Graphing Y = 2√(3x - 1) + 1: A Step-by-Step Guide

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Graphing y = 2√(3x - 1) + 1: A Step-by-Step Guide

Hey math enthusiasts! Today, we're diving deep into the world of graphing functions, and our star for the day is y = 2√(3x - 1) + 1. You might look at this and think, "Whoa, what's going on here?" But trust me, guys, once we break it down, it's totally manageable and even pretty cool. We'll be going through this step-by-step, so grab your calculators, your notebooks, and maybe a comfy seat, because we're about to conquer this radical function!

Understanding the Building Blocks of Our Function

Before we start plotting points, let's get familiar with the pieces that make up our function, y = 2√(3x - 1) + 1. Think of it like taking apart a cool gadget to see how it works. Our function is built upon the basic square root function, y = √x. This is our foundation. Remember what the graph of y = √x looks like? It starts at the origin (0,0) and curves upwards and to the right. It's the starting point for all square root graphs.

Now, let's look at the transformations happening. We've got a few things going on inside and outside the square root. First, inside the radical, we have (3x - 1). This part affects the horizontal positioning and stretching/compressing of our graph. The coefficient '3' in front of the 'x' is a horizontal compression. It means the graph will be squeezed horizontally compared to a simple y = √x. The '-1' inside the parentheses causes a horizontal shift. Remember, with horizontal shifts, it's the opposite of what you see. So, '-1' actually means we shift the graph one unit to the right. This is super important, guys, don't get tricked by the sign!

Outside the radical, we have the '2' multiplying the square root. This is a vertical stretch. Our graph will be stretched upwards by a factor of 2. If you compare it to y = √x, the same x-value will give you a y-value that's twice as big (before considering the other transformations). Finally, the '+1' outside the entire square root is a vertical shift. This one's straightforward: it shifts the entire graph one unit upwards. So, all the y-values are increased by 1.

Putting it all together, our function y = 2√(3x - 1) + 1 is essentially the basic y = √x graph that has been horizontally compressed by a factor of 3, shifted 1 unit right, stretched vertically by a factor of 2, and then shifted 1 unit up. It sounds like a lot, but by understanding each transformation individually, we can predict the final shape and position of our graph. The domain and range will also be affected by these transformations, which we'll talk about next.

Determining the Domain and Range

Alright, before we plot any points for y = 2√(3x - 1) + 1, let's nail down the domain and range. These are like the boundaries of our graph. For any square root function, the expression inside the square root must be greater than or equal to zero. This is because you can't take the square root of a negative number and get a real number, right? So, for our function, we need 3x - 1 ≥ 0. Let's solve this inequality for x. Add 1 to both sides: 3x ≥ 1. Now, divide by 3: x ≥ 1/3. This tells us that the smallest possible x-value we can plug into our function is 1/3. Therefore, our domain is all real numbers greater than or equal to 1/3. We can write this in interval notation as [1/3, ∞). This is crucial for knowing where our graph starts on the horizontal axis.

Now, let's talk about the range. The range refers to all the possible y-values our function can produce. Remember the basic y = √x graph? Its range is y ≥ 0. Now, consider the transformations we applied to y = √x to get y = 2√(3x - 1) + 1. We have a vertical stretch by a factor of 2 and a vertical shift upwards by 1. The square root part, √(3x - 1), will always output a non-negative number (0 or positive). When we multiply it by 2, it's still non-negative. Then, we add 1. This means the smallest possible value for our function will occur when the square root part is at its minimum, which is 0. So, the minimum y-value is 2 * 0 + 1 = 1. Since the square root can grow infinitely large, and we're multiplying by 2 and adding 1, the y-values can also grow infinitely large. Thus, our range is all real numbers greater than or equal to 1. In interval notation, this is [1, ∞). Knowing the domain and range helps us sketch a more accurate graph and understand the function's behavior. It's like having a roadmap before you start your journey!

