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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=x is shown below.
Which of the following is the graph of y=x+35?
Choose 1 answer:

Practice question 2: Cube-root function

The graph of y=Ax3 is shown below.
Which of the following is the graph of y=Ax+23+5?
Choose 1 answer:

Want to join the conversation?

  • piceratops seedling style avatar for user chandlerlyon9772
    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)
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  • leafers tree style avatar for user Amy
    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)
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    • male robot hal style avatar for user Michael
      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)
  • leafers seed style avatar for user Emma
    Is there a secret method to graphing the cubic root and the square root without a graphing calculator?
    (6 votes)
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  • aqualine ultimate style avatar for user Ryan Thammakhoune
    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)
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    • duskpin ultimate style avatar for user Sarah Shores
      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)
  • aqualine seed style avatar for user risharocks0
    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)
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  • duskpin seed style avatar for user Mariam Saidou
    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)
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    • female robot grace style avatar for user loumast17
      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)
  • leafers seedling style avatar for user Nat
    What in the word is “g(x)” what does it mean
    (4 votes)
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  • male robot johnny style avatar for user Mohamed Ibrahim
    What's the order of operation for function ?
    (8 votes)
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  • stelly green style avatar for user Rosemary Monson
    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)
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  • leafers sapling style avatar for user Emma Hesselton
    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)
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    • mr pink green style avatar for user David Severin
      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)