chapter 7 absolute value and reciprocal functions
TRANSCRIPT
Chapter 7
ABSOLUTE VALUE AND RECIPROCAL
FUNCTIONS
EXAMPLE
How many triangles are in the diagram below?
Chapter 77.1 – ABSOLUTE
VALUE
ABSOLUTE VALUE
For a real number a, the absolute value is written as |a| and is a positive number. For example:
|5| = 5|–5| = 5
Absolute value can be used to represent the distance of a number from zero on a real-number line.
Evaluate:
|3| |–7|
EXAMPLE
Evaluate the following:a) |4| – |–6| b) 5 – 3|2 – 7| c) |–2(5 – 7)2 + 6|
a) |4| – |–6| = 4 – 6 = –2
b) 5 – 3|2 – 7| = 5 – 3|–5| = 5 – 3(5) = 5 – 15 = –10
c) Try it!
Independent Practice
PG. 363-367 #1, 6, 7(A,C,E), 11, 16
HANDOUT
Answer the questions on the “Investigating Absolute Value Functions” worksheet to the best of your ability.
Chapter 77.2 – ABSOLUTE
VALUE FUNCTIONS
ABSOLUTE VALUE FUNCTIONS
For what values of x is the function y = |x| equivalent to y = x?
when x ≥ 0When x < 0, what is the function represented by y = |x|?
y = –x
We can write this as a piecewise function:
EXAMPLE
Consider the absolute value function y = |2x – 3|.a) Determine the y-intercept and the x-intercept.b) Sketch the graph.c) State the domain and range.d) Express as a piecewise function.
a) The y-intercept is at x = 0. y = |2x – 3|y = |2(0) – 3|y = |–3|y = 3The y-intercept is (0, 3).
The x-intercept is at y = 0.0 = |2x – 3|0 = 2x – 3x = 3/2The x-intercept is at (3/2, 0).
b) x y
-1 5
0 3
3/2 0
3 3
4 5
EXAMPLE
Consider the absolute value function y = |2x – 3|.a) Determine the y-intercept and the x-intercept.b) Sketch the graph.c) State the domain and range.d) Express as a piecewise function.
x y
-1 5
0 3
3/2 0
3 3
4 5
b) c) D: {x | x E R} R: {y | y ≥ 0, y E R}
d) The equation on the right is just y = 2x – 3.
What’s the one on the left?
It’s just –(2x – 3)!
The x-intercept is call an invariant point because it’s a part of both functions.
EXAMPLE
Consider the absolute value function f(x) = |–x2 + 2x + 8|.a) Determine the y-intercept and the x-intercepts.b) Sketch the graph.c) State the domain and range.d) Express as a piecewise function
a) The y-intercept is at x = 0.
f(0) = |–(0)2 + 2(0) + 8| = |8| = 8
The x-intercepts are when y = 0. 0 = |–x2 + 2x + 8| 0 = –x2 + 2x + 8 0 = –(x – 4)(x + 2) x = 4 x = –2
b) What’s the vertex of the function
f(x) = –x2 + 2x + 8
Use your calculator, or complete the square:
f(x) = –(x2 – 2x) + 8 f(x) = –(x2 – 2x + 1 – 1 ) + 8 f(x) = –(x2 – 2x + 1) + 1 + 8 f(x) = –(x – 1)2 + 9 The vertex is (1, 9)
EXAMPLE
Consider the absolute value function f(x) = |–x2 + 2x + 8|.a) Determine the y-intercept and the x-intercepts.b) Sketch the graph.c) State the domain and range.d) Express as a piecewise function
Recall: y-intercept is (0, 8) x-intercepts are (4, 0) and
(–2, 0) Vertex is (1, 9)c) D: {x | x E R} R: {y | y ≥ 0, y E R}
d)
Independent Practice
PG. 375-379, #2, 5, 7, 10, 12-14.
Chapter 77.3 - ABSOLUTE
VALUE EQUATIONS
ABSOLUTE VALUE EQUATIONS
When solving equations that involve absolute value equations you need to consider two cases:
Case 1: The expression inside the absolute value symbol is positive or zero.
