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

Hey guys! Today, we're diving into the awesome world of graphing functions, specifically $f(x)=-\sqrt{x}$. We'll be using our trusty graphing calculators to figure out which set of points truly represents this function. So, grab your calculators and let's get this party started!

Understanding $f(x)=-\sqrt{x}$

Alright, let's break down the function $f(x)=-\sqrt{x}$. The first thing you'll notice is the negative sign out front. This is a super important clue! When we graph basic functions, we usually start with something familiar, like $y=\sqrt{x}$. For $y=\sqrt{x}$, we know that the square root function only accepts non-negative numbers (zero and positive numbers) because you can't take the square root of a negative number and get a real result. So, the domain for $y=\sqrt{x}$ is $x \ge 0$. The output, or the $y$-values, will always be non-negative too, meaning $y \ge 0$. For instance, if $x=4$, then $y=\sqrt{4}=2$. If $x=9$, then $y=\sqrt{9}=3$. Easy peasy, right?

Now, let's bring in that minus sign for $f(x)=-\sqrt{x}$. This negative sign outside the square root flips the graph of $y=\sqrt{x}$ vertically across the x-axis. Think of it like a mirror image. So, if $y=\sqrt{x}$ gives us positive $y$-values (or zero), then $f(x)=-\sqrt{x}$ will give us negative $y$-values (or zero). The domain doesn't change; it's still $x \ge 0$ because we can only plug in non-negative values into the square root. However, the range does change. For $f(x)=-\sqrt{x}$, the $y$-values will be less than or equal to zero, meaning $y \le 0$. So, if $x=4$, then $f(4)=-\sqrt{4}=-2$. If $x=9$, then $f(9)=-\sqrt{9}=-3$. See the difference? The $x$-values are positive, but the $y$-values are negative.

When you're using your graphing calculator, you'll input this function as $Y_1 = -\sqrt{X}$. Make sure you're using the negative sign (usually located near the subtraction button) and not the subtraction sign itself. Also, ensure your $X$ variable is correctly entered. Once you graph it, you should see a curve that starts at the origin $(0,0)$ and goes downwards into the fourth quadrant. It will look like the top half of a sideways parabola opening to the right, but flipped upside down. The key takeaway here is that for any positive $x$-value you plug in, you will get a negative $y$-value. This is the fundamental behavior of $f(x)=-\sqrt{x}$.

Using the Graphing Calculator

Alright, let's put our graphing calculators to work! This is where the magic happens, guys. We need to input the function $f(x)=-\sqrt{x}$ into our calculator and see what it looks like. Most graphing calculators have a button for functions, often labeled 'Y=' or 'f(x)'. Press that button, and you'll see a list of equations you can enter. We want to enter our function into one of these slots, let's say $Y_1$.

So, in $Y_1$, you'll type: - sqrt( X ). The sqrt button is usually found under a 'math' or 'catalog' menu. It's crucial to use the negative sign (often a small '-' in parentheses or separate) and not the subtraction sign. Using the subtraction sign will lead to incorrect results. After entering -sqrt(X), you'll want to set your viewing window. For this function, a standard window is usually a good start. Try setting Xmin to -5, Xmax to 10, Ymin to -5, and Ymax to 5. This should give you a decent view of the function's behavior. Hit the 'graph' button, and voilà! You should see a curve that starts at the origin $(0,0)$ and extends into the fourth quadrant, meaning $x$-values are positive and $y$-values are negative.

Now, let's test the points given in the options. We can do this in a couple of ways. The most straightforward method is to use the 'TABLE' feature on your calculator. Go to the table setup (usually by pressing '2nd' then 'graph' to get to the TABLE screen) and make sure 'Auto' is selected for the Independent variable (X) and Dependent variable (Y). Then, go to the main table screen. You can manually enter $x$-values to see the corresponding $f(x)$ values, or if your table is set to auto, you can scroll through values. Alternatively, you can use the 'TRACE' function after graphing. Press 'TRACE', and then you can type in specific $x$-values and see the corresponding $y$-values that the calculator calculates for the function $f(x)=-\sqrt{x}$.

