Graphing $f(x) = -\sqrt{x}$: Find 3 Points

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Hey guys! Today, we're diving into the fascinating world of graphing calculators and square root functions. Specifically, we're going to explore the function f(x)=−xf(x) = -\sqrt{x}. Our main goal is to use the graphing calculator to visualize this function and then pinpoint three points that confidently sit on its graph. So, grab your calculators, and let's get started!

Understanding the Function f(x)=−xf(x) = -\sqrt{x}

First, let’s break down what this function actually means. The square root function, x\sqrt{x}, takes a non-negative number xx and returns the non-negative number that, when multiplied by itself, equals xx. For example, 9=3\sqrt{9} = 3 because 3∗3=93 * 3 = 9. Now, the negative sign in front of the square root, i.e., −x-\sqrt{x}, simply means that we take the result of the square root and flip its sign. So, −9-\sqrt{9} would be −3-3. Essentially, this negative sign reflects the standard square root function over the x-axis.

Before we jump into using the graphing calculator, it’s important to understand that the domain of this function is x≥0x \geq 0. Why? Because you can only take the square root of non-negative numbers and get a real number result. Trying to take the square root of a negative number introduces us to the world of imaginary numbers, which isn't what we're focusing on today. Remember, the domain is all the possible input values (x-values) that you can plug into the function, and the range is all the possible output values (y-values) that the function can produce. For f(x)=−xf(x) = -\sqrt{x}, the range is f(x)≤0f(x) \leq 0, meaning the function will only output zero or negative values.

When we talk about points on the graph of a function, we're referring to coordinate pairs (x,y)(x, y) that satisfy the function's equation. In other words, if you plug in the xx-value into the function, you should get the corresponding yy-value. For example, the point (4,−2)(4, -2) lies on the graph of f(x)=−xf(x) = -\sqrt{x} because f(4)=−4=−2f(4) = -\sqrt{4} = -2. This is the fundamental concept we'll use to check if a given point lies on the graph.

Using the Graphing Calculator

Now, let’s fire up our graphing calculators! Here’s how to graph the function f(x)=−xf(x) = -\sqrt{x}:

  1. Turn on your calculator: This might seem obvious, but it’s always the first step!
  2. Press the "Y=" button: This button allows you to enter the functions you want to graph. You’ll see a screen with lines like Y1=, Y2=, etc.
  3. Enter the function: Type in "-" (the negation sign, not the subtraction sign), then press the "2nd" button followed by the "x²" button (which usually has the square root symbol above it). Then, type in "X" (usually found near the ALPHA button). So, you should have Y1 = -√(X).
  4. Adjust the window (if necessary): Sometimes, the default window settings won’t show the part of the graph you're interested in. Press the "WINDOW" button to adjust the x-min, x-max, y-min, and y-max values. For this function, a window with x-min = 0, x-max = 10, y-min = -4, and y-max = 1 might be a good starting point.
  5. Press the "GRAPH" button: This will display the graph of the function on your screen.

Once the graph is displayed, you’ll see a curve that starts at the origin (0, 0) and extends downwards to the right. This is the visual representation of the function f(x)=−xf(x) = -\sqrt{x}. The graph confirms our earlier understanding that the function only outputs zero or negative values and is only defined for non-negative x-values. It's a reflection of the standard square root function across the x-axis. To find specific points, you can use the "TRACE" function or the "TABLE" function on your calculator. The "TRACE" function allows you to move a cursor along the graph and see the coordinates of the points as you move. The "TABLE" function displays a table of x and y values for the function, which can be very helpful for identifying points. Always remember that your graphing calculator is a powerful tool for visualizing functions and finding points on their graphs.

Verifying the Points

Now that we have the graph, let’s check which of the given lists contain three points that lie on the graph of f(x)=−xf(x) = -\sqrt{x}.

Option A: (1, -1), (4, -2), (9, -3)

  • For (1, -1): f(1)=−1=−1f(1) = -\sqrt{1} = -1. This point lies on the graph.
  • For (4, -2): f(4)=−4=−2f(4) = -\sqrt{4} = -2. This point lies on the graph.
  • For (9, -3): f(9)=−9=−3f(9) = -\sqrt{9} = -3. This point lies on the graph.

All three points in option A satisfy the function, so this list seems promising.

Option B: (-9, 3), (-4, 2), (-1, 1)

Wait a minute! Remember that the domain of our function is x≥0x \geq 0. This means we can only plug in non-negative values for xx. Since all the x-values in this list are negative, none of these points can lie on the graph of f(x)=−xf(x) = -\sqrt{x}. We can stop right here and eliminate this option.

Option C: (1, 1), (4, 2), (9, 3)

  • For (1, 1): f(1)=−1=−1f(1) = -\sqrt{1} = -1. This point does not lie on the graph (it should be (1, -1)).
  • For (4, 2): f(4)=−4=−2f(4) = -\sqrt{4} = -2. This point does not lie on the graph (it should be (4, -2)).
  • For (9, 3): f(9)=−9=−3f(9) = -\sqrt{9} = -3. This point does not lie on the graph (it should be (9, -3)).

None of the points in option C satisfy the function, so we can eliminate this option as well.

Conclusion

Therefore, the correct answer is Option A: (1, -1), (4, -2), (9, -3). These are the three points that confidently lie on the graph of the function f(x)=−xf(x) = -\sqrt{x}. Always remember to check the domain of the function and carefully evaluate each point to make sure it satisfies the function's equation. Keep practicing with your graphing calculator, and you'll become a pro in no time!