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

Question

Accepted Answer

Welcome back. I am so glad you're here. We're told for the following function sketch its graph, our function is G of X equals negative square root of and then all inside the radical negative 25 X. All right, let's start off by drawing a really rough sketch of a coordinate plane. We have a vertical Y axis, horizontal X axis, they come together at the origin in the middle and we will keep going with this a little bit to get some more details for how to scale our axes and label our units. So first thing when we are graphing a function, we want to look at it and see what its base function is. And here the base function is taking the square root of X. So we can use our, our base function of Y equals the square root of X. And we recall from previous lessons that that is going to be kind of half of a parabola. It starts at the origin and it's going to remain in the first quadrant. It's going to curve up and head toward the positive infinity for the X axis more quickly than positive infinity for the Y axis. And that's what our Y equals the square root of X looks like. Now, what's the difference between that and our G of X function? There are three transformations happening with our G of X function. First, if we look inside the radical that X is being multiplied by 25 and it's also being multiplied by negative, we look at those two separately. And then after we take the square root, so outside the radical, we're also multiplying by a negative. Each of those three things mean something different. So let's start with multiplying X by 25 inside the radical because that's happening inside the radical directly to the X that indicates a horizontal shift. And more specifically because 25 is greater than one, it indicates horizontal shrinking. Now, the second transformation is inside of that radical. We're multiplying the X by a negative when that's happening directly to the X. It's before we take the square root of it, that is going to indicate reflection over the Y axis. All right. The third transformation taking place is multiplying it by the negative after we take the square root. And again, that indicates reflection. But this time because it happens after we've taken the square re of X, it's reflection over the X axis. So if we think about this with our horizontal shrinking, it's gonna get squished up a little bit and then reflection over the Y axis. So it's as though it's flipping over and heading to the left toward negative infinity for the X values, but positive infinity for the Y values. But then as we reflect it over the X axis, it's now heading toward negative infinity for both the X and Y values. So from the origin, we still have this curve, but now it's a little more squished and we're heading toward negative infinity for both X and Y. Now that is not a precise graph, but it tells us approximately what's going on. Now, we can get a more precise graph. Since we know essentially what's going on by creating an XY table, we use the rough sketch of a graph to see what X values we can choose, we plug those in to our function, negative square root of negative 25 X come up with the corresponding Y values and then plot those points. So let's create a new graph. We have a vertical Y axis and a horizontal X axis. And we'll be focusing on the third quadrant where our graph will be, we can see from our sketch that we'll have a domain for our X axis from negative infinity to zero. So we want to choose X values within that domain. So I'm going to choose X values of zero, negative one and negative four. So keeping those X values at between zero and negative five, I can label my X axis. So from heading to the left of the origin will have a negative one, a negative two, a negative three, a negative four and a negative five. And then the X axis continues toward negative infinity. And we could label the right hand side of the X axis similarly, but I am just going to focus on that third quadrant. Now we'll figure out what our Y values will be and then we can label our Y axis. So let's plug zero N for X in our function, we have a negative square of negative 25 multiplied not by X but by zero. Well, negative 25 multiplied by zero is zero. So we have the negative square root of zero, the square root of zero is zero. So we have a negative zero, which is just zero. So our first point is at the origin. Now let's find the next Y value. We have a negative square root of negative 25 multiplied not by X but by negative one, negative 25 multiplied by negative one is positive 25. So we have negative square root of 25. The square root of 25 is five. So this equals a negative five. So our next point is at negative one, negative five. Let's figure out the last Y value and then we can draw our scale. So we've got a negative square of negative 25 multiplied by our X value of negative four. Negative 25 multiplied by negative four is positive 100. So we have negative square rut of 100 while the square rut of 100 is 10. So we have negative 10. All right. So for our Y axis, if we're just going to label beneath the origin, for our negative Y values, we can have units um skip counting by twos to get to negative 10. So we'll have a negative two, negative four, negative six, negative eight, negative 10. All right. And so now we can plot these other two points. We have one at negative one, negative five from the origin, we go to the left one and we go down five. And for the final point negative four, negative 10 from the origin, we go to the left four and we go down 10 and we can connect these as smoothly as possible beginning at the origin going through our point negative one, negative five and then our point negative four, negative 10 almost going through it. I was close and then heading toward negative infinity for both the X and Y values. And there is our G of X equals negative square root of negative X. Well done. We'll catch you on the next one.

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

Key Concepts

Domain of a Function

Square Root Function and Transformations

Graphing Functions with Negative Inputs

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