Finding Inverses of Quadratic and Square Root Functions
Description/Explanation/Highlights
Video Description
This video explains how to find inverses of quadratic and square root functions.
Steps and Key Points to Remember
To find inverses of quadratic and square root functions, follow these steps:
- To find the inverse from a graph or table of a quadratic or square root equation, make a table of key coordinates from the graph or use the table given if available.
- Now make a new table and write the y-values in the x column and the x-values in the y column. In other words switch x & y.
- Graph the values in the new table and connect the points to form a smooth curve. You have graphed the inverse of the original function!
- This new inverse graph should be a reflection of the original graph over the y = x line.
- If you graph the inverse of the quadratic function and want the resulting graph of the inverse to also be a function, you will need to remove the lower half of the resulting graph which results from only using positive values of the square root of x. This is how the square root function is created…by finding the inverse of the quadratic parent.
- You can solve the equation algebraically for its inverse equation by switching x & y and solving the new equation for y. In the example: \(y=3x^2-6\), switch x and y as follows \(x=3y^2-6\) then solve for the new y by adding 6 to both sides, dividing by 3 and taking the square root to get rid of \(y^2\). The result is the inverse equation which written in function notation is \(f^{-1}(x)=\sqrt{\frac{1}{3}x+2}\)
- To check, you can find the inverse of the inverse equation and it should be the original equation!
Here are some key points to keep in mind when finding inverses:
- All inverse graphs, including quadratics and square roots, reflect over the y = x line.
- To graph an inverse, make a table of the coordinates of key points and switch the x & y values. Graph the switched coordinates to graph the inverse.
- To find the inverse equation, switch x and y in the original equation and solve the new equation for Y. This is the inverse equation.
- If you are graphing the inverse of the quadratic function, it will be the square root function as long as you restrict the domain of the original function and use only the top half of the graph.
- \(f^{-1}(x)\) is used as the function notation for inverse.
Video Highlights
- 00:00 Introduction
00:14 Inverse example of a quadratic graph
02:55 Finding inverse of \(y = 3x^2-6\) algebraically
04:54 Finding inverse of \(y = (x + 5)^2-3\)
06:30 Check by finding inverse of the inverse
08:30 Conclusion
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