1. Graphs of Radical Functions

2. The Graph of a Square Root Function
On your calculator, graph
The graph should look like half of a sideways parabola with the vertex at the origin.
In fact, it is a sideways parabola.
If you start with and square both sides of the equation, you get
The graph of y2 = x is the same as y = x2. They are both parabolas. The only difference is the y2 = x opens right instead of up.

3. The Graph of a Square Root Function
The graph of does not include the bottom half of the parabola. This part is excluded, so that the graph will be a function.
What is the domain of this function?
What is the range of this function?

4. Shifting the Graph of a Square Root Function
The graph of has a vertex at the origin and opens right.
What does the graph of look like? Where’s the vertex?
What is the domain of this function?
What is the range of this function?
The + 1 shifted the graph vertically up one.

5. Shifting the Graph of a Square Root Function
What does the graph of look like? Where’s the vertex?
What is the domain of this function?
What is the range of this function?
The - 5 shifted the graph vertically down five.

6. Shifting the Graph of a Square Root Function
What does the graph of look like? Where’s the vertex?
What is the domain of this function?
What is the range of this function?
The + 2 shifted the graph horizontally two to the left.

7. Shifting the Graph of a Square Root Function
What does the graph of look like? Where’s the vertex?
What is the domain of this function?
What is the range of this function?
The -4 shifted the graph horizontally four to the right.

8. Shifting the Graph of a Square Root Function
What does the graph of look like? Where’s the vertex?
What is the domain of this function?
What is the range of this function?
The minus sign in front made the graph go down.

9. Shifting the Graph of a Square Root Function
What does the graph of look like? Where’s the vertex?
What is the domain of this function?
What is the range of this function?
The 2 in front made the graph go up twice as fast (made it steeper).

10. Shifting the Graph of a Square Root Function
What does the graph of
look like? Where’s the vertex?
What is the domain of this function?
What is the range of this function?
The 1/2 in front made the graph go up half as fast (made it less steep).

11. Shifting the Graph of a Square Root Function
What does the graph of
look like? Where’s the vertex?
What is the domain of this function?
What is the range of this function?
The horizontal shift is 3 and the negative in front of the x makes it open left.

12. Shifting the Graph of a Square Root Function
What does the graph of
look like? Where’s the vertex?
What is the domain of this function?
What is the range of this function?
The horizontal shift is 2 to the left and the negative in front of the x makes it open left.

13. Vertex Form of a Radical Function
The vertex form of a radical function is:
Whatever makes the inside of the radical equal zero is the horizontal shift.
The sign of the x makes it open left or right.
The coefficient determines the steepness of the graph (like a slope)
The sign of the coefficient makes it go up or down.
The k is the vertical shift.

14. Vertex Form of all the Functions that you have learned.
The vertex form of a absolute value function is:
The vertex form of a quadratic function is:
The vertex form of a radical function is:

15. Graph the equation. Then state the domain and range.
What is the domain of this function?
What is the range of this function?

16. Graph the equation. Then state the domain and range.
What is the domain of this function?
What is the range of this function?

17. Graph the equation. Then state the domain and range.
What is the domain of this function?
What is the range of this function?

18. Graph the equation. Then state the domain and range.
What is the domain of this function?
What is the range of this function?

19. Graphs of Cube Roots Functions
What does the graph of look like?
Why does the cube root function have points on both sides of the y axis but the square root function does not?
Domain:
(all real numbers)
Range:
(all real numbers)

20. Graph & state the domain and range.
vertex at (3,5)
the coefficient is positive so the graph goes up
the 2 makes it twice as steep as a regular graph.
Domain:
(all real numbers)
Range:
(all real numbers)

21. Graphs of cube root functions
Vertex form of a cube root function is:
Whatever makes the inside of the radical equal zero is the horizontal shift.
The sign of the coefficient of x makes it open left or right.
The coefficient determines the steepness of the graph (like a slope)
The sign of the coefficient makes it go up or down.
The k is the vertical shift.
Note: the signs of the 2 coefficients may cancel either out.

22. Graph & state the domain and range.
vertex at (-4,-6)
the coefficient is negative so the graph goes down
the 1/2 makes it half as steep as a regular graph.
Domain:
(all real numbers)
Range:
(all real numbers)

23. Graph & state the domain and range.
vertex at (5,2)
the coefficient is negative so the graph goes down
the negative in front of the x makes the direction of the graph flip left to right.
Domain:
(all real numbers)
Range:
(all real numbers)

24. Higher Degree Roots

25. Sketch the graph

26. Sketch the graph