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5.05 Graphs and characteristics of square root functions

Lesson

The parent function $f\left(x\right)=\sqrt{x}$f(x)=x

The basic square root function is given by $y=\sqrt{x}$y=x.

The name 'root' derives from an old Scandinavian word meaning the source or origin of something. The Latin form of the word is Radix, and square roots are often called radicals for that reason.

The square root $\sqrt{x}$x can be interpreted as the side length that is needed to create a square of area $x$x as shown in this diagram.

 

Characteristics of the square root function

Although no geometric square exists with length and area of zero, we still define $\sqrt{0}$0 as being a valid element of the domain and range of the function. However, negative values of $x$x are excluded from the domain as we can't have a negative length.

Let's work towards graphing $f\left(x\right)=\sqrt{x}$f(x)=x by starting with a table of values.

$x$x $0$0 $1$1 $4$4 $9$9 $16$16 $25$25
$f(x)=\sqrt{(}x)$f(x)=(x) $0$0 $1$1 $2$2 $3$3 $4$4 $5$5

Note from the table that it takes a jump of $3$3 from $x=1$x=1 to $x=4$x=4 to take $y$y from $1$1 to $2$2. It takes a jump of $5$5 to take $y$y from $2$2 to $3$3 and a jump of $7$7 to take $y$y from $3$3 to $4$4. The pattern continues indefinitely - the size of the increases in $x$x to move $y$y up by $1$1 unit continues to climb. As the $x$x-value increases, the $y$y-value also increases, but the rate of change is decreasing as seen in the graph below.

Using the graph we can identify the key characteristics of the square root function, $f\left(x\right)=\sqrt{x}$f(x)=x.

Characteristic $f\left(x\right)=\sqrt{x}$f(x)=x
Domain

Inequality form: $x\ge0$x0
Interval form: $\left[0,\infty\right)$[0,)

Range Inequality form: $y\ge0$y0
Interval form: $\left[0,\infty\right)$[0,)
Vertex/endpoint/extrema $\left(0,0\right)$(0,0)
$x$x-intercept $\left(0,0\right)$(0,0)
$y$y-intercept $\left(0,0\right)$(0,0)
Increasing/decreasing Increasing over its domain
End behavior As $x\to\infty$x, $y\to\infty$y

 

Practice question

Question 1

Consider the function $y=\sqrt{x}$y=x.

  1. State the $y$y value of the $y$y-intercept.

  2. Complete the table of values.

    $x$x $1$1 $4$4 $9$9 $16$16
    $y$y $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$
  3. Which of the following statements is correct?

    As $x$x increases, $y$y increases at a constant rate.

    A

    As $x$x increases, $y$y increases at a decreasing rate.

    B

    As $x$x increases, $y$y increases at an increasing rate.

    C
  4. Graph the function.

    Loading Graph...

 

Transformations of $y=\sqrt{x}$y=x

The basic root function can be dilated, reflected and translated in a similar way to other functions.

The root function $f(x)=\sqrt{x}$f(x)=x can be transformed to $g(x)=a\sqrt{b(x-h)}+k$g(x)=ab(xh)+k by dilating and/or reflecting it using the factors $a$a and $b$b, translating it, first horizontally $h$h units to the right, and then $k$k units upward.

Notation

Using function notation, if $f\left(x\right)=\sqrt{x}$f(x)=x then, for example, after a dilation of $2$2, and translations of  $5$5 units to the right and $3$3 units down, we create a new transformed function, say $g\left(x\right)$g(x) ,where:

$g\left(x\right)=2\cdot f\left(x-5\right)-3=2\sqrt{x-5}-3$g(x)=2·f(x5)3=2x53

Exploration

The best thing to do is to experiment with the first applet below showing how the variables $a,h$a,h and $k$k change the basic curve. Try both negative and positive values of $a$a.  

Focusing on the domain and range of the transformed function $y=a\sqrt{x-h}+k$y=axh+k, note that we need $x-h\ge0$xh0 and thus $x\ge h$xh.

The new range, because of the translation up or down caused by $k$k, has also changed to the interval $k\le y<\infty$ky< for values of $a>0$a>0. If however $a$a is negative the new range becomes $-\infty<yk. Experiment with the applet, taking careful note of the natural domains and ranges.