Case 2: The expression inside the absolute value symbol is negative.
EXAMPLE
Solve: |x – 3| = 7
Consider the equation as a piecewise function:
Case 1:
x – 3 = 7 x = 10
Case 2:
–(x – 3) = 7 x – 3 = –7 x = –4
The solution is x = 10 or x = –4.
TRY IT
Solve |6 – x| = 2
EXAMPLE
Solve |2x – 5| = 5 – 3x
Consider:
What is the x-intercept of y = 2x – 5? 0 = 2x – 5 5 = 2x x = 5/2
Case 1: (x ≥ 5/2)
2x – 5 = 5 – 3x 5x = 10 x = 2
Case 2: (x < 5/2)
–(2x – 5) = 5 – 3x –2x + 5 = 5 – 3x x = 0
EXAMPLE
Solve: |3x – 4| + 12 = 9
|3x – 4| = –3
Is there any possible way that the absolute value of something is equal to –3?
No solution.
EXAMPLE
Solve: |x – 10| = x2 – 10x
Independent Practice
PG. 389-391, #4, 5, 6, 9, 11, 22, 23
Chapter 77.4 – RECIPROCAL
FUNCTIONS
EXAMPLE
Sketch the graphs of y = f(x) and its reciprocal function y = 1/f(x), where f(x) = x. Examine how the functions are related.
x y = x y = 1/x
–5 –5 –1/5
–2 –2 –1/2
–1 –1 –1
–1/2 –1/2 –2
–1/10 –1/10 –10
0 0 Undef.
1/10 1/10 10
1/2 1/2 2
1 1 1
2 2 1/2
RECIPROCAL FUNCTIONS
An asymptote is a line whose distance from a curve approaches zero.
This graph has two pieces, that both approach the vertical asymptote, which is defined by the non-permissible value of domain of the function, and a horizontal asymptote, defined by the value that is not in the range of the function.
What is the vertical asymptote?
What is the horizontal asymptote?
EXAMPLE
Consider f(x) = 2x + 5.a) Determine its reciprocal function y = 1/f(x).b) Determine the equation of the vertical asymptote of the reciprocal
function.c) Graph the function y = f(x) and its reciprocal function y = 1/f(x).
a) The reciprocal function is:
b) The vertical asymptote is always the non-permissible values of the function.
2x + 5 = 0 2x = –5 x = –5/2
There is a vertical asymptote at x = –5/2
c)Characteristic
f(x) = 2x +5
f(x)=1/(2x + 5)
x-intercept/asymptotes
x-intercept at x = –5/2
Asymptote at x = –5/2
Invariant points
2x + 5 = 1 x = –2 (–2, 1)
(–2,1)
2x + 5 = –1 x = –3 (–3,–1)
(–3,–1)Invariant points are at y = 1 and y = –1.
EXAMPLE
Characteristic
f(x) = 2x +5
f(x)=1/(2x + 5)
x-intercept/asymptotes
x-intercept at x = –5/2
Asymptote at x = –5/2
Invariant points
2x + 5 = 1 x = –2 (–2, 1)
(–2,1)
2x + 5 = –1 x = –3 (–3,–1)
(–3,–1)
Consider f(x) = 2x + 5.a) Determine its reciprocal function y = 1/f(x).b) Determine the equation of the vertical asymptote of the reciprocal
function.c) Graph the function y = f(x) and its reciprocal function y = 1/f(x).
EXAMPLE
Consider f(x) = x2 – 4.a) What is the reciprocal function of f(x)?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) Graph the functions.
a)
b) What are the non-permissible values? x2 – 4 = 0 (x – 2)(x + 2) = 0 x = 2 x = –2
The vertical asymptotes are at x = ±2
c) How can I find the x-intercept of the the reciprocal? Let f(x) = 0
There is no solution, so there is no x-intercept.
y-intercept:y-intercept is y = –1/4
Independent practice
PG. 403-408, #3, 5, 7, 8, 9, 10, 12