The square root function is given by yyyy xxxx (or y xy xy x12).
Power functions are functions of the form f (ff x) = xn, n∈ R. The value of the power, n, determines the type of function. We saw earlier that when n= 1, f (ff x) = x and the function is linear. When xx n= 2, f (
ff x) = x2 and the function is quadratic. When n= 3, f (ff x) = x3 and the function is cubic. When n= 4, f (ff x) = x4 and the function is quartic.
When n=−1, f (ff x) = x−1 and the power function produces the graph of
a hyperbola. When n=−2, f (ff x) = x−2 and the power function produces the graph of a truncus.
The power function that produces the graph of the square root function has a value of n=1 2. Thus, the function f xf xf x( )= x can also be expressed as the power function f xf xf x( )= f xf xf xf xf xf xf xf xf xf xf x( )( )( )( )=x12.
The function is defi ned for x≥ 0; that is, the domain is R+∪ {0}, or [0, ∞).
2E
y
0 x
As can be seen from the graph, the range of the square root function is also R+∪ {0}, or [0, ∞).
Throughout this section we will refer to the graph of yyyy xxxx as ‘the basic square root curve’.
Let us now investigate the effects of various transformations on the basic square root curve.
Consider the function y a x b cy ay a x b== x b−− + , or oror y a x by ay a(=== x bx bx bx bx bx b−−− )1
Let us now investigate the effects of various transformations on the basic square root curve.
Let us now investigate the effects of various transformations on the basic square root curve.
2 + c.
Dilation
The value a is a dilation factor; it dilates the graph from the x-axis.
The domain is still [0, ∞).
Refl ection
If a is negative, the graph of a basic square root curve is refl ected in the x-axis. The range becomes (−∞, 0]. The domain is still [0, ∞).
If x is replaced with xx −x, the graph is refl ected in the y-axis.
For example, the graphs with equations yyyy xxxx and For example, the graphs with equations
y x
y x
y x
y −x are refl ected across the y-axis. The domain becomes (−∞, 0] and the range is [0, ∞).
Translation
Horizontal translation
The value b translates the graph horizontally. If b > 0, the graph is translated to the right, and if b < 0, the graph is translated to the left. The graph with the equation yyyyyy= xxxxxx 3 results when the − basic curve is translated 3 units to the right. This translated graph has domain [3, ∞) and range [0, ∞). If the basic curve is translated 2 units to the left, it becomes yyyyyy== xxxxxx 2 and has domain [++ −2, ∞) and range [0, ∞). The domain of a square root function after a translation is given by [b, ∞).
Vertical translation
The value c translates the graph vertically. If c > 0, the graph is translated vertically up, and if c < 0, the graph is translated vertically down. If yyyy xxxx is translated 2 units vertically up, the graph obtained is yyyyyy== xxxxxx 2, with domain [0, ++ ∞) and range [2, ∞).
If the basic curve is translated 4 units down, it becomes yyyyyy= xxxxxx 4, − with domain [0, ∞) and range [−4, ∞).
The range of the square root function is [c, ∞) for a > 0.
y
0 x
y = a x a= 1 a= 2 a= 3
a=1–2
x y
(0, 0) (1, 1)
y= √x
y=−√x (1, −1)
√√
√√
x y
(0, 0) (−1, 1) (1, 1)
y= √x
y= √√√−x √√
y
0 x (−2, 0) (3, 0)
y = x − b x x x b= −2
b= 3
y
0 x
(0, −4) (0, 2)
y = x + c x x x c= 2
c= −4
Combination of transformations
The graph of y a x b cy ay a x b== x b−− + shows the combination of these transformations.
The point (b, c) is the end point of the square root curve. For example, the end point of yyyyyy= xxxxxx 2 1− ++ is (2, 1).
It is always good practice to label the end point with its coordinates. Make sure it is an open circle if the x-value is not in the required domain and a closed circle if its x-value is within the function’s domain.
Consider the function y a b x cy ay a b x== b x−− + .
The graph of y a x b cy ay a x b== x b++ + has (−b, c) as its end point. If this function is refl ected in the y-axis, it becomes y a== x b c++
3 State the effect of a on the graph. The graph is dilated by a factor of 3 from the x-axis and refl ected in the x-axis.