 

Summary

Dilations

$g(x)=a\sqrt{x}$g(x)=ax is the graph of $f(x)=\sqrt{x}$f(x)=x dilated vertically

$\left(1,1\right)$(1,1) $\to$ $\left(1,a\right)$(1,a)

  • If $a>1$a>1, then vertical dilation (stretch) away from the $x$x-axis
  • If$00<a<1, then vertical dilation (compression) toward the $x$x-axis

$g(x)=\sqrt{ax}$g(x)=ax is the graph of $f(x)=\sqrt{x}$f(x)=x dilated horizontally

$\left(1,1\right)$(1,1) $\to$ $\left(\frac{1}{a},1\right)$(1a,1)

  • If $a>1$a>1, then horizontal dilation (compression) toward the $y$y-axis

  • If $00<a<1, then horizontal dilation (stretch) away from the $y$y-axis

Reflections

$g(x)=-\sqrt{x}$g(x)=x is the graph of $f(x)=\sqrt{x}$f(x)=x reflected vertically

$\left(1,1\right)$(1,1) $\to$ $\left(1,-1\right)$(1,1)

  • Reflected over the $x$x-axis
  • Range becomes $y\le0$y0

$g(x)=\sqrt{-x}$g(x)=x is the graph of $f(x)=\sqrt{x}$f(x)=x reflected horizontally

$\left(1,1\right)$(1,1) $\to$ $\left(-1,1\right)$(1,1)

  • Reflected over the $y$y-axis
  • Domain becomes $x\le0$x0

Translations

$g(x)=\sqrt{x}+k$g(x)=x+k is the graph of $f(x)=\sqrt{x}$f(x)=x translated vertically

$\left(0,0\right)$(0,0) $\to$ $\left(0,k\right)$(0,k)

  • If $k>0$k>0, then translated $k$k units up
  • If $k<0$k<0, then translated $k$k units down
  • Range becomes $y\ge k$yk

$g(x)=\sqrt{x-h}$g(x)=xh is the graph of $f(x)=\sqrt{x}$f(x)=x translated horizontally

$\left(0,0\right)$(0,0) $\to$ $\left(h,0\right)$(h,0)

  • If $h>0$h>0, then translated $h$h units right
  • If $h<0$h<0, then translated $h$h units left
  • Domain becomes $x\ge h$xh

 

Practice questions

QUESTION 2

Use the graph of $y=f\left(x\right)$y=f(x) to graph $y=f\left(x-3\right)+4$y=f(x3)+4.

  1. Loading Graph...

QUESTION 3

Consider the function $y=2\sqrt{x}+3$y=2x+3.

  1. Is the function increasing or decreasing from left to right?

    Decreasing

    A

    Increasing

    B
  2. Is the function more or less steep than $y=\sqrt{x}$y=x?

    More steep

    A

    Less steep

    B
  3. What are the coordinates of the vertex?

  4. Hence graph $y=2\sqrt{x}+3$y=2x+3

    Loading Graph...

QUESTION 4

Consider the function $y=-2\sqrt{x-2}$y=2x2.

  1. Is the function increasing or decreasing from left to right?

    Decreasing

    A

    Increasing

    B
  2. Is the function more or less steep than $y=\sqrt{x}$y=x?

    More steep

    A

    Less steep

    B
  3. What are the coordinates of the vertex?

  4. Plot the graph $y=-2\sqrt{x-2}$y=2x2

    Loading Graph...

 

Finding the equation given its graph

We specifically focus on the transformed basic function given as $y=a\sqrt{x-h}+k$y=axh+k, which has the three unknown constants $a,h$a,h and $k$k.

As we have found from previous work, the starting point for such a function is the point $\left(h,k\right)$(h,k), a coordinate position completely independent of the dilation factor $a$a.

By independent we mean that changing the dilation factor has no effect on $\left(h,k\right)$(h,k).  The following graph shows various dilated root functions of the form $y=a\sqrt{x-3}+2$y=ax3+2

What the independence means is that it now becomes straightforward to find the correct dilation factor $a$a once the starting point has been determined.

For example, suppose we established that the root function $y=a\sqrt{x-3}+2$y=ax3+2 that starts at the point $\left(3,2\right)$(3,2), also passes through the point $\left(7,-4\right)$(7,4)

Since $\left(7,-4\right)$(7,4) is on the curve, it must satisfy the equation, so we proceed as follows:

$y$y $=$= $a\sqrt{x-3}+2$ax3+2
$-4$4 $=$= $a\sqrt{7-3}+2$a73+2
$-4$4 $=$= $a\sqrt{4}+2$a4+2
$-4$4 $=$= $2a+2$2a+2
$-6$6 $=$= $2a$2a
$a$a $=$= $-3$3

Therefore the correct equation is given by $y=-3\sqrt{x-3}+2$y=3x3+2, which is shown in the above graph. Notice that it does indeed passes through $\left(7,-4\right)$(7,4)

Worked example

Question 5

Consider the function of the form $y=a\sqrt{-x}+k$y=ax+k shown below. Can we identify the specific graph from the sketch? 

Think:

From the graph we see that the starting point is $\left(0,10\right)$(0,10) and this explains why the general form given does not show the presence of the horizontal translation $h$h - there is no shift left or right from the origin.

So the function we are dealing with is given by $y=a\sqrt{-x}+10$y=ax+10, and all we have to do is determine the size of the dilation. For this we need a point on the curve, we can see that $\left(-16,0\right)$(16,0) is on the curve.