Let's try an $x$-value like $x=4$. Inputting $x=4$ into the function $f(x)=-\sqrt{x}$ gives us $f(4)=-\sqrt{4}=-2$. So, the point $(4,-2)$ should be on the graph. If we try $x=9$, $f(9)=-\sqrt{9}=-3$, giving us the point $(9,-3)$. And for $x=1$, $f(1)=-\sqrt{1}=-1$, giving us the point $(1,-1)$. Notice a pattern? For positive $x$-values, we get negative $y$-values. This is exactly what we expect from $f(x)=-\sqrt{x}$.

Analyzing the Point Options

Now, let's look at the options provided and see which list of points matches our findings. We've already established that for $f(x)=-\sqrt{x}$, we need non-negative $x$-values and non-positive (negative or zero) $y$-values. This immediately helps us eliminate some options. Remember, the domain for $\sqrt{x}$ is $x \ge 0$, and the range for $-\sqrt{x}$ is $y \le 0$.

Let's examine each option:

A. $(-9,3),(-4,2),(-1,1)$: Right away, we see negative $x$-values here. Since the domain of $\sqrt{x}$ requires $x \ge 0$, these points cannot possibly be on the graph of $f(x)=-\sqrt{x}$. The calculator wouldn't even compute a real value for $f(-9)$ or $f(-4)$ or $f(-1)$. So, this option is out.

B. $(1,1),(4,2),(9,3)$: In this option, all the $x$-values are positive, which is good. However, all the $y$-values are also positive. We know that for $f(x)=-\sqrt{x}$, the $y$-values must be negative (or zero). For example, if $x=1$, $f(1)=-\sqrt{1}=-1$, not 1. If $x=4$, $f(4)=-\sqrt{4}=-2$, not 2. This option represents the function $g(x)=\sqrt{x}$, not $f(x)=-\sqrt{x}$. So, this option is also out.

C. $(-9,-3),(-4,-2),(-1,-1)$: Similar to option A, this option includes negative $x$-values. As we discussed, the square root function is not defined for negative inputs in the real number system. Therefore, these points cannot be part of the graph of $f(x)=-\sqrt{x}$. Eliminated.

D. $(1,-1),(4,-2),(9,-3)$: Let's check these points against our function $f(x)=-\sqrt{x}$.

• For the point $(1,-1)$: If $x=1$, then $f(1)=-\sqrt{1}=-1$. This matches! The point $(1,-1)$ is on the graph.
• For the point $(4,-2)$: If $x=4$, then $f(4)=-\sqrt{4}=-2$. This also matches! The point $(4,-2)$ is on the graph.
• For the point $(9,-3)$: If $x=9$, then $f(9)=-\sqrt{9}=-3$. This matches perfectly! The point $(9,-3)$ is on the graph.

All the $x$-values in this option are non-negative, and all the $y$-values are non-positive, which is exactly what we expect from the function $f(x)=-\sqrt{x}$. This option aligns with our calculator's graphing and the properties of the function.

Conclusion: The Correct Points!

So, after using our graphing calculators and carefully analyzing the behavior of the function $f(x)=-\sqrt{x}$, we can confidently say which list of points represents it. We looked at the domain ($x \ge 0$) and the range ($y \le 0$) and how the negative sign affects the graph. We also tested specific points using the calculator's features like the table and trace functions. Options A and C were eliminated because they had negative $x$-values, which are not in the domain of the square root function. Option B was incorrect because it showed positive $y$-values, contradicting the function $f(x)=-\sqrt{x}$.

This leaves us with Option D: $(1,-1),(4,-2),(9,-3)$. These points satisfy the condition $y = -\sqrt{x}$ for non-negative $x$-values, resulting in non-positive $y$-values. When you graph $f(x)=-\sqrt{x}$ on your calculator, you'll see the curve passing through these exact points. It's a beautiful demonstration of how mathematical functions behave and how our calculators can help us visualize them. Keep practicing, and you'll be graphing like a pro in no time! Happy graphing, everyone!