62
Chapter 2 Radical Functions
Chapter 2
Radical Functions
Square Roots• In mathematics, a square root of a number a
is a number y such that y2 = a• For example, 4 is a square root of 16 because
42 = 16• And so is -4 because (-4) 2 = 16
So what is ……
Why isn’t it 2 and -2?Because means the principal square root ... … the one that isn't negative!There are two square roots, but the radical symbol means just the principal square root
4 = 2
• The square roots of 36 are …• … 6 and -6 • But = …• …6
• When you solve the equation x2 = 36, you are trying to find all possible values that might have been squared to get 36
6
636
362
2
x
xx
x
What about … = ?There is no real number that when squared is negativeFor the purposes of Grade 12 Pre-Calculus you cannot take the square root of a negative number
Radical Function• A radical function is a function that has a
variable in the radicand
7
3
16)(
92
xxg
xy
xy
,0
,0
Order???• Stretches and Reflections, performed in any order,
followed by translations.• If is a point on then …… is a point on
Graphing Radical Functions Using a Table of Values
1)1(23 xy• A good place to start
is to determine the domain• The radicand must be
greater than or equal to zero
0)1(2 x
Remember to reverse the inequality when multiplying or dividing by a negative number
20
2)1(2
x
01x
1x
Graphing Radical Functions Using a Table of Values
1)1(23 xy 1x
1)1(23 xyThe Domain is or
The Range is or
Graphing Radical Functions Using Transformations
1)1(23 xy
khxbay )(1123
khba
Not an invariant point!! It does not map to itself!!!(0,0) maps to (1,1)(1,1) maps to (0.5, 4)
1123
khba
1)1(23 xy
xy
1123
khba
1)1(23 xy
Mapping Notation
kayhxb
yx ,1,
How points on this map to that
xy khxbay )(
13,121, yxyx
What do you notice?
xy
xy
xy
4
42
16
has been horizontally stretched by a factor of
has been vertically stretched by a factor of 4
has been horizontally stretched by a factor of and vertically stretched by a factor of 2
These all have the same graph!! They are identical!
So…“a” can do anything that “b” can do and vice versa … right? Can’t we just get rid of one of them?
Wrong!!! Without “a” you can’t do reflections in the x-axis and without “b” you can’t do reflections in the y-axis.
khxbay )(
Because the origin is an invariant point as far as stretching and reflecting is concerned we know that if the starting point hasn’t moved then no translations were involved.
If the starting point of the graph hasn’t moved horizontally then “h” must be zero and if the starting point hasn’t shifted vertically then “k” must be zero. No amount of stretching or reflecting can change that.
If the starting point has moved, then the values of h and k are just the coordinates of the place where the starting point has moved to.
Remember this!!
xy khxbay )(
00??
khba Since the graph has not been reflected in
either the x or y axis we know that a and b must be positive.
xy khxbay )(
00??
khba It can be viewed as either a purely vertical
stretch or purely horizontal stretch.
xy khxbay )(
00??
khba
xay bxy
xy 2
xy 4
Viewed as a vertical stretch Viewed as a horizontal stretch
12 a2a
)4(4 b
441644
22
bbb
xy 2
xy 4
xy 2
xy 4
xy 2
xy 4
xy 2
xy 4
xy 2
xy 4
khxbay )(54
kh 5,4
Since the graph has been reflected in both the x or y axis we know that both a and b must be negative. So we can set a = -1 and find b or set b = -1 and find a.
khxbay )(
541?
khba
5)4( xay
5)4(4 xy
5)43(1 a
451
aa
khxbay )(
54?1
khba
5)4( xby
5)4(16 xy
5)43(1 b51 b
224 b
16 b16b
015.273
1 T The Domain is
or [-273.15, +∞)The only transformations that can change the range as compared to the base function are vertical translations and reflections over the x-axis. Neither of these occur.
The Range is or [0, +∞)
115.273
T
15.273T
15.27313.331 Ts
15.27315.27315.2733.331 Ts
15.27315.2733.331 Ts
15.27315.2733.331 Ts
Ts 15.27305.20
15.27305.20 Ts
khxbay )(
The graph of has been stretched vertically by a factor of about 20 and then translated horizontally about 273 units to the left.