Do:

$0$0 $=$= $a\sqrt{-\left(-16\right)}+10$a(16)+10
$0$0 $=$= $a\sqrt{16}+10$a16+10
$-10$10 $=$= $a\sqrt{16}$a16
$-10$10 $=$= $4a$4a
$a$a $=$= $\frac{-10}{4}$104
$a$a $=$= $-\frac{5}{2}$52

The specific graph we are looking for is given by $y=-\frac{5}{2}\sqrt{-x}+10$y=52x+10.

 

Practice questions

QUESTION 6

The square root function that has been graphed has an equation of the form $y=a\sqrt{x}$y=ax.

Loading Graph...

  1. Which of the following are points that lie on the graphed function? Select the two that apply.

    $\left(1,2\right)$(1,2)

    A

    $\left(3,3\right)$(3,3)

    B

    $\left(4,4\right)$(4,4)

    C

    $\left(2,3\right)$(2,3)

    D

    $\left(4,8\right)$(4,8)

    E
  2. Solve for the value of $a$a.

  3. Hence state the equation of the function.

QUESTION 7

The diagram shows the graph of the square root function $f\left(x\right)$f(x).

Loading Graph...

A number plane with axes labeled x and y. The x-axis is horizontal with positive values increasing to the right, and the y-axis is vertical with positive values increasing upwards. A smooth, continuous curve starts at the point (0,2) in the y-axis. As the value of x increases, the curve increases at a decreasing rate. It can also be observed that the curve is only available on values of x greater than 0.
  1. What is the smallest $y$y-value of the function?

  2. Hence determine which transformation of $y=\sqrt{x}$y=x has occurred to create the function $f\left(x\right)$f(x).

    vertical shift $2$2 units up

    A

    horizontal shift $2$2 units to the right

    B
  3. Hence state the equation of the function.

 

Applications of the square root function

 

Worked example

Question 8

On the Earth, when a rock is dropped from a $100$100 meter cliff face the elapsed time $t$t in seconds after it has fallen $d$d meters will be approximately given by $t_e=0.452\sqrt{d}$te=0.452d . On the Moon however, from a $100$100 meter cliff face, the rock would fall so that the elapsed time  after $d$d meters would be given by $t_m=1.11\sqrt{d}$tm=1.11d. Compare the elapsed times for the Earth and Moon for a rock falling from $20$20 m and $80$80 m.

Think: Perhaps the best way to answer this question is to graph both functions using technology.

Do: For $20$20 m, we can read off the graph that the Moon rock takes $5$5 seconds, whereas the Earth rock only takes $2$2 seconds.

For $80$80 m, we can read off the graph that the Moon rock takes $10$10 seconds, whereas the Earth rock only takes $4$4 seconds. 

Reflect: It seems that for both the Earth and the Moon it takes twice as long to fall four times the distance. Isaac Newton discovered a universal law of falling 'bodies'. Ignoring air resistance, for any planet, if a body (such as a rock) takes $t$t seconds to fall $d$d meters, then it will take  $kt$kt seconds to fall $k^2d$k2d meters.

This applet will allow you to compare the falling times from any planet (and the moon) with Earth. Remember that when comparing gravitational attractions, its not really about the size of a planet, but rather more about the density of its matter. The gas giant Uranus for example has almost the same fall curve as the Earth, but there is a vast difference between their sizes.  

 

Practice questions

QUESTION 9

The length of a blue whale calf in its first few months is modeled approximately by the equation $l=1.5\sqrt{t+4}$l=1.5t+4, where $l$l represents its length in meters at $t$t months of age.

  1. In this practical context, what are the possible values of $t$t? Enter your expression as an inequality.

  2. Complete the table of values.

    Months ($t$t) $0$0 $60$60 $96$96
    Length ($l$l) $\editable{}$ $\editable{}$ $\editable{}$
  3. Graph the length function.

    Loading Graph...

  4. Is the calf’s length increasing by the same amount each month?

    Yes

    A

    No

    B
  5. According to the model, what will be the length of the whale at $7$7 years of age? Give your answer correct to one decimal place.

  6. An orca whale calf is born shorter than a blue whale calf. If its growth can be modeled by the equation $L=k\sqrt{t+4}$L=kt+4, which of the following could be a possible value of $k$k?

    $0.6$0.6

    A

    $1.5$1.5

    B

    $2.4$2.4

    C

QUESTION 10

The radius $r$r of a sphere with surface area $A$A is $r=\sqrt{\frac{A}{4\pi}}$r=A4π.

Solve for the surface area, $A$A, of a moon with a radius of $1080$1080 kilometers. Give your answer to the nearest square kilometer.

Outcomes

A2.4.E

Formulate quadratic and square root equations using technology given a table of data

A2.4.G

Identify extraneous solutions of square root equations

A2.7.I

Write the domain and range of a function in interval notation, inequalities, and set notation

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