Download Chapter 7 notes - Winston Knoll Collegiate

7.1 – Absolute Value & 7.2A – Graphing Absolute Value Functions
Date:
Key Ideas:
absolute
value
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| =____. –3 is 3 jumps from zero, so |−3| =____. So if|𝑥| = 3, 𝑥 could have been
_____ OR _____. For every absolute value solution, there is a positive and negative
possibility.
Examples - Evaluate
a) |5|
b) |−7|
c) |−0.34|
5
3
d) |6|
e) |−6 8|
Examples - What are the possible values of x?
a) |𝑥| = 6
b) |𝑥| = 9.7
c) |𝑥| = −2
Example – Write the real numbers in order from least to greatest
−13
|3.5|, −2, |−5.75|, 1.05, |
4
1
| , |−0.5|, −1.25, |−3 3|
Absolute value symbols should be treated in the same manner as brackets when
applying order of operations (BEDMAS).
Examples - Evaluate the following
a) |−4| − |−3|
b) 2 − 3|−12 + 8|
c) −2|−2(5 − 7)2 + 6|
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 36.1C. The
coldest temperature was -16C. What is the total temperature difference?
7.2A
7.2A – Graphing Absolute Value Functions of the Form 𝒚 = 𝒂|𝒙 + 𝒑| + 𝒒
Example – Graph 𝑦 = 𝑥. On the same grid, make a table of values and graph 𝑦 = |𝑥|.
𝑦=𝑥
𝑦 = |𝑥|
x
-3
-2
-1
0
1
2
3
x
-3
-2
-1
0
1
2
3
y
y
How did the y values change?
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:
Example – Graph 𝑦 = |𝑥 − 2| − 3
without making a table of values.
domain &
range
State the domain and range:
Example – Graph 𝑦 = 4 − 2|𝑥 + 5|
State the domain and range:
1
Example – Graph 𝑦 = 2 |𝑥 + 1| − 1
Domain & Range:
7.2B – Graphing Absolute Value Functions of the Form 𝒚 = |𝒂𝒙 + 𝒃|
Date:
Key Ideas:
Notice the ‘𝑎’ 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 𝑦 = |3𝑥 + 1|
Step 1 – Sketch the graph of 𝑦 = 3𝑥 + 1
y-intercept:
slope:
Step 2 – Calculate the 𝑥-intercept by
setting 𝑦 = 0 in the absolute value
equation. The 𝑥-intercept can also be
located graphically from step 1.
invariant
points
The 𝑥-intercept of 𝑦 = |3𝑥 + 1| is also
the 𝑥-intercept of 𝑦 = 3𝑥 + 1. This is
called an invariant point.
Step 3 – Reflect in the 𝑥-axis the part of
the graph of 𝑦 = 3𝑥 + 1 that is below
the 𝑥-axis. The V-shaped graph that
results is the graph of 𝑦 = |3𝑥 + 1|.
Invariant points are points that remain
unchanged when a transformation is
applied. What other points on the
graph are invariant points?
Step 4 – State the domain and range:
The right branch was first constructed
by graphing 𝑦 = 3𝑥 + 1. If the left
branch was constructed in this manner,
what would the equation be?
Since the right branch is from the equation 𝑦 = 3𝑥 + 1 and the left branch is from the
equation
, the graph 𝑦 = |3𝑥 + 1| can be described as a piecewise
function:
1
Example – Sketch the graph of 𝑦 = |− 2 𝑥 − 4|. State the domain and range, and
express as a piecewise function.
7.2C – Absolute Value Functions of Degree 2
Date:
Key Ideas:
quadratic
absolute
functions
Graphing an Absolute Value Function of the Form 𝑓(𝑥) = |𝑎𝑥 2 ± 𝑏𝑥 ± 𝑐|
Example – Sketch the graph of 𝑓(𝑥) = |𝑥 2 − 𝑥 − 2| by first sketching the graph of
𝑓(𝑥) = 𝑥 2 − 𝑥 − 2. Then state the domain and range of the absolute value graph only.
1)
2)
3)
4)
Factor the corresponding quadratic equation to find the roots (x-intercepts).
Complete the square on the function to find the vertex.
Graph the parabola.
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.
