In the following animation, experiment with the parameters |a,| |b,| |h,| and |k| of the square root function and observe their effects on the function’s properties. Afterwards, read the concept sheet to learn more about the function’s properties.
The following table summarizes the properties of the square root function various properties algebraically.
|
If... |
|a>0| and |b>0| |
|a>0| and |b<0| |
|a<0| and |b>0| |
|a<0| and |b<0| |
|---|---|---|---|---|
|
Domain and range |
|dom\ f= [h,+\infty[| |
|dom\ f = ]-\infty,h]| |
|dom\ f = [h,+\infty[| |
|dom\ f = ]-\infty,h]| |
|
Increasing and decreasing intervals |
Increasing |
Decreasing |
Decreasing |
Increasing |
|
Minimum |
|min\ f = k| |
No |
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Maximum |
No |
|max\ f = k| |
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|x|-intercept (zero of the function) |
|x_1,| solution of:||0=a\sqrt{b(x-h)}+k|| |
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|y|-intercept (initial value) |
If it exists, it will be the value of |f(0).| |
|||
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Positive and negative intervals |
|f| is positive on |[x_1,+\infty)| and is negative on |[h,x_1]| |
|f| is positive on |(-\infty,x_1]| and is negative on |[x_1,h]| |
|f| is positive on |[h,x_1]| and is negative on |[x_1,+\infty)| |
|f| is positive on |[x_1,h]| and is negative on |(-\infty,x_1]| |
Determine the properties of the following function. ||f(x)=-2\sqrt{1.2(x+1)}-2||
To successfully find the different properties, it is useful to draw a graph.
It is now easier to determine the different properties of the previous table, the vertex of the function being |(-1,-2).|
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The domain of the function is |[-1,+\infty)| and its range is |(-\infty,-2].|
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Increasing and decreasing intervals: the function is decreasing over its domain, i.e. over |[-1,+\infty).|
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Extrema: |f(x)=-2\sqrt{1.2(x+1)}-2| is decreasing, so it only has a maximum which is given by the parameter |k,| or |-2.|
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Moreover, the graph makes it easy to see that the function does not have an |x|-intercept. However, it can be helpful to do the math to understand this better. So, replace |f(x)| by |0| and isolate |x.|
||\begin{align} 0 &= -2\sqrt{1.2(x+1)}-2 \\ 2 &= -2\sqrt{1.2(x+1)} \\ -1 &= \sqrt{1.2(x+1)} \end{align}||
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We can stop at this point, because there is no solution.
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Compute the |y|-intercept by replacing |x| by |0| in the equation.
||\begin{align} f(0) &= -2\sqrt{1.2(0+1)}-2 \\ f(0) &= -2\sqrt{1.2}-2 \\ f(0) &= -4.19 \end{align}||
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Thus, the value of the |y|-intercept is |-4.19.|
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Positive and negative intervals: since there is no |x|-intercept at the origin and the function is below the |x|-axis, it is negative over its entire domain, i.e., on |[-1,+\infty).|
What are the different properties of the function |f(x)=\dfrac{4}{3}\sqrt{x-1}-4| ?
To help determine the properties, it is strongly suggested to make a graph. It is very important to identify the vertex, which has coordinates |(1,-4).|
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The domain of the function is |[1,+\infty)| and its range is |[-4,+\infty).|
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The interval: the function is increasing over its entire domain and therefore over |[1,+\infty).|
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Extrema: as the function is increasing, it only has a minimum which is given by the parameter |k| , i.e. |-4.|
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The |x|-intercept is calculated by replacing |f(x)| with |0| and isolating |x|. ||\begin{align} 0 &= \frac{4}{3}\sqrt{x-1}-4 \\ 4 &= \frac{4}{3}\sqrt{x-1} \\ 4 \times \frac{3}{4} &= \sqrt{x-1} \\ 3 &= \sqrt{x-1} \end{align}|| At this point, square both sides of the equal sign. ||\begin{align} 9 &= x-1 \\ 10 &= x \end{align}|| So, the conclusion is that the |x|-intercept is |10.|
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For the |y|-intercept, it is unnecessary to do a calculation since its domain clearly indicates that it is undefined at |x=0.|
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For the positive and negative intervals, using the |x|-intercept, the conclusion is that the function is positive on |[10,+\infty)| and that it is negative over |[1,10]|.