Main content

### Course: Algebra 2 > Unit 9

Lesson 6: Graphs of square and cube root functions# Radical functions & their graphs

Practice some problems before going into the exercise.

## Introduction

#### In this article, we will practice a couple of problems where we should match the appropriate graph to a given radical function.

### Practice question 1: Square-root function

The graph of $y=\sqrt{x}$ is shown below.

### Practice question 2: Cube-root function

The graph of $y=\sqrt[3]{\phantom{A}x}$ is shown below.

## Want to join the conversation?

- I am confused as how to graph a cube root equation when there is a negative exponent outside of the radical. Can someone help?(41 votes)
- (cbrt(a+5))^(-8) does not equal (cbrt(a+5))^(1/8)

(cbrt(a+5))^(-8) = 1/((cbrt(a+5))^8) which then equals 1/((a+5)^(8/3))

****cbrt means cube root****(7 votes)

- I'm really confused with the cube roots part; Sal didn't talk about it any of the videos, I'm not finding the "Show Answer" here helpful, and I can't figure it out.(16 votes)
- Cube roots are pretty similar to square roots, except that their value is the number that, when multiplied by itself three times, is equal to the number under the radical, just as the square root of a number is the number that, when multiple by itself twice, is equal to the number under the radical. For example, the cube root of 8 is 2, because 2 x 2 x 2 is 8, just as the square root of 4 is 2, because 2 x 2 is 4. So, to graph a cube root function, you find the perfect cubes (numbers like 1 (1 x 1 x 1), 8 (2 x 2 x 2), 27 (3 x 3 x3), -1 (-1 x -1 x -1), -8 (-2 x -2 x -2), -27 (-3 x -3 x -3) etc.) and plot them on the graph. Then, just "connect the dots" and you have the graph (or at least a good approximation.) All the rules of shifting and stretching functions that apply to square root functions apply to cube root functions as well. (Note, however, that cube root functions give value outputs for negative values for x, since you are multiplying it three times, ensuring a real number value.)

I hope that helps.(41 votes)

- Is there a secret method to graphing the cubic root and the square root without a graphing calculator?(6 votes)
- You can graph just about anything by hand by evaluating the function at enough points and plotting all of them.(9 votes)

- Is there a video on cube root functions? Or do you not need to know about cube root functions for the question? Basically, I'm asking if the question wants you to know anything about cube root graphs.(7 votes)
- For this question, knowledge of cube-root functions is not required. The question is simply trying to show the connection between square and cube root functions. If you take the graph of a y = x^3 function and reflect it over the line y = x, it will look like a sideways y = x^3 graph (or cube-root graph), like how a "sideways" parabola (y = x^2) is a radical function (well, half of a sideways parabola, anyway, because of domain issues.)

Basically, just imagine the graph of y = x^3, turn it 90 degrees clockwise, and do translations as necessary.(7 votes)

- How do I graph a cube root function that has x as a negative as opposed to the negative being outside the radical?(10 votes)
- I am confused on how we are supposed to change the graph when the x on the inside of the radical is negative.(8 votes)
- sqrt(-x) is reflected over the y axis, in fact any function with a -x inside of it (like (-x)^2 or 1/(-x) ) is reflected over the y axis.

You want to be careful though, if you had something like sqrt(-5x+25) you may think it is moved to the left by 25, but this is not the case. If you have a number multiplying x you want to factor it out. so it becomes sqrt(-5(x-5)) so it is moved to the right by 5. the -5 means it is flipped over the y axis, because it's negative, and it is squished horizontally by a factor of 5.

when you have a function like this you want to do the stretching/ shrinking first, then the shifting. In fact with all graph transformations you want to start witht he parent function, in this case that's sqrt(x), then in oder you want to apply the vertical stretch, horizontal shrink, horizontal shift and finally verical shift. The main point is doing the shifts after the stretching/ shrinking. so in sqrt(-5(x-5) you want to imagine sqrt(x) and squish it horizontally by 5 after flipping it over the y axis. this means you take all points and divide the x terms by -5. so (1,1) becomes (-1/5, 1) then you do the horizontal shift of -5, which is 5 to the right. this adds 5 to all x values, so (-1/5, 1) becomes (24/5, 1).

I want to repeat, it's super important to do the stretches/ shrinks first then the shifts.

Let me know if this didn't help.(5 votes)

- What in the word is “g(x)” what does it mean(4 votes)
- g(x) means the same as f(x). Both are function notation.

-- f(x) tells you the equation is a function called "f" with input values "x"

-- g(x) tells you the equation is a function called "g" with input values "x".

See lessons at: https://www.khanacademy.org/math/cc-eighth-grade-math/cc-8th-linear-equations-functions/8th-functions-and-function-notation/v/what-is-a-function(10 votes)

- What's the order of operation for function ?(8 votes)
- In the previous quiz the function: f(x)=∣x+3∣−3 and it graph were given. Then the next equation: g(x)=3∣x+3∣−9.The question said to pick the graph of g(x). the answer was that the graph of g(x) is f(x) stretched vertically by a factor of three. but what about the -9? should one of the answer choices include the shift down by 6 units as well as the vertical stretch?(5 votes)
- The entire function f(x) is being multiplied by 3 to do the stretch.

g(x) = 3*f(x) = 3|x+3|+3(-3)

Simplify and you get: g(x) = 3}x+3|-9(5 votes)

- How come for question 1 the graph is shifted 3 units to the left instead of 3 units to the right? The 3 is positive . . . I know there is a negative in front of the radical but I'm still a little confused.(5 votes)
- two ways to think about it.

Formula is y=sqrt(x-h) + k, so if h is positive (right shift), you end up with a x-h in radical. For a left shift where h is negative, you end up with x - (-h) or x+h under radical.

Second way to think about it is what will make inside the radical 0? If you have x+3, you need x to be -3 to make it zero which is a shift left.(4 votes)