7.1 –Absolute Value & 7.2A – Graphing Absolute Value Functions Date:
Key Ideas:
Absolute Value is “how many jumps the number is from zero”. Stated another way, it is the distance from zero on the number line, regardless of direction. Distances are always POSITIVE values. 3 is 3 jumps from zero, so the absolute value of 3, or ____. –3 is 3 jumps from zero, so ____. So if, could have been _____ OR _____. For every absolute value solution, there is a positive and negative possibility.
Examples - Evaluate
a) b) c) d) e)
Examples - What are the possible values of x?
a) b) c)
Example – Write the real numbers in order from least to greatest
Absolute value symbols should be treated in the same manner as brackets when applying order of operations (BEDMAS).
Examples - Evaluate the following
a) b) c)
When subtracting two numbers, the difference is most often represented as a positive number. To ensure a positive result, put the subtracting inside the absolute value symbol.
Example – The hottest temperature ever recorded in Victoria was 36.1C. The coldest temperature was -16C. What is the total temperature difference?
7.2A – Graphing Absolute Value Functions of the Form
Example – Graph . On the same grid, make a table of values and graph .
x / y-3
-2
-1
0
1
2
3
x / y
-3
-2
-1
0
1
2
3
Describe what is similar and what is different about the two graphs.
Basic count for an absolute value graph of degree 1:
General form for an absolute value graph of degree 1:
State the domain and range:
Example – Graph
State the domain and range:
Domain & Range:
7.2B – Graphing Absolute Value Functions of the Form Date:
Key Ideas:
Notice the ‘’ value is inside the absolute value. If this is the case, the absolute value should be graphed using a completely different method.
Example – Sketch the graph of
Example –Sketch the graph of . State the domain and range, and express as a piecewise function.
7.2C – Absolute Value Functions of Degree 2 Date:
Key Ideas:
Graphing an Absolute Value Function of the Form
Example – Sketch the graph of by first sketching the graph of . Then state the domain and range of the absolute value graph only.
1)Factor the corresponding quadratic equation to find the roots (x-intercepts).
2)Complete the square on the function to find the vertex.
3)Graph the parabola.
4)Reflect in the x-axis the part of the graph that lies below the x-axis in order to build the absolute value graph. The negative y values in the original parabola will have the absolute value applied to them, thereby making them positive.
The absolute value graph above is actually a combination of two parabolas. What are the quadratic functions of the two parabolas?
We can define the absolute value function as a piecewise function of the two quadratic functions:
7.3 – Absolute Value Equations Date:
Key Ideas:
Example – Solve
Steps for solving an absolute value equation:
1)Get the absolute value by itself on one side (everything not in the absolute value should be on the other side).
2)Set up two cases: the positive case and the negative case. Solve for each case.
3)Check each solution to see if it is an actual or extraneous solution.
Example – Solve
Solve the same example by graphing:
Example – Solve algebraically
Example – Solve
Example – Solve
Example – Solve algebraicially
Example – Solve
7.4A – Reciprocals of Linear Functions Date:
Key Ideas:
A reciprocal of a number can be found by…
Plot the following points on the vertical number line: 4, 3, 2, 1
Plot and label their reciprocals:
Observations:
The reciprocal of 1 is _____.
The bigger the number is, the…
For negative numbers:
The reciprocal of -1 is ______.
The smaller the number is (the more negative it is), the…
What is the reciprocal of 1000?
What is the reciprocal of 1 000 000?
As numbers increase, how do their reciprocals behave?
What is the reciprocal of ?
What is the reciprocal of ?
As numbers decrease, how do their reciprocals behave?
Example – Graph and its reciprocal on the same coordinate plane
An asymptote is a straight line that is approached, but never reached by a curve. They are identified by a dashed line on the graph.
In the previous graph, there is a vertical asymptote and a horizontal asymptote.
Vertical Asymptotes are at any values that are the non-permissible values to the function - in the previous case, the non-permissible value of is , so the vertical asymptote is the line (the y-axis).
For the reciprocals of linear graphs, the horizontal asymptote will always be the line (the axis). This is because if the numerator is always 1, there is no way to make the entire fraction (hence y) equal 0. So there are no values that make .
no x value can make y = 0
Example – Graph , then determine and graph its reciprocal. Label the asymptotes, the invariant points, and the intercepts of the reciprocal.
7.4B – Reciprocal of Quadratic Functions Date:
Key Ideas:
Example – Sketch the graph of
a)What is the reciprocal function?
b)State the non-permissible values of x and the equation(s) of the vertical asymptote(s) of the reciprocal function.
c)What are the x-intercepts and y-intercepts of the reciprocal function?
d)What are the invariant points?
e)Graph the reciprocal function.
Example – Sketch the graph of
a)What is the reciprocal function?
b)State the non-permissible values of x and the equation(s) of the vertical asymptote(s) of the reciprocal function.
c)What are the x-intercepts and y-intercepts of the reciprocal function?
d)What are the invariant points?
e)Graph the reciprocal function.
Example – Graph and its reciprocal