Finding the Starting Point (Vertex)

Every good graph needs a starting point, and for our function y = 2√(3x - 1) + 1, this starting point is super important. It's often referred to as the "vertex" for radical functions, similar to parabolas. Where do we find this magical point? It's determined by the domain and the vertical shift. We already found that the domain starts at x = 1/3. This is the x-coordinate of our starting point. To find the y-coordinate, we plug this x-value back into our function:

y = 2√(3 * (1/3) - 1) + 1 y = 2√(1 - 1) + 1 y = 2√(0) + 1 y = 2 * 0 + 1 y = 0 + 1 y = 1

So, our starting point (vertex) is at (1/3, 1). This is the point where the curve of our square root function begins. It's the lowest point on the graph (considering the combined effects of the vertical stretch and shift). Make sure you get this point right, guys, because everything else builds off of it. It's the anchor of our entire graph. If you got the domain and range correct, you should already know this point because the minimum x from the domain corresponds to the minimum y from the range!

Plotting Key Points

Now that we have our starting point (1/3, 1) and our domain (x ≥ 1/3), it's time to find a few more points to help us sketch the curve of y = 2√(3x - 1) + 1. The best way to do this is to pick x-values that are greater than or equal to 1/3 and make the expression inside the square root a perfect square. This makes the calculation much cleaner, trust me! Let's try some values:

  1. When x = 1/3: We already know this gives us y = 1. Our point is (1/3, 1).
  2. Let's choose an x that makes (3x - 1) equal to 1. 3x - 1 = 1 3x = 2 x = 2/3 Now, plug x = 2/3 into the function: y = 2√(3 * (2/3) - 1) + 1 y = 2√(2 - 1) + 1 y = 2√(1) + 1 y = 2 * 1 + 1 y = 3 So, another point is (2/3, 3).
  3. Let's choose an x that makes (3x - 1) equal to 4. 3x - 1 = 4 3x = 5 x = 5/3 Now, plug x = 5/3 into the function: y = 2√(3 * (5/3) - 1) + 1 y = 2√(5 - 1) + 1 y = 2√(4) + 1 y = 2 * 2 + 1 y = 4 + 1 y = 5 So, we have the point (5/3, 5).
  4. Let's choose an x that makes (3x - 1) equal to 9. 3x - 1 = 9 3x = 10 x = 10/3 Now, plug x = 10/3 into the function: y = 2√(3 * (10/3) - 1) + 1 y = 2√(10 - 1) + 1 y = 2√(9) + 1 y = 2 * 3 + 1 y = 6 + 1 y = 7 Our next point is (10/3, 7).

We've got our starting point (1/3, 1) and three more points: (2/3, 3), (5/3, 5), and (10/3, 7). These points give us a good sense of the curve's shape and direction. Remember, since our domain is x ≥ 1/3, we only need to consider points to the right of x = 1/3.

Sketching the Graph

Now for the fun part – putting it all together and sketching the graph of y = 2√(3x - 1) + 1! Grab your graph paper or open up your graphing software. First, set up your x and y axes. Since our domain starts at x = 1/3 and goes to infinity, and our range starts at y = 1 and goes to infinity, you'll want to make sure your axes accommodate these values.

  1. Plot the Starting Point: Mark the point (1/3, 1). This is where your graph begins. Remember, 1/3 is about 0.33, so it's a little to the right of the y-axis.
  2. Plot the Additional Points: Now, plot the other points we found: (2/3, 3), (5/3, 5), and (10/3, 7). Again, estimate the fractions: 2/3 is about 0.67, 5/3 is about 1.67, and 10/3 is about 3.33. These points should look like they're curving upwards and to the right.
  3. Draw the Curve: Starting from (1/3, 1), draw a smooth curve that passes through the other points. Remember that the square root function is not a straight line; it has a gentle, increasing curve. The curve should continue to move upwards and to the right indefinitely, reflecting the domain and range going to infinity.
  4. Consider the Transformations: Keep in mind the transformations we discussed. The horizontal compression by 3 means the graph rises a bit faster than y = √x. The vertical stretch by 2 makes it rise even more steeply. The shifts move the entire graph. Your plotted points should visually confirm these effects.
  5. Labeling: Don't forget to label your axes (x and y) and mark the scale. It's also a good idea to label the key points you plotted, especially the starting point (1/3, 1). If you want to be extra fancy, you can even label the equation of the function next to the curve: y = 2√(3x - 1) + 1.

