Square Root Functions
Transformations
Square root functions can also be written in h,k form. A Square root function contains a square root with the independent variable (x) under the radical. The parent function is f(x) = √x. The graph and table of the parent function is show to the right. Notice there are no negative x values in the parent function. This is because taking the square root of a negative number results in a non-real number. This is why the square root function has a point of origin (starting point). To the right you see the point of origin is (0, 0).
Transformations: Recall that the parent function of a quadratic is y = x^2 and the transformations applied to this parent function in h,k form, is what determines the parabola after the transformations. Square root functions are very similar. Notice from the diagram on the right, the a, h, and k values preform the same transformations as they did in the quadratic function. Horizontal Compressions and Stretches (b) The b value controls the horizontal stretch or horizontal compression. A horizontal compression pushes the graph closer to the y-axis , while a horizontal stretch pulls them away from the y-axis. A horizontal compression will give the same outcome as a vertical stretch (by the same factor), and a horizontal stretch produces the same outcome as a vertical compression. Reflection over the y-axis (-b) If the value of b is negative the function is reflected across the y-axis. Quadratic Functions could also be shown with a b value, however since a parabola has horizontal symmetry it is usually not included. Click on the buttons below for more details on each transformation. |
Square Root Parent Function
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Watch each example below and take notes. Then complete the worksheet.
Transformations WS Answer Key
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Domain and Range
Remember that the domain is all the x values possible within a function. With quadratic functions, the domain was always all real numbers because the set of x values that can be inputted into a quadratic function rule can be any real number.
Square root functions do not have an unlimited domain. Anytime the x value produces a negative number under the radical, the result will be a non-real number. Therefore these x values will not be part of the domain. To find the domain, focus on the h value (x coordinate of the point of origin). If the graph is going to the right (like i the example below) the domain will be x ≥ h. If the graph is going to the left the domain will be x ≤ h. The range will be all the possible outputs within the function, so focus on the k (y value of the point of origin). Below the graph is going up so the range will be y ≥ k. If the graph is going down the range will be y ≤ k. |
Intervals of Increasing and Decreasing
Remember when identifying intervals of increasing and decreasing you are finding the x values that cause the graph to go up and the x values that cause the graph to go down. Square root functions will only do one or the other. The graph to the right is increasing only. It will increase when x ≥ 2 or in the interval [2, ∞). This will always match the domain. You will just need to determine if it is increasing or decreasing. |
End Behavior
End behavior shows what is happening at the ends of the graph. Since the square root function is not continuous (stops at one end), it is different than the quadratic function you are familiar with. One side will stay as x approaches positive or negative infinity, this will be determined by the arrow. See the example to the right.
However since the left side of the graph stops at (2, 0), the end behavior here will be, as x approaches 2, y approaches 0. These values will be determined from the point of origin (h, k). |
Systems
You have seen systems of non-linear equations before dealing with quadratics. Systems of equations can involve any combination of function rules. You will use the same system rules you always have, the only difference is the way you solve the resulting equation. Select the example button to see an example worked out algebraically.
REMEMBER!
The solution to a system is the point of intersection. This shows the x and y values the two function rules have in common. If you graph the 2 functions you can see the point of intersection. Below is the graphs of the system from the example.
The calculator works too!
Hit 2nd, trace #5. Move the curser to the point of intersection and hit enter 3 times.
Hit 2nd, trace #5. Move the curser to the point of intersection and hit enter 3 times.
Complete the problems below using any method. Solve at least 2 algebraically.