Graph each function. See Examples 1 and 2.

ƒ(x) = -√-x

Question

Graph each function. See Examples 1 and 2.

ƒ(x) = -√-x

Accepted Answer

Welcome back. I am so glad you're here. We are asked to plot the given function and our given function is G of X equals negative square root of negative 11 X. Then we're given a graph, we have a vertical Y axis, a horizontal X axis. They come together at the origin. The domain for what's drawn for our X axis is from negative 25 to positive five. And the range for what's drawn for our Y axis is from negative 20 to positive five. All right. So looking at our function, we want to ask ourselves what's our base function here? What's happening to the X? And we are taking the square root of that X. So the base function is Y equals the square rut of X. And we recall from previous lessons that that base function of Y equals the square rut of X has its beginning at the origin. And then it curves up into the first quadrant toward positive infinity for both X and Y but toward positive infinity for X faster than toward positive infinity for Y. And that's our Y equals the square root of X. Now, what's the difference between that and our given function, well, there are three main differences here. Our X is being multiplied by 11. And I mean, it's being multiplied by negative 11, but we're going to look at the negative and the 11 separately. So that's the second transformation here, it's being multiplied by a negative. And then outside of the radical after we've taken the square root, we also uh have a negative being multiplied by our number. So those three things mean three different transformations multiplying the X by the 11. It's happening directly to the X before we take the square rot. So that means that it's going to indicate a horizontal change. And specifically because that 11 looking just at the 11 and not the negative, the 11 is greater than one, it indicates horizontal shrinking. Now looking at that negative that's being multiplied by that X, it's happening inside the radical directly to the X before we take the square root. So that's going to indicate reflection over the Y axis. The last transformation here, we're multiplying that negative after we've taken the square root, that's also going to indicate reflection. But because it's not directly to that X, it's after we take the square red, it's reflection over the X axis. So those are the three shifts. So we can look at our Y equals the square root of X, we shrink it a bit horizontally, reflect it over the Y axis and then reflect it over the X axis. And we have something starting at the origin that is going to curve toward negative infinity for both X and Y. But we can be a little bit more specific with our final answer of a graph. By creating an XY chart, we're going to select some values for X. Plug those into our function into negative square root of negative 11 X find the corresponding Y values and plot those points because we understand what's going on with our function. We don't have to choose too many X values. And because we can see what the domain of our function is, it's from negative infinity to zero, we can select X values within that domain. So I'm going to select zero negative one and negative 11 for my X values. Now plugging those in will have a negative square root of negative 11 multiplied by zero. Well, negative 11 multiplied by zero is zero. The square root of zero is zero and negative zero is zero. So that first point is just going to be at the origin. So we can plot that point next. For negative one, we have a negative square root of negative 11 multiplied by negative one, negative 11 multiplied by negative one is positive 11. So this simplifies two, the negative square root of 11 and the negative square root of 11 looking at just the square root of 11, that's going to be a number between three and four. So we're looking at Y values between negative three and negative four from the origin, we go to the left one for that X value of negative one. And then we go down between three and four for that Y value of negative square foot 11. All right. Last X value of negative 11, we have negative square it of negative 11 multiplied by negative 11, negative 11 multiplied by negative 11 is positive 121. So we have a negative square root of 121. The square root of 121 is 11. So this is negative 11 from the origin. We're going to the left 11 and I'll do my best here and might not hit it exactly 14, 13, 12, 11 and then down 11, which I think is about right here. And now we've got those three points plotted. We can connect them as smoothly as possible from the origin going through our other two points and heading toward negative infinity for both the X and Y values. And there is our G of X equals negative squarer of negative 11 X. Well done. We'll catch you on the next one.

Graph each function. See Examples 1 and 2. ƒ(x) = -√-x

Key Concepts

Square Root Function

Domain and Range

Graphing Transformations

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