smaller than the absolute value of -10). It crosses the y-axis at (0, -1).

Now that we have a good grasp of what these lines look like individually, we can start thinking about how to transform one into the other. The key here is to consider how the slope and y-intercept change between the two functions. Specifically, we need to figure out what we can do to the slope of -10 in f(x) to make it -2 in g(x), and what we can do to the y-intercept of -5 to make it -1.

Identifying Potential Transformations

Okay, so we know we're going from a steeper line (f(x)) to a less steep line (g(x)). This immediately suggests some kind of compression or shrink. But is it happening vertically or horizontally? That's the million-dollar question! To figure this out, let's consider the options presented in the original problem:

  • A. Vertical Stretch: A vertical stretch would make the line steeper, which is the opposite of what we want. So, we can rule this one out.
  • B. Vertical Shrink: A vertical shrink compresses the graph vertically, making it less steep. This sounds promising!
  • C. Horizontal Stretch: A horizontal stretch stretches the graph horizontally. While this can affect the perceived steepness of the line, it does so in a different way than a vertical transformation. We'll need to investigate further to see if this is the best fit.
  • D. Horizontal Shrink: A horizontal shrink compresses the graph horizontally. Similar to a horizontal stretch, it can impact steepness, but we need to think carefully about how.

So, right now, vertical shrink is looking like our prime suspect. But let's not jump to conclusions just yet. We need to quantify the transformation – that is, figure out how much the graph is being shrunk or stretched.

Determining the Scale Factor

This is where the math gets really interesting! To determine the scale factor of the transformation, we need to compare the slopes of the two lines. Remember, the slope tells us how much the line rises (or falls) for every unit we move to the right.

  • The slope of f(x) is -10.
  • The slope of g(x) is -2.

What do we need to multiply -10 by to get -2? The answer is 1/5 or 0.2. This means the slope of g(x) is 1/5 the slope of f(x). This is a crucial piece of information! It tells us that the vertical change in g(x) is 1/5 of the vertical change in f(x). This strongly suggests a vertical shrink by a factor of 1/5.

But wait, there's more! We also need to consider the y-intercepts. The y-intercept of f(x) is -5, and the y-intercept of g(x) is -1. A vertical shrink by a factor of 1/5 would transform the y-intercept of -5 to -1 (since -5 * (1/5) = -1). This further solidifies our suspicion that a vertical shrink is the correct transformation.

Visualizing the Transformation

Sometimes, the best way to understand a transformation is to see it. Imagine the graph of f(x) = -10x - 5. It's a steep line plunging downwards. Now, imagine squeezing the graph vertically, like you're pressing it down towards the x-axis. This would make the line less steep, bringing it closer to the slope of g(x) = -2x - 1. The y-intercept would also move closer to the x-axis, from -5 to -1.

If we were to perform a horizontal stretch or shrink, the line would pivot around the y-intercept, changing its steepness in a different way. It wouldn't directly compress the vertical distance between the line and the x-axis, which is what we observe in this case.

The Final Verdict

Alright, guys, we've analyzed the slopes, the y-intercepts, and even visualized the transformation. Based on our findings, the transformation that converts the graph of f(x) = -10x - 5 into the graph of g(x) = -2x - 1 is a B. vertical shrink. Specifically, it's a vertical shrink by a factor of 1/5.

Why This Matters

Understanding graph transformations isn't just about solving textbook problems. It's a fundamental skill in mathematics and has applications in various fields, including physics, engineering, and computer graphics. Being able to visualize how functions change when they're stretched, shrunk, or shifted allows you to:

  • Predict the behavior of systems: In physics, for example, transformations can help you understand how the amplitude of a wave changes over time.
  • Design efficient algorithms: In computer graphics, transformations are used to manipulate objects in 3D space.
  • Solve complex equations: By understanding how transformations affect the solutions of equations, you can simplify problems and find solutions more easily.

Practice Makes Perfect

The best way to master graph transformations is to practice! Try working through different examples, experimenting with different functions, and visualizing the results. You can also use graphing calculators or online tools to help you see the transformations in action.

Here are a few exercises to get you started:

  1. What transformation converts the graph of f(x) = x² into the graph of g(x) = 4x²?
  2. What transformation converts the graph of f(x) = |x| into the graph of g(x) = |x - 3|?
  3. Describe the transformations that convert the graph of f(x) = sin(x) into the graph of g(x) = 2sin(x + π/2) - 1.

By tackling these problems, you'll not only solidify your understanding of graph transformations but also develop your problem-solving skills and mathematical intuition.

Wrapping Up

So, there you have it! We've successfully transformed one line into another and learned a valuable lesson about graph transformations along the way. Remember, the key is to analyze the changes in slope and y-intercept and to visualize the transformation in your mind's eye.

Keep practicing, keep exploring, and keep transforming! You've got this!