Pozo, Damone
Math SL 11
What is the absolute value?
In math, the absolute value of A of a real number is the numerical value of A
without considering its sign. E.g. the absolute value of 5 is 5, and the absolute
value of -5 is also 5. In other words, it is the distance it covers to the value of 0.
The basic shape of the graph
What is the Basic equation?
The equation of the absolute value function is f(x)=|x|.
Any types of values can be added such as m=slopes, h= y-intercepts and k=xintercepts, depending on the function that is intended to represent.
The equation uses |x| because in the graph, there are two sides, the positive
and the negative. It is important to mention that the line can be located in any
part of the Cartesian plane, it being on the positive or negative side. What does
not change and what is characteristic about this function is the V shape of the
line. Changing the slope would determine the proximity or opening of the Vshape of the line. Changing the y-intercept or x-intercept would only affect in a
Pozo, Damone
Math SL 11
way that the coordinates can be located in any part of the plane, (+,+) (+,-)(-,-)(,+) but the V-shape always facing upwards.
F(x)= abs(x+2)-9
What are the domains and ranges?
The domain of the equation f(x)= |x| are all the real numbers. The range is all
real numbers greater than or equal to zero.
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E.g. In equations were you have m*abs(x – h) + k, were m is the
slope, h is the y-intercept, and k is the x-intercept, the domain would
be assigned different values. When changing h, the y-intercept, the v
shape would only move horizontally, depending on the value that is
assigned to x, since y will always equal to 0 (in this case). But as the
k value is changes, the opposite will vary. Changing the k value will
affect the shape of the graph in a way that the V shape of the line can
only move vertically. X-intercept is zero (not altering the equation) and
the value of y varies as positive and negative numbers are plugged in.
The vertices are the origins, the point where both lines meet and start their
aperture. This varies according to the function, the different values assigned to
the h and k values, x-intercept and y-intercepts.
The vertex of the parent function, f(x) = |x| is the origin, in this case is
coordinate (0,0.)
Pozo, Damone
Math SL 11
The domain for most absolute value functions are all real numbers the y
coordinate of the vertex is the boundary point for the range of most absolute
value function.
The y coordinate of the vertex on this graph is 0. The range is all real numbers
greater or equal to zero.
What is the inverse?
Inverse functions only work when the function is one to one.
The formula of the inverse function is y = x (x ≥ 0) and the inverse range of abs
(x) is x>=0, the domain of the inverse is x>=0, but x=|y| is not a function
because in each x value is two y values, so this doesn’t agree with the rules of a
function. The function of the absolute value function is not one-to-one so there
is no inverse.
Pozo, Damone
Math SL 11
Possible applications for real-world situations:
Time can never have a negative value, that is why they have absolute values
for seconds, minutes, hours etc. As well as distance. There is no such thing as
a negative distances.
4
The roof of a building can be modeled by the function 3 |x − 9| + 12 where x and
y are measured in feet and the x-axis represents the base of the roof.
Construction can be applied by the technique of reflection.
The flying route of planes, it being the reflection, can also be some real life
examples of absolute values functions.
What are the different types of Transformations?
1. Translation
A translation is a transformation that shifts a graph horizontally or vertically,
but doesn’t change the overall shape or orientation. This means that the xintercepts or y-intercepts change in an equation as you assess a different value,
but not altering the slope.
2. Shifting
A translation in which the size and shape of a graph of a function is not
changed, but the location of the graph does change.
A transformation can be when y=|x|+a. if a is great than zero then the graph
shifts units up and if a is less than zero the graph shifts units down.
Ex: y=|x|+3 is a great than zero so the graph shift unit up.
other way is when y=|x-b|, then the expression (x - b) is equal it to zero (x - b =
0) and the result is x = b. when b is great that zero shift the graph units to the
right and when b is less than zero shifts the graph units to the left.
Pozo, Damone
Math SL 11
Ex: y=|x-6|
if y=|x+1|+3 the graph shift 3 units up since 3 > 0, then the expression x + 1
equal it to zero and the result is x = -1, because –1 is less than zero , so the
graph shift 1 unit to the left.
when y=c|x| then the if: |c| great than 1 then the graph is closer to the y-axis, but
when |c| is equal to 1 then the graph remains the same, also if zero is less than
|c| and |c| is less than 1 the graph is further from the y-axis. if c is a negative
number, then the graph will reflect on the x-axis.
Ex: y=5|x| and y=|x|
Pozo, Damone
Math SL 11
3. Reflections
Another interesting transformation is a reflection, like a mirror image. The
reflection of a point across a line sends the point p onto the point q, where the
reflecting line is the perpendicular line (opposite) of the line segment between p
and q.
Reflecting a graph across a line reflects each point of the graph across the line.
Remember that the absolute values will always be positive! So this would
mean that the values of the reflection v-shape line would have negative values
in the case were the vertex is (0,0).
Resources used:
http://people.richland.edu/james/lecture/m116/functions/translations.html
http://www.shodor.org/interactivate/activities/FunctionFlyer/
http://www.purplemath.com/modules/graphabs.htm
http://www.mathsisfun.com/sets/function-absolute-value.html