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Absolute value graphs review

The general form of an absolute value function is f(x)=a|x-h|+k. From this form, we can draw graphs. This article reviews how to draw the graphs of absolute value functions.
General form of an absolute value equation:
f, left parenthesis, x, right parenthesis, equals, start color #e07d10, a, end color #e07d10, vertical bar, x, minus, start color #11accd, h, end color #11accd, vertical bar, plus, start color #11accd, k, end color #11accd
The variable start color #e07d10, a, end color #e07d10 tells us how far the graph stretches vertically, and whether the graph opens up or down. The variables start color #11accd, h, end color #11accd and start color #11accd, k, end color #11accd tell us how far the graph shifts horizontally and vertically.
Some examples:
A coordinate plane. The x- and y-axes both scale by one. The graph is of the function y equals the absolute value of x. The vertex is at the point zero, zero. The points negative one, one and one, one can be found on the graph.
Graph of y=|x|
A coordinate plane. The x- and y-axes both scale by one. The graph is of the function y equals three times the absolute value of x. The vertex is at the point zero, zero. The points negative one, three and one, three can be found on the graph.
Graph of y=3|x|
A coordinate plane. The x- and y-axes both scale by one. The graph is of the function y equals negative one times the absolute value of x. The vertex is at the point zero, zero. The points negative one, negative one and one, negative one can be found on the graph.
Graph of y=-|x|
A coordinate plane. The x- and y-axes both scale by one. The graph is of the function y equals the absolute value of the sum of x plus three minus two. The vertex is at the point negative three, negative two. The points negative two, negative one and negative four, negative one can be found on the graph.
Graph of y=|x+3|-2

Example problem 1

We're asked to graph:
f, left parenthesis, x, right parenthesis, equals, vertical bar, x, minus, 1, vertical bar, plus, 5
First, let's compare with the general form:
f, left parenthesis, x, right parenthesis, equals, start color #e07d10, a, end color #e07d10, vertical bar, x, minus, start color #11accd, h, end color #11accd, vertical bar, plus, start color #11accd, k, end color #11accd
The value of start color #e07d10, a, end color #e07d10 is 1, so the graph opens upwards with a slope of 1 (to the right of the vertex).
The value of start color #11accd, h, end color #11accd is 1 and the value of start color #11accd, k, end color #11accd is 5, so the vertex of the graph is shifted 1 to the right and 5 up from the origin.
Finally here's the graph of y, equals, f, left parenthesis, x, right parenthesis:
A coordinate plane. The x- and y-axes both scale by one. The graph is of the function y equals the absolute value of the difference of x minus one plus five. The vertex is at the point one, five. The points zero, six and two, six can be found on the graph.

Example problem 2

We're asked to graph:
f, left parenthesis, x, right parenthesis, equals, minus, 2, vertical bar, x, vertical bar, plus, 4
First, let's compare with the general form:
f, left parenthesis, x, right parenthesis, equals, start color #e07d10, a, end color #e07d10, vertical bar, x, minus, start color #11accd, h, end color #11accd, vertical bar, plus, start color #11accd, k, end color #11accd
The value of start color #e07d10, a, end color #e07d10 is minus, 2, so the graph opens downwards with a slope of minus, 2 (to the right of the vertex).
The value of start color #11accd, h, end color #11accd is 0 and the value of start color #11accd, k, end color #11accd is 4, so the vertex of the graph is shifted 4 up from the origin.
Finally here's the graph of y, equals, f, left parenthesis, x, right parenthesis:
A coordinate plane. The x- and y-axes both scale by one. The graph is of the function y equals negative two times the absolute value of x plus four. The vertex is at the point zero, four. The points negative one, two and one, two can be found on the graph.
Want to learn more about absolute value graphs? Check out this video.
Want more practice? Check out this exercise.

Want to join the conversation?

  • blobby green style avatar for user torresmc8760
    is there any easier steps to explain this type of lesson
    (7 votes)
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    • female robot grace style avatar for user loumast17
      Maybe I can better explain

      when you have an absolute value function you want to look at what are in the places of a, h and k. a|x-h|+k Specifically you want to look at h and k first. Normally the tip of the V shape is at (0,0) this changes depending on h and k. specifically it moves the tip to (h,k) so if you have |x+5|-7 then the tip of the V shape goes to (-5,-7). if you wonder why it is -5 even though we are adding 5, you just need to look at the original a|x-h|+k if we had -5 then it would be just like that, but since it is +5, we have to look at it as - -5, minus negative 5. so if it helps, the x coordinate is kinda backwards.

      After the V tip you then look at a. treat it like a linear equation where a is the slope. so if a was -3 that's down 3 right 1 using rise over run. then, since it's an absolute value function you need to know that the same line goesalong the left to make that V shape, so -5 would mean on the left down 3 and left 1.

      if you ever have something like a|bx-h|+k where there is a number in front of the x you need to get rid of it if you are not aware of factoring this is what it would look like a|b||x - h/b|+k where a|b| becomes the new "a" and h/b becomes the new h, then you would solve it normally. The point being you always want x by itself for this. Also, keep in mind that even if inside the absolute value bars if b was negative, outside it becomes positive.

      Let me know if that didn't help, or if there is a specific function you are struggling with, or maybe would even like some to try out.
      (48 votes)
  • hopper cool style avatar for user TheAvenger1621
    In example problem 1, why isn’t the graph shifted 1 unit to the left instead of to the right?
    (8 votes)
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    • spunky sam green style avatar for user Karmanyaah Malhotra
      It is shifted to the right because x-1 would make it 0 when x=1 because
      x=1
      1-1=0
      So, we always want the absolute value part of the equation to be equal to 0 when we use x as the horizontal shifting.

      While, the vertical part goes up with + not down because when,
      y=a∣x−h∣+k
      y-k=a|x-h|
      So basically we transpose it to make it easier to distinguish.
      (18 votes)
  • leafers seed style avatar for user jhuang
    So is h is positive that means that it is actually negative? Is that why if its x + 3 on the graph you go to negative 3?
    (3 votes)
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    • mr pink green style avatar for user David Severin
      No, if it is positive it means I move in the negative direction, but if h is negative I move in the positive direction, it does not change the sign of h. The idea is that what value of x would make the inside of the absolute value (or other function) 0, so if you have x + 3, it would require x = -3 to be 0, thus causing a shift in the negative direction. The other idea is that since the formula has | x - h | + k where (h,k) is the vertex, then using x+ 3 would actually be x - (- 3) so -3 would cause a shift to the left.
      (5 votes)
  • aqualine ultimate style avatar for user TAYLOR191
    I am confused on how you know if the vertex is a minimum or maximum point.
    (1 vote)
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    • stelly blue style avatar for user Kim Seidel
      The general form of the absolute value function is:
      f(x) = a|x-h|+k
      When "a" is negative, the V-shape graph opens downward and the vertex is the maximum.
      When "a" is positive, the V-shape graph opens upward and the vertex is a minimum.
      Hope this helps.
      (6 votes)
  • starky seedling style avatar for user Maddy S
    i dont know what made the diffrence to make it go up or down
    (1 vote)
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  • stelly green style avatar for user Sarah
    What if there's something in front of the x? In my homework, there's a problem that says a(x) = |5x|. How would I graph that?
    (2 votes)
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  • blobby green style avatar for user dannni2019
    I have a table where the coordinates are in a table, the coordinates are
    (4,7) (5,6) (6,5) (7,4) (8,5)
    it is wanting me to find the domain which I found, but I don't know how to find the range, intercepts, vertex, or the max or min.
    How do I find the range, intercepts, vertex, and the max or min?
    (0 votes)
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    • mr pants teal style avatar for user VincentTheFrugal
      The range refers to the possible y values. Based on the table you gave us, it looks like there is a minimum y value. It looks like the y values start at 7, go down, and then start going up again. The lowest y value is going to be the turning point of the variable. The turning point is always going to be the minimum or the maximum. The turning point is your vertex.

      Since there appears to be a lowest y value, the graph probably doesn't have a highest y value. It depends a little on the question.

      The vertex is where the graph turns. It should happen at the lowest y value (or the highest, if that makes more sense for the problem. If the graph goes up forever, then it should be the lowest y value). The vertex is also where the min, or max of the function is... but remember, whatever the vertex is (max or min), the other one (min or max) is probably infinity.

      For intercepts... based on the table, does it look like it ever crosses the x axis? Use the slope of the line (before it turns, or after it turns), to figure out where the line goes beyond the points that it gave you. For the Y-intercept, what is the Y value when X is 0?
      (4 votes)
  • piceratops seed style avatar for user annabethschuyler
    I just need to know how to graph absolute value with a table!
    (1 vote)
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    • mr pink green style avatar for user David Severin
      To graph the absolute value, you need to know where the vertex is, so the numbers will be going down, then turn and start going back up, or going up and then turn and start going down, the middle point will be the vertex and should be the center point of the table. if you learned vertex form in quadratics, it is the same as in absolute value, so vertex form gives f(x) = a | x - h | + k has a vertex at (h,k), example: F(x) = | x - 4 | - 3 has a vertex at (4,3).
      (4 votes)
  • blobby green style avatar for user Sambhav Jain
    y = modulus of x - x, where x is less then 0
    (2 votes)
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  • piceratops tree style avatar for user Jean
    This article says that the general form of absolute value equations is f (x) = a |x - h| + k.
    I have the problem y = 4|x|.
    How do I solve it when I have a different form?
    I would really appreciate help.
    (1 vote)
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