When you're done, your graph should look like a curve starting at (1/3, 1) and moving upwards and to the right. It will resemble the right half of a parabola opening upwards, but it's specifically a square root function's shape. It's crucial to ensure the curve starts precisely at (1/3, 1) and doesn't extend to the left of x = 1/3, as that would violate the domain. The rate at which it rises is influenced by the '2' and the '3' in the equation. You've successfully graphed a transformed square root function, guys! High five!

Common Pitfalls and How to Avoid Them

Even with the best guides, sometimes we stumble, right? When graphing functions like y = 2√(3x - 1) + 1, there are a few common traps. Let's talk about them so you can steer clear!

  • Confusing Horizontal Shifts: The biggest one, hands down! Remember, with (x - h), you shift right by h. With (x + h), you shift left by h. In our function, we have (3x - 1). It's tempting to just say "shift right by 1" because of the '-1'. But here's the catch: the '3' is also playing a role. To properly isolate the shift, we often factor out the coefficient of x: 3(x - 1/3). Now it's clearer! The '-1/3' inside the parentheses indicates a shift 1/3 unit to the right. Always factor out the coefficient of x inside the radical if you're unsure about the horizontal shift. This makes the transformation explicit and much easier to handle correctly.
  • Getting the Domain Wrong: This is directly linked to the horizontal shift confusion. If you miscalculate the starting point of your domain (e.g., saying x ≥ 1 instead of x ≥ 1/3), your entire graph will be in the wrong place. Always set the expression inside the square root to be greater than or equal to zero (≥ 0) and solve for x carefully. Double-check your algebra, especially with fractions.
  • Ignoring Vertical Stretch/Compression: The coefficient '2' outside the square root means our graph is stretched vertically. If you forget this, you might draw a graph that looks like a basic square root function but is in the wrong position vertically. Remember, for any given x in the domain, the y-value of y = 2√(...) + 1 will be twice as large as the y-value of y = √(3x - 1) + 1 (before the +1 shift). This means the curve should rise more steeply.
  • Mistaking the Starting Point: This often happens if you get the domain wrong or miscalculate the y-coordinate. The starting point (or vertex) is the anchor. If it's off, your whole graph is off. Always calculate the y-coordinate by plugging the minimum x-value from the domain back into the original function. This ensures consistency between your domain, range, and the graph itself.
  • Thinking it's a Straight Line: Square root functions are curves! They have a characteristic shape that starts at a point and curves outwards. Don't draw a straight line segment. Pay attention to the points you plot; they should guide you towards a smooth, curved line.

By being mindful of these common mistakes, guys, you'll be much more confident in accurately graphing any square root function. It's all about careful calculation and understanding how each part of the equation transforms the basic graph. Keep practicing, and you'll be a graphing pro in no time!

Conclusion: You've Mastered the Graph!

And there you have it, math whizzes! We've successfully tackled the graph of y = 2√(3x - 1) + 1. We broke it down, identified the transformations, figured out the domain and range, found our crucial starting point, plotted key points, and finally, sketched the graph. Remember, this process of understanding transformations applies to all sorts of functions, not just square roots. By recognizing the basic function (y = √x in this case) and then systematically analyzing the shifts, stretches, and compressions, you can confidently graph even complex functions.

This function, y = 2√(3x - 1) + 1, is a beautiful example of how algebra and geometry intersect. The domain [1/3, ∞) tells us where the function exists horizontally, and the range [1, ∞) tells us where it exists vertically. The starting point (1/3, 1) is our anchor, and the curve beautifully illustrates the combined effects of vertical stretch and horizontal compression. Keep practicing these steps, and soon you'll be able to visualize and sketch these graphs almost instinctively. Math is all about practice and understanding the underlying principles, so don't shy away from more problems like this. You guys totally crushed it!