First let’s look at …
xxf )(
Square Root of a function
xxfy )(
xxf )(
xxfy )(
The domain of consists only of the values in the domain of for which
The range of consists of the square roots of all the values in the range of for which is defined.
You can use values of to predict values of and to sketch the graph of .
Invariant points occur at a because at these values .
F athe graph of i athe graph of i
xxf 23)(
i
The Domain of is or
xxfy 23)(
xxfy
xxf
23)(
23)(
23
23
22
32023
x
xxx
Invariant points occur at a because at these values .
122
22
22123
x
xxx
x
0 0
1 1 1
4 2
-3 9 3
xxfy
xxf
23)(
23)(
xxfy
xxf
23)(
23)(
The Domain of is and the Range of is
The Domain of is and the Range is
xxfy
xxf
23)(
23)(
Invariant points occur at (1, 1) and (1.5, 0)
2
2
5.02)(
5.02)(
xxfy
xxf
First find key points of 2)0(5.05.02)( 22 xxxf
Vertex at (0, 2) and y-int = 2
Find x-ints (set :
2
2
5.02)(
5.02)(
xxfy
xxf
Set
Invariant points occur at a because at these values .
Invariant points occur at (-2, 0) , (2, 0) , ( , 1), and ( , 1)
2
2
5.02)(
5.02)(
xxfy
xxf
The y-coordinates of the points on are the square roots of the corresponding points on
5.025.0
41.12
2
2
5.02)(
5.02)(
xxfy
xxf
The Domain of is and the Range of is
The Domain of is and the Range is
The y-coordinates of the points on are the square roots of the corresponding points on
Roots, zeroes, x-intercepts, solutions …??? What is the relationship between these things?
The following phrases are equivalent:"find the zeroes of f(x)""find the roots of f(x)""find all the x-intercepts of the graph of f(x)""find all the solutions to f(x)=0"
They are the same!!!!!
Example: The roots of are the zeroes of the function are the solutions to the equation and are the x-intercepts of the graph.
Determine the root(s) of algebraically.
05xFirst consider any restrictions on the variable in the radical.
035 x5x35 x
2235 x95x4x
Determine the root(s) of algebraically.
495
35
35
035
22
xxx
x
x Algebraic solutions to radical equations sometimes produce extraneous roots
In mathematics, an extraneous solution represents a solution that emerges from the process of solving the problem but is not a valid solution to the original problem.
You must always check your solution in the original equation.
Left Side Right Side
03339
354
35
x 0
One of the basic principles of algebra is that one can perform the same mathematical operation to both sides of an equation without changing the equation's solutions. However, strictly speaking, this is not true, in that certain operations may introduce new solutions that were not present before.The process of squaring the sides of an equation creates a "derived" equation which may not be equivalent to the original radical equation. Consequently, solving this new derived equation may create solutions that never previously existed. These "extra" roots that are not true solutions of the original radical equation are called extraneous roots and are rejected as answers.
Solve the equation algebraically.
35 xx
505
xx
2235 xx965 2 xxx0452 xx
014 xx0104 xx14 xx
Check:
Solve the equation algebraically. 5x
14 xx
Left Side Right Side
11
54
5
x
134
3
x
Left Side Right Side
24
51
5
x
231
3
x
The solution is x = -1
Solve the equation graphically. Express your answer to the nearest tenth.
First consider any restrictions on the variable in the radical.
053 2 x
3.13.1 xandx
135
135
xandx53 2 x
352 x
35
x
35
35
xandx
35
35
xandx
Solve the equation graphically. Express your answer to the nearest tenth.
Method 1:
Graph each side of the equation as a function:
Then determine the values of x at the point(s) of intersection.
The solutions are and
Solve the equation graphically. Express your answer to the nearest tenth.
Method 2:
Rearrange the radical equation so that one side is equal to zero:
Graph the corresponding functionAnd determine the x-intercepts of the graph.
The solutions are and
Algebraic solutions sometimes produce extraneous roots, whereas graphical solutions do not produce extraneous roots.
Algebraic solutions are generally exact while graphical solutions are often approximate.
