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### Course: Algebra 1 (Eureka Math/EngageNY)>Unit 4

Lesson 14: Topic C: Lessons 20-22: Scaling and transforming graphs

# Transforming the square-root function

Sal shows various examples of functions and their graphs that are a result of shifting and/or flipping y=√x. Created by Sal Khan.

## Want to join the conversation?

• What if it 's like 2x+3? with that radical sign over it ? How would you graph that?
• If you have the function y = sqrt(2x+3), you can rewrite the right hand side of this as:
y = sqrt(2(x+3/2)).

Then, using the property that sqrt(ab) = sqrt(a)*sqrt(b), you can rewrite this again as:
y = sqrt(2) * sqrt(x+3/2).

Now, notice that sqrt(2) is no more than a constant, you all you've done is stretched the graph vertically byu a factor of sqrt(2). Then, notice that under the second radical sign, you've got a shift to the left by 3/2. To show how this process makes sense, try graphing both y = sqrt(2x+3) and y = sqrt(2) * sqrt(x+3/2). You should get the same thing.

To graph it, know what the graph of y = sqrt(x) looks like first (its a parabola on its side with only the top half). Then, notice that you've shifted the graph to the left by 3/2 and stretched the entire graph by sqrt(2). All done!
• Why am I not getting any of this.
• lol so it's not just me
• Sal writes sqrt - (x + 3)

I can see the principle of reflection this radical function along line of x = -3. However will this not yield complex numbers? Thank you
• It will not yield imaginary numbers as long as "x" is chosen carefully. We can find exactly for which values of x no complex numbers result. We do this by finding the domain of the function:
ƒ(x) = √[-(x + 3)]
The radicand must be greater than or equal to 0 in order for the function to yield only real numbers:
-(x + 3) ≥ 0
-x - 3 ≥ 0
-x ≥ 3
x ≤ -3
Therefore, all values of "x" that produce real numbers in the aforementioned equation are all numbers lesser than or equal to -3. This is also why the graph of the function starts at -3 on the x-axis and keeps going in the negative direction, but does not go in the positive direction at all. Comment if you have questions.
• Another way to say why x is positive instead of negative, you could say that y=√x -> or x=y^2. Because you want to move x three spaces to the left, you would do x=y^2 -3. Then, to convert it back, it would become y^2=x+3, or y=√x+3. idk just another way to understand it I guess.
• Your logic is good up to here: x=y^2 -3
y^2 = x+3
Then take the square root of both sides (the entire side)
y = √(x+3)
You can't just do the square root of "x"
• You said you have to make sure that whatever is under the radical will equal 0 at the origin like if we want to shift 2 to left we +2 to make sure that the number under radical is zero,but i want to ask why is that necessary to make it 0.?
• Radicals are tricky things. To evaluate points on a radical function, you want to think about values for x that make the radicand (value under the radical) equal to a perfect square, like 0, but also 1, 4, 9, 16, etc.
• how do you compute √-(x+3) ? I thought you couldn't have negatives under the radical
• does anyone know a video or link that describes how to find the x intercept of a transformation of the function y = cube root of x^2?
(1 vote)
• I don't know of a video or link, but if you express the cube root of x^2 as x^(2/3) power, I'm sure you can find the x-intercepts by substituting 0 for y and solving for x with exponent rules.
• What would the graph of y = 3* sqrt(x) or y = sqrt(3x) or y = -3*sqrt(x), etc.?
(1 vote)
• y=3*sqrt(x) would be the a slightly steeper version of y=sqrt(x). Think of it this way: each value for sqrt(x) is multiplied by three, making it it much steeper.

y=-3*sqrt(x) would be the exact mirror along the y axis of y=3*sqrt(x).

Hope this helped.
• At , what is y when sqrt(x+3) is negative? I don't understand!
(1 vote)
• sqrt(x) means positive (principal) sqrt always. If someone wants the negative sqrt they must write -sqrt(x). This is a rule.

When -(x+3) is negative (that is, to the right of -3), y = sqrt(-(x+3)) is undefined. At -3, -(x+3) is 0. As x moves to the left of -3, -(x+3) increases from 0, and so its sqrt, y, increases.
(1 vote)
• When you graph a radical function how do you tell whether the x-value is negative or positive? I get that the y-value maintains it's negativity/positivity from the equation but the x seems to switch at random.
(1 vote)
• All the possible x values come from the domain of the function. It's not random.
(1 vote)