Lesson

We now turn to the task of finding the equation of a root function given its graph.

We specifically focus on the transformed basic function given as $y=a\sqrt{x-h}+k$`y`=`a`√`x`−`h`+`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.

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`=`a`√`x`−3+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`=`a`√`x`−3+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 |
$=$= | $a\sqrt{x-3}+2$a√x−3+2 |

$-4$−4 | $=$= | $a\sqrt{7-3}+2$a√7−3+2 |

$-4$−4 | $=$= | $a\sqrt{4}+2$a√4+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`=−3√`x`−3+2, which is shown in the above graph. Notice that it does indeed passes through $\left(7,-4\right)$(7,−4).

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

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`=`a`√−`x`+10, and all we have to do is determine the size of the dilation.

Three points have been circled on the graph. These look to be, *by observation*, points that lie on the graph. However looks can be deceiving! We will attempt to find a using all three points, one at at time.

If we select $\left(-16,0\right)$(−16,0) then we can write $0=a\sqrt{-\left(-16\right)}+10$0=`a`√−(−16)+10 and solve for $a$`a`. The expression simplifies to $4a=-10$4`a`=−10 and so $a=-\frac{5}{2}$`a`=−52 or as a decimal $a=-2.5$`a`=−2.5.

If we select $\left(-10,2\right)$(−10,2) then we can write $2=a\sqrt{-\left(-10\right)}+10$2=`a`√−(−10)+10. This slightly more difficult equation can be solved as follows:

$2$2 | $=$= | $a\sqrt{-\left(-10\right)}+10$a√−(−10)+10 |

$2$2 | $=$= | $a\sqrt{10}+10$a√10+10 |

$-8$−8 | $=$= | $$ |

$a$a |
$=$= | $-\frac{8}{\sqrt{10}}$−8√10 |

$a$a |
$=$= | $-2.5298...$−2.5298... |

This is a worrying result, because it disagrees with our first derived value of $a$`a`. A quick look back at the graph puts doubt in our mind that the second point $\left(-10,2\right)$(−10,2) perhaps might just miss the graph. This is a great reminder that forming equations from a graph has its problems.

We need some confirmation on our first answer of $a=-2.5$`a`=−2.5.

If we select the final point $\left(-4,5\right)$(−4,5) then we can write $5=a\sqrt{-\left(-4\right)}+10$5=`a`√−(−4)+10 and solve for a. The expression easily simplifies to $2a=-5$2`a`=−5 and so $a=-\frac{5}{2}$`a`=−52 or as a decimal $a=-2.5$`a`=−2.5.

Thus, with some sureness we can conclude with reasonable confidence that the specific graph we are looking for is given by $y=-\frac{5}{2}\sqrt{-x}+10$`y`=−52√−`x`+10.

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

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By using a point on the graph, solve for the value of $a$

`a`.Hence state the equation of the function.

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

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State the value of $b$

`b`.By using a point on the graph, solve for the value of $a$

`a`.Hence state the equation of the function.

Consider the square root function that has been graphed.

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For a square root function whose equation is of the form $y=a\sqrt{x-h}+k$

`y`=`a`√`x`−`h`+`k`, which of the following are the coordinates of the endpoint (where the function is not continuous)?$\left(a,k\right)$(

`a`,`k`)A$\left(k,a\right)$(

`k`,`a`)B$\left(k,h\right)$(

`k`,`h`)C$\left(h,k\right)$(

`h`,`k`)D$\left(a,k\right)$(

`a`,`k`)A$\left(k,a\right)$(

`k`,`a`)B$\left(k,h\right)$(

`k`,`h`)C$\left(h,k\right)$(

`h`,`k`)DState the coordinates of the endpoint of the function that has been graphed.

Using a point on the curve, solve for the value of $a$

`a`.Hence state the equation of the square root function.

Display the graphs of linear and non-linear functions and connect the structure of the functions with their graphs

Apply graphical methods in solving problems