EXAMPLE 1

1. Compare the given equation to the standard form of an equation of a circle. You can see that the graph is a circle with center at (h, k) = (2, – 3)and radius r= = 3. 9 EXAMPLE 1 Graph the equation of a translated circle Graph(x – 2)2 + (y + 3) 2 = 9. SOLUTION STEP 1

2. EXAMPLE 1 Graph the equation of a translated circle STEP 2 Plot the center. Then plot several points that are each 3units from the center: (2 + 3, – 3) = (5, – 3) (2 – 3, – 3) = (– 1, – 3) (2, – 3 + 3) = (2, 0) (2, – 3 – 3) = (2, – 6) STEP 3 Draw a circle through the points.

3. (y – 3)2 (x + 1)2 4 9 EXAMPLE 2 Graph the equation of a translated hyperbola Graph – = 1 SOLUTION STEP 1 Compare the given equation to the standard forms of equations of hyperbolas. The equation’s form tells you that the graph is a hyperbola with a vertical transverse axis. The center is at (h, k) = (– 1, 3). Because a2 = 4 and b2 = 9, you know that a = 2 and b = 3.

4. Plot the center, vertices, and foci. The vertices lie a = 2 units above and below the center, at (21, 5)and (21, 1). Because c2 = a2 + b2 = 13, the foci lie c = 13 3.6 units above and below the center, at (– 1, 6.6)and (– 1, – 0.6). EXAMPLE 2 Graph the equation of a translated hyperbola STEP 2

5. EXAMPLE 2 Graph the equation of a translated hyperbola STEP 3 Draw the hyperbola. Draw a rectangle centered at (21, 3)that is 2a = 4 units high and 2b = 6 units wide. Draw the asymptotes through the opposite corners of the rectangle. Then draw the hyperbola passing through the vertices and approaching the asymptotes.

6. Compare the given equation to the standard form of an equation of a circle. You can see that the graph is a circle with center at (h, k) = (– 1, 3)and radius r= = 2. 4 for Examples 1 and 2 GUIDED PRACTICE 1. Graph(x + 1)2 + (y – 3) 2 = 4. SOLUTION STEP 1

7. for Examples 1 and 2 GUIDED PRACTICE STEP 2 Plot the center. Then plot several points that are each 2units from the center: (– 1 + 2, 3) = (1, 3) (– 1 – 2, 3) = (– 3, 3) (– 1, 3 + 2) = (– 1, 5) (– 1, 3 – 2) = (– 1, 1) STEP 3 Draw a circle through the points.

8. for Examples 1 and 2 GUIDED PRACTICE 2. Graph(x – 2)2 = 8 (y + 3) 2. SOLUTION STEP 1 Compare the given equation to the standard form of an equation of a parabola . You can see that the graph is a parabola with vertex at (2, – 3) ,focus (2, – 1)and directrix y=– 5

9. for Examples 1 and 2 GUIDED PRACTICE STEP 2 Draw the parabola by making a table of value and plot y point. Because p > 0, he parabola open to the right. So use only points x- value STEP 3 Draw a circle through the points.

10. (y – 4)2 9 for Examples 1 and 2 GUIDED PRACTICE 3. Graph (x + 3)2 – = 1 SOLUTION STEP 1 Compare the given equation to the standard forms of equations of hyperbolas. The equation’s form tells you that the graph is a hyperbola with a vertical transverse axis. The center is at (h, k) = (– 3, 4). Because a2 = 1 and b2 = 4, you know that a = 1 and b = 2.

11. Plot the center, vertices, and foci. The vertices lie a = 1 units above and below the center, at (–2, 4) and (–4, 4). Because c2 = a2 + b2 = 5, the foci lie c = 5 units above and below the center, at (– 3, + ,4 ) and (– 3, – , 4). 5 5 for Examples 1 and 2 GUIDED PRACTICE STEP 2

12. for Examples 1 and 2 GUIDED PRACTICE STEP 3 Draw the hyperbola. Draw a rectangle centered at ( –3, 4) that is 2a = 2 units high and 2b = 4 units wide. Draw the asymptotes through the opposite corners of the rectangle.

13. (y – 1)2 9 for Examples 1 and 2 GUIDED PRACTICE (x – 2)2 4. Graph + = 1 16 SOLUTION STEP 1 Compare the given equation to the standard forms of equations of hyperbolas. The equation’s form tells you that the graph is a hyperbola with a vertical transverse axis. The center is at (h, k) = (2, 1). Because a2 = 16 and b2 = 9, you know that a = 4 and b = 3.

14. for Examples 1 and 2 GUIDED PRACTICE STEP 2 Plot the center, vertices and co-vertices. Vertices are at (h+ a ,k) are (6,1) and ( – 2,1)and co-vertices are at (h,k +b) are (2,4) and (2,– 2) STEP 3 Draw the ellipse that panes through each vertices and co-vertices