Pythagorus’ theorem tells us that for a right angled triangle, the total area of the squares drawn to touch the two short sides, must equal the area of the square drawn on the long side (the hypotenuse), symbolically:
Pythagorus’ Theorem
a 2 b 2 c 2 a a a a b b b b SHOW ME
Absolute Values
Pythagorus’ theorem can be applied and generalized to points including as many coordinates as we like. We define:
The absolute value of a coordinate , is written with vertical bars and is just the is the distance from the origin to the point.
In other words, it’s the distance we’d have to walk to get to our point. So for points of different dimension, we can define it as:
The absolute value of points in different dimensions
Therefore, if we have some number , we can write it as a one dimensional coordinate. So we define the absolute value of some number as:
Absolute Value
Squaring will make it positive, and then square rooting it will return it’s length to the original size.
Graphing
To graph the absolute value of a function (), you just need to take every coordinate on the graph, and make any negative y values positive as shown:
0 0 xfyxfy f(x)
If you want to graph the absolute value of a straight line:
Start by actually graphing the straight line.
It’s y intercept is just 2
It’s x intercept is just
Graph of 3x+2
Now you just take all of the negative parts and flip their sign as shown:
Graph of |3x+2|
As you can see, graphing this function just gives you a standing v shape
Algebra
Finally, when you are solving algebraic equations that involve absolute values, you need to make sure you remember the definition:
If you’re solving You remove the absolute value by adding a plus minus
This happens because replacing for , means that you have to move the square over to the other side. When you move squares over, you gain a .
You can also solve these expressions graphically as well by setting:
Then you can graph the two equations and think of where they intersect as the solution to your problem.
Simultaneous equations with absolute values
Suppose you wanted to solve this equation simultaneously:
By graphing the two sides:
Graph the pair of functions
We can see that the intersects occur at , and . But we only want the values, which are -1, 3 and 5. Let’s see how we can solve for them.
Solving the Problem
To remove the absolute value, we need to add a to both sides:
Now this is going to be quite hard to solve with the , so we consider both cases individually.
Positive Case
If we ignore the on the , we get:
We can now solve it for as a quadratic:
Now we can factorize this quadratic directly to get:
From this we notice that we have two of our solutions. We now know that and are solutions. To get the last solution, we need to consider the negative case as well.
Negative Case
This time, we assume that we are ignoring the positive part of the , so we get:
Now we repeat the same steps as we performed above:
Now from the product rule we get and . It turns out that is a solution for both the positive and negative cases.
Lesson Loading
Interactive description
Difficulty
00
Time
SOLUTION
Difficulty
3 Mins
Time
Solve the following equation for :
SOLUTION
Case 1 - Positive Case
Case 2 - Negative Case
Testing - You must always test the answers you get to an absolute value problem as some of them may not be valid solutions
Testing
Therefore is a valid solution.
Testing
Therefore is also a valid solution
Difficulty
2 Mins
Time
Solve the following equation for :
SOLUTION
Case 1 - Positive case
Case 2 - Negative case
Therefore,
Difficulty
2 Mins
Time
Solve the following equation for :
SOLUTION
To solve an absolute value problem, we must solve the positive and negative case.
Case 1 - Positive Case
Case 2 - Negative Case
Testing - Whenever solving absolute value questions, we must always verify the solutions via substitution.
Testing
Simplifying using a calculator gives:
Therefore, is a valid solution.
Testing
Therefore, is a valid solution.
Difficulty
2 Mins
Time
Solve the following equation for :
SOLUTION
Case 1 - Positive case
Note - Never forget to add the preceding plus-minus when square-rooting,
Case 2 - Negative case
Note - There is a difference of two squares
Testing - If we substitute both of these solutions into the original problem we will find that they are both valid solutions. Hence:
Difficulty
3 Mins
Time
Sketch the function showing all intercepts
SOLUTION
Step 1 - Determine intercepts
Determine the -intercept;
Note: Since absolute values cannot produce negative values, the above statement is fundamentally incorrect and thus there is no -intercept.
Determine the intercept;
Note - The function’s graph can be thought of as a series of translations applied to , more specifically, translated to the right by 1 unit and up by 1 unit.
Step 2 - Sketch the function (Be mindful of the vertex coordinates)
Difficulty
3 Mins
Time
Sketch the function showing all asymptotes
This question is geared towards extension 1 students.
SOLUTION
Note - When doing a question like this, the an easy way to tackle it is by graphing and then applying the absolute values.
Step 1 - Determine asymptotes
Determine the vertical asymptote
Since cannot equal zero, there is a vertical asymptote at
Determine the horizontal asymptote
Dividing each term by the highest degree of in the denominator.
Apply the limit,
Therefore the horizontal asymptote occurs at
Step 2 - Sketch the function without applying the absolute value
Step 3 - Sketch the function applying the absolute value
Note - The absolute value function only outputs positive values. Since the -coordinates are the output of the original function, all negative values of will be converted to positive values. Visually, this means that everything below the x-axis will be reflected up the axis as shown below.
Difficulty
30 Mins
Time
Given , sketch
Note: This question is geared towards extension one students
SOLUTION
Note - When doing a question like this, the an easy way to tackle it is by graphing and then applying the absolute values.
Step 1 - Determine intercepts
Determine the -intercepts;
Determine the -intercept;
Let ,
Step 2 - Determine the vertex
Method 1 (Hard way):
Expand the function,
Find the axis of symmetry using the formula ,
Find the coordinate on the axis of symmetry (substitute ),
Therefore the coordinates of the vertex are,
Method 2 (Easy way):
Find axis of symmetry
We know that the axis of symmetry occurs at the midpoint of the roots (-intercepts) for a parabola. So instead of expanding and using the formula , we just find the midpoint/average of and .
Find the coordinate on the axis of symmetry (substitute )
Therefore the coordinates of the vertex are,
Step 3 - Sketch the function without the absolute value
Step 4 - Sketch the function applying the absolute value
Difficulty
2 Mins
Time
Given and
Find an expression for
Evaluate
SOLUTION
Part A
Find an expression for by replacing every in for the value of , which is just :
Part B
Substitute into ,
Apply the absolute value and simplify the expression
Difficulty
3 Mins
Time
Show algebraically that
SOLUTION
Start with the left hand side,
Factor out
We can use the fact that . This gives us:
Hence we can now conclude that:
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Presets
The Absolute Value Function
4D - Cambridge Advanced Year 11
Question 1 (b, c, d, f, h)
Question 2 (c, e)
Question 3 (c, e, g)
Question 5 (b, e, f)
Question 6 (a, c, e, f)
Question 7 (b, e, f, h)
Question 9 (b{i, ii})
Question 13 (a, b)
Question 14 (a, b, c)
Lesson Loading
×
×
The Absolute Value
Review Questions
What is the absolute value of ?
-3
3
-9
9
What does the absolute value do to the value of ?
It forces it to be positive without changing its size
It leaves positive values unchanged but squares negative values
It keeps the sign, but removes the magnitude
It finds the distance to
Which letter or symbol does the graph of an absolute value look closest to?
V
~
^
M
/
W
×
The Absolute Value
Review Questions
What is the absolute value of ?
-3
3
9
-9
What does the absolute value do to the value of ?
It forces it to be positive without changing its size
It leaves positive values unchanged but squares negative values
It keeps the sign, but removes the magnitude
It finds the distance to
Which letter or symbol does the graph of an absolute value look closest to?
M
V
W
/
~
^
×
Graphs of Absolute Values
Review Questions
What transformation do we apply to to graph
It moves the graph 3 units downwards
It moves the graph 3 units to the right
It moves the graph 3 units to the left
It moves the graph 3 units upwards
What transformations do we apply to |x| to graph
We flip the graph horizontally and move it downwards by 2 units
We flip the graph vertically and move it upwards by 2 units
We flip the graph horizontally and move it upwards by 2 units
We flip the graph vertically and move it downwards by 2 units
What transformations do we apply to to graph
We stretch the graph horizontally by 2 units and move down by 3 units
We stretch the graph vertically by 2 units and move right by 3 units
We stretch the graph horizontally by 2 units and move left by 3 units
We stretch the graph vertically by 2 units and move down by 3 units
×
Graphs of Absolute Values
Review Questions
What transformation do we apply to to graph
It moves the graph 3 units upwards
It moves the graph 3 units to the right
It moves the graph 3 units downwards
It moves the graph 3 units to the left
What transformations do we apply to |x| to graph
We flip the graph horizontally and move it downwards by 2 units
We flip the graph horizontally and move it upwards by 2 units
We flip the graph vertically and move it upwards by 2 units
We flip the graph vertically and move it downwards by 2 units
What transformations do we apply to to graph
We stretch the graph vertically by 2 units and move down by 3 units
We stretch the graph horizontally by 2 units and move down by 3 units
We stretch the graph horizontally by 2 units and move left by 3 units
We stretch the graph vertically by 2 units and move right by 3 units