4 Identify the value of b. b=−5
5 State the effect of b on the graph. The graph is translated 5 units to the left.
6 Identify the value of c. c= 3
7 State the effect of c on the graph. The graph is translated 3 units up.
WORKED EXAMPLE 16 eBookplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplus eBoo eBookkkplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplus
Worked example 16
For each of the following functions fi nd the domain and range.
a yyyyyyyyyyyyyyyy==2222 xxxxxxxxxxxxxxxx−−3++1111 b yyyyyyyyyyyyyyyyyyyyy===========−44444 3xxxxxxxxxxxxxxxxxxxxx+++++++++++2−−44444
THINK WRITE
Write the general formula. y a x b cy ay a x b== x b−− + a 1 Write the question. a yyyyyyyyyyyyyyyy==22 xxxxxxxxxxxxxxxx−−33+1
2 Identify the values of b and c. b = 3, c = 1
3 State the domain x≥ b. The domain is [3, ∞).
4 State the range (y≥ c for a > 0). The range is [1, ∞).
b 1 Write the question. b yyyyyyyy −4 34 3xxxxxxxx 2 4+ −+ −+ −
2 Factorise the expression under the square root sign.
y x
y= x+
y x
y==−4 34 3 x++ −
y x
y x
y x
y==== x++++
y x
y x
y x
y x
y x
y x
y x
y ((x 23)) 4
y x
y== x++
y x
y x
y x
y x 2233
3 State the domain. The domain is [ ,[ ,[ ,[ ,[ ,[ , )−32 ∞∞∞ .
4 Identify the value of c and check whether a is
positive or negative. c =−4, a < 0
5 State the range. The range is (−∞, − 4].
c 1 Write the question. c yyyyyyyyyyyy 4 xxxxxxxxxxxx++2
2 Identify the values of b and c. b = 4, c = 2
3 Since the function is of the form y a b x c== −−
y a
y a b xb x+ , the domain is x ≤ b.
The domain is (−∞, 4].
4 State the range (y≥ c). The range is [2, ∞).
y = a b − x + c a b a b a b y = a x − b + c a x a x a x (b, c) a< 0 a< 0
a> 0 a> 0
To sketch the graph of the square root function, we need to compare the given formula with
y a x b c== −− y a
y a x bx b+ . This will give us an idea of the changes required to transform the basic square root curve into the one we want. It will also let us know the way the curve will look. The diagram above illustrates the idea.
Once the coordinates of the end point and the direction of the curve are known, the intercepts with the axes (if any) should be found before sketching.
WORKED EXAMPLE 17
Sketch the graph of yyyyyyyyyyyyyy −2222 xxxxxxxxxxxxxx− +− +− +− +− +− + , clearly marking intercepts and the end points.3 1111
THINK WRITE/DRAW
1 Write the equation. yyyyyyyyyyyyyyyy== −22 xxxxxxxxxxxxxxxx−−33+1
2 Write the coordinates of the end point. End point: (3, 1)
3 State the shape of the graph. Shape:
4 Find the x-intercept by letting y= 0. x-intercept: y = 0
0 2 3 1
2 3
2 3 1
3 3
3
1 21
2 2 1 41
4
= −
= −
0 2
0 2 3 13 1
− =
2 3
2 3
− =33
− =33
=
=
0 2
0== 2 x−−
2 3
2 3
x x
x ( ) ( )1122
5 Find the y-intercept if there is one. There is no y-intercept.
6 Sketch the graph by plotting the end point, marking the x-intercept, and drawing the curve so that it starts at the end point and passes through the x-intercept.
y
0 x
14
1 (3, 1)
( , 0)3311––44
3
WORKED EXAMPLE 18
Given f : [0, ff ∞) → R, where f xf xf x( )==== x and g(x(( ) = af (afaf x ( ( ) + b, where a and b are positive real constants, consider the effect on g(x(( ) as a and b increase individually.
THINK WRITE
1
2 Write your description in words. As a increases, the graph is dilated away from the x-axis, with the graph stretched further from the x-axis.
As b increases, the graph is translated up parallel to the y-axis.
1. The graph of the function y a x b cy ay a x b== x b−− + is the graph of yyyy xxxx, dilated by the factor of a from the x-axis and translated b units along the x-axis and c units along the y-axis.
If
2. a< 0, the basic graph is refl ected in the x-axis.
The end point of the graph is (
3. b, c).
The domain is
4. x≥ b.
The range is
5. y ≥ c for a > 0, or y ≤ c for a < 0.
If
6. y a b x cy ay a b x== b x−− + , the domain is x ≤ b; the graph of y a xy ay a is refl ected in the y-axis.
y
0 x
(b, c)
y = a x − b + c REMEMBER
On a Graphs page, complete the function entry line as:
f x1 vv1 xx v f x
f x( ) f x f x11 11 f x f x f x
f x11 ==vvvvvvvv11 xxxxxxxx++ 2 Insert a slider by pressing:
• MENU b
• 1:Actions 1
• A:Insert Slider A Then press ENTER ·.
Repeat to insert a second slider.
Grab the slider for each variable and move it back and forth taking note of the effect of each variable increasing.