1. 1.5 Transformations of Some Basic Curves 1 In previous sections, we have graphed equations such as f(x)=x 2 +3 by either translating the basic function f(x)=x 2 or by finding the vertex and then choosing two points to the left and two points to the right of the x value of the vertex. In this section, procedures will be used to graph four new basic functions. The method of transformations and reflections can be used to graph these new functions. Another method which is choosing points to graph the functions. However, when choosing points, the correct values for x must be chosen. If the correct values for x are not chosen, an accurate graph of the function can not be obtained. 1.Determine the function’s domain. Since the third function above has a radical, set x – h greater than or equal to zero. (Always Necessary) 2. Find the y-intercept and any x-intercepts. Let x equal zero to find the y intercept, and let y or f(x) equal to zero and solve for x to find the x intercept. 4. Set up a table of ordered pairs that satisfies the function. When choosing values for x, start with ‘h’. This is similar to the vertex of a parabola. With the first two functions above, pick two points to the left and two points to the right of ‘h’. For the third function above, choose values according to the domain of the function. 5. Plot the points and connect them with a smooth curve. 3.Determine if the function possesses any type of symmetry. If f(-x)=f(x), then the function has y-axis symmetry. If f(-x)=-f(x), then the function has x-axis symmetry.

2. 1.5 Transformations of Some Basic Curves 2 1. Find the domain of this function: Graphing the cubic function: f(x)=x 3 The domain is all real numbers, or (-∞,∞). 2. Set up a table of ordered pairs. x y 0 0 1 1 2 8 -2 -8 x axis y axis 3. Plot the points and sketch the curve. This graph has a stretched out s- shape. 1. Find the domain of this function:(-∞,∞). 2. Set up a table of ordered pairs. Start with ‘h’, or the value which will give zero inside the parentheses. Then two points to the left and two points to the right. x y 2 -1 3 0 4 7 1 -2 0 -8 Example 1. Graph f(x)= (x – 2) 3 – 1 Solution: x axis y axis 3. Plot the points and sketch the curve.

3. 1.5 Transformations of Some Basic Curves 3 Your Turn Problem #1 x y -2 1 -1 2 0 9 -3 0 -4 -7 x axis y axis

4. 1.5 Transformations of Some Basic Curves 4 Since x does not appear under a radical, nor does it appear in a denominator, there are no restrictions on what values x can become. The domain of this function is: (-∞,∞). Graphing the function: f(x)=x 4 1. Find the domain of this function: 2. Set up a table of ordered pairs. x y 0 0 1 1 2 16 -1 1 -2 16 x axis y axis 3. Plot the points and sketch the curve. This graph has a parabola-like shape. Next Slide

5. 1.5 Transformations of Some Basic Curves 5 1. Find the domain of this function:(-∞,∞). 2. Set up a table of ordered pairs. Start with ‘h’, or the value which will give zero inside the parentheses. Then two points to the left and two points to the right. x y -1 -4 0 -3 1 12 -2 -3 -3 12 3. Plot the points and sketch the curve. Example 2. Graph f(x)= (x+1) 4 – 4 Solution: x axis y axis Next Slide

6. 1.5 Transformations of Some Basic Curves 6 Your Turn Problem #2 x y 2 -3 3 -2 4 13 1 -2 0 13 x axis y axis

7. 1.5 Transformations of Some Basic Curves 7 Since x does appear under an even radical, we must take measures to ensure that we only take the square root of nonnegative numbers. Therefore we set the radicand to be greater than or equal to zero. The numbers that satisfy the inequality will be elements of the domain. So, our domain is [0,∞). 1. Find the domain of this function: 2. Set up a table of ordered pairs. Choose values for x which is in our domain and values which will give perfect square roots. x y 0 0 1 1 4 2 9 3 16 4 3.Plot the points and sketch the curve. This graph is half a parabola opening to the right. x axis y axis Next Slide

8. 1.5 Transformations of Some Basic Curves 8 1. Find the domain of this function: 2.Set up a table of ordered pairs. Start with -2. Then choose values which will give perfect square roots. Solution: So, our domain is [-2,∞). x y -2 0 -1 1 2 2 7 3 14 4 3. Plot the points and sketch the curve. x axis y axis

9. 1.5 Transformations of Some Basic Curves 9 Your Turn Problem #3 x y 0 2 1 3 4 4 9 5 16 6 x axis y axis

10. 1.5 Transformations of Some Basic Curves 10 Since x does not appear under a radical, nor does it appear in a denominator, there are no restrictions on what values x can become. The domain of this function is: (-∞,∞). 1. Find the domain of this function: 2. Set up a table of ordered pairs. x y 0 0 1 1 2 2 -1 1 -2 2 3. Plot the points and sketch the curve. This graph has a v-shape. x axis y axis

11. 1.5 Transformations of Some Basic Curves 11 1. Find the domain of this function: 2.Set up a table of ordered pairs. Start with h=1. Then choose two values to the right of 1 and two values to the left of 1. Solution: x y 1 0 2 1 3 2 0 1 -1 2 (-∞,∞). Next Slide 3. Plot the points and sketch the v-shape curve. x axis y axis

12. 1.5 Transformations of Some Basic Curves 12 Your Turn Problem #4 x y 4 0 5 -1 6 -2 3 -1 2 -2 x axis y axis Please Note: This lesson emphasized the technique of obtaining the points by choosing certain points for x. The starting point is the value which will give zero in the parentheses, absolute value or under the radical sign. In the first two cases, pick two points to the right and two points to the left of the starting point. This will give enough points to sketch the curve. If the equation has the radical sign, after the starting point, choose values which will be in our domain. The End. B.R. 8-3-04