1)
2)
Domain:
4)
Range:
The absolute value graph above is actually a combination of two parabolas. What are the
quadratic functions of the two parabolas?
piecewise
functions
We can define the absolute value function as a piecewise function of the two quadratic
functions:
Example:
a) Sketch the graph of 𝑦 = 2𝑥 2 + 3𝑥 − 9 and 𝑦 = |2𝑥 2 + 3𝑥 − 9|
b) State the domain and range of the absolute value graph.
c) Express the absolute value as a piecewise function.
Domain:
Range:
Piecewise Function:
7.3 – Absolute Value Equations
Date:
Key Ideas:
Example – Solve |𝑥| = 2
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 |𝑥 − 2| + 3 = 9
Positive Case:
Negative Case:
Solve the same example by graphing:
Check:
Example – Solve |3𝑥 − 2| = 1 − 𝑥 algebraically
Check:
Example – Solve |𝑥 − 3| + 7 = 4
Check:
no
solutions
An Absolute Value Equation with No Solution:
Example – Solve |4𝑥 − 5| + 9 = 2
Example – Solve |𝑥 + 5| = 4𝑥 − 1 algebraicially
Check:
quadratic
absolute
value
equations
Example – Solve |𝑥 2 − 7𝑥 + 2| = 10
Check:
7.4A – Reciprocals of Linear Functions
Date:
Key Ideas:
A reciprocal of a number can be found by…
4
Plot the following points on the vertical number line: 4, 3, 2, 1
Plot and label their reciprocals:
Observations:
3
The reciprocal of 1 is _____.
The bigger the number is, the…
For negative numbers:
2
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?
1
As numbers increase, how do their reciprocals behave?
1
What is the reciprocal of 100?
1
What is the reciprocal of 100 000?
0
As numbers decrease, how do their reciprocals behave?
basic
reciprocal
graph for a
linear
function
Example – Graph 𝑦 = 𝑥 and its reciprocal on the same coordinate plane
𝑥
-10
-5
-2
𝑦
1
𝑦=𝑥
𝑥
1
10
1
5
-1
1
2
−1
2
1
−1
5
2
−1
10
0
5
10
𝑦
1
𝑦=𝑥
What is unique
about the
reciprocals of -1
and 1 in this
example?
In this example,
(1, 1) and (-1, -1)
are called
invariant points,
as they are the
same for the
original and
reciprocal.
Invariant points
are always where
𝑓(𝑥) = ±1
asymptotes
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 𝑥 values that are the non-permissible values to the
1
function - in the previous case, the non-permissible value of 𝑥 is 𝑥 ≠ 0, so the vertical
asymptote is the 𝑥 = 0 line (the y-axis).
For the reciprocals of linear graphs, the horizontal asymptote will always be the line 𝑦 =
0 (the 𝑥 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 𝑥 values that make 𝑦 = 0.
1
𝑦 = 𝑥  no x value can make y = 0
Example – Graph 𝑓(𝑥) = 2𝑥 − 3, then determine and graph its reciprocal. Label the
asymptotes, the invariant points, and the intercepts of the reciprocal.
Graph 𝑦 = 2𝑥 − 3:
y-intercept:
slope:
reciprocal:
non-permissible value:
(notice the non-permissible value is the
x-intercept from the original line)
vertical asymptote:
horizontal asymptote:
invariant
points
x-intercept of the reciprocal graph:
invariant points: the line and the recip
graph are equal where y = 1 and y = -1, so
solve 2𝑥 − 3 = 1 and 2𝑥 − 3 = −1.
y-intercept of the reciprocal graph:
invariant points are (
, 1) & (
, -1)
7.4B – Reciprocal of Quadratic Functions
Date:
Key Ideas:
Example – Sketch the graph of 𝑓(𝑥) = 𝑥 2 − 4
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.
a)
b)
c) x-intercept(s):
c) y-intercept(s):
d) invariant points
Example – Sketch the graph of 𝑓(𝑥) = 𝑥 2 + 6𝑥 + 9
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.
a)
b)
c) x-intercept(s):
c) y-intercept(s):
d) invariant points
graphing a
reciprocal
quickly
Example – Graph 𝑦 = −(𝑥 + 3)2 + 5 and its reciprocal
Tips to graphing a reciprocal quickly: