Warm-Up Exercises ANSWER 17 ANSWER 13 Evaluate each expression. 1. (10 – 2) 2 + (8 + 7) 2 2. (–3 – 2) 2 + (–8 – 4) 2

Warm-Up Exercises 3. In a map on a coordinate grid, two towns are at coordinates (2, 3) and (–1, 4). If each unit on the grid is 1 kilometer, what is the distance between the towns? Evaluate each expression. ANSWER 10 km or about 3.16 km

Warm-Up Exercises EXAMPLE 1 Write an equation of a circle Write the equation of the circle shown. The radius is 3 and the center is at the origin. x 2 + y 2 = r 2 x 2 + y 2 = 3 2 x 2 + y 2 = 9 Equation of circle Substitute. Simplify. ANSWER The equation of the circle is x 2 + y 2 = 9

Warm-Up Exercises EXAMPLE 2 Write the standard equation of a circle Write the standard equation of a circle with center (0, –9) and radius 4.2. SOLUTION (x – h) 2 + ( y – k) 2 = r 2 (x – 0) 2 + ( y – (–9)) 2 = x 2 + ( y + 9) 2 = Standard equation of a circle Substitute. Simplify.

Warm-Up Exercises GUIDED PRACTICE for Examples 1 and 2 Write the standard equation of the circle with the given center and radius. 1. Center (0, 0), radius 2.5 x 2 + y 2 = 6.25 ANSWER

Warm-Up Exercises GUIDED PRACTICE for Examples 1 and 2 Write the standard equation of the circle with the given center and radius. 2. Center (–2, 5), radius 7 (x + 2) 2 + ( y – 5) 2 = 49 ANSWER

Warm-Up Exercises EXAMPLE 3 Write the standard equation of a circle The point (–5, 6) is on a circle with center (–1, 3). Write the standard equation of the circle. SOLUTION To write the standard equation, you need to know the values of h, k, and r. To find r, find the distance between the center and the point (–5, 6) on the circle.

Warm-Up Exercises EXAMPLE 3 Write the standard equation of a circle r =[–5 – (–1)] 2 + (6 – 3) 2 = (–4) = 5 Standard equation of a circle Simplify. Substitute (h, k) = (–1, 3) and r = 5 into the standard equation of a circle. (x – h) 2 + (y – k) 2 = r 2 [x – (–1)] 2 + (y – 3) 2 = 5 2 (x +1) 2 + (y – 3) 2 = 25 Substitute. Simplify. The standard equation of the circle is (x +1) 2 + (y – 3) 2 = 25. ANSWER

Warm-Up Exercises GUIDED PRACTICE for Example 3 3. The point (3, 4) is on a circle whose center is (1, 4). Write the standard equation of the circle. The standard equation of the circle is (x – 1) 2 + (y – 4) 2 = 4. ANSWER

Warm-Up Exercises GUIDED PRACTICE for Example 3 4. The point (–1, 2) is on a circle whose center is (2, 6). Write the standard equation of the circle. The standard equation of the circle is (x – 2) 2 + (y – 6) 2 = 25. ANSWER

Warm-Up Exercises EXAMPLE 4 Graph a circle The equation of a circle is (x – 4) 2 + (y + 2) 2 = 36. Graph the circle SOLUTION Rewrite the equation to find the center and radius. (x – 4) 2 + (y +2) 2 = 36 (x – 4) 2 + [y – (–2)] 2 = 6 2 The center is ( 4, –2 ) and the radius is 6. Use a compass to graph the circle.

Warm-Up Exercises EXAMPLE 5 Use graphs of circles EARTHQUAKES The epicenter of an earthquake is the point on Earth’s surface directly above the earthquake’s origin. A seismograph can be used to determine the distance to the epicenter of an earthquake. Seismographs are needed in three different places to locate an earthquake’s epicenter. Use the seismograph readings from locations A, B, and C to find the epicenter of an earthquake.

Warm-Up Exercises EXAMPLE 5 Use graphs of circles The epicenter is 7 miles away from A ( –2, 2.5 ). SOLUTION The set of all points equidistant from a given point is a circle, so the epicenter is located on each of the following circles. The epicenter is 5 miles away from C ( 3, –2.5 ). The epicenter is 4 miles away from B ( 4, 6 ).

Warm-Up Exercises EXAMPLE 5 Use graphs of circles A with center (–2, 2.5) and radius 7 To find the epicenter, graph the circles on a graph where units are measured in miles. Find the point of intersection of all three circles. ANSWER The epicenter is at about ( 5, 2 ). C with center (3, –2.5) and radius 5 B with center (4, 6) and radius 4

Warm-Up Exercises GUIDED PRACTICE for Examples 4, and The equation of a circle is (x – 4) 2 + (y + 3) 2 = 16. Graph the circle. SOLUTION

Warm-Up Exercises GUIDED PRACTICE for Examples 4, and The equation of a circle is (x + 8) 2 + (y + 5) 2 = 121. Graph the circle. SOLUTION

Warm-Up Exercises GUIDED PRACTICE for Examples 4, and Why are three seismographs needed to locate an earthquake’s epicenter? SOLUTION Two circles intersect in two points. You would not know which one is the epicenter, so you need the third circle to know which one it is.

Warm-Up Exercises Daily Homework Quiz 1. Write the standard equation of a circle with center (–3, 5) and radius 2. ANSWER (x + 3) 2 + (y – 5) 2 = 4 2. Graph the circle (x + 1) 2 + (y – 3) 2 = 9. ANSWER

Warm-Up Exercises Daily Homework Quiz 3. The line x = 2 is tangent to the circle in Exercise 2. Find the coordinates of the point of tangency. ANSWER (2, 3)