warm-up exercises answer 17 answer 13 evaluate each expression. 1. (10 – 2) 2 + (8 + 7) 2 2. (–3...
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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 atcoordinates (2, 3) and (–1, 4). If each unit on the gridis 1 kilometer, what is the distance betweenthe towns?
Evaluate each expression.
ANSWER 10 km or about 3.16 km
Warm-Up ExercisesEXAMPLE 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.
x2 + y2 = r2
x2 + y2 = 32
x2 + y2 = 9
Equation of circle
Substitute.
Simplify.
ANSWER
The equation of the circle is x2 + y2 = 9
Warm-Up ExercisesEXAMPLE 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 = r2
(x – 0)2 + ( y – (–9))2 = 4.22
x2 + ( y + 9)2 = 17.64
Standard equation of a circle
Substitute.
Simplify.
Warm-Up ExercisesGUIDED 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
x2 + y2 = 6.25
ANSWER
Warm-Up ExercisesGUIDED 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 ExercisesEXAMPLE 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 ExercisesEXAMPLE 3 Write the standard equation of a circle
r = [–5 – (–1)]2 + (6 – 3)2
= (–4)2 + 32
= 5
Standard equation of a circle
Simplify.
Simplify.
Substitute (h, k) = (–1, 3) and r = 5 into the standard equation of a circle.
(x – h)2 + (y – k)2 = r2
[x – (–1)]2 + (y – 3)2 = 52
(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 ExercisesGUIDED 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 ExercisesGUIDED 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 ExercisesEXAMPLE 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 = 62
The center is (4, –2) and the radius is 6. Use a compass to graph the circle.
Warm-Up ExercisesEXAMPLE 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 ExercisesEXAMPLE 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 ExercisesEXAMPLE 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 ExercisesGUIDED PRACTICE for Examples 4, and 5.
5. The equation of a circle is (x – 4)2 + (y + 3)2 = 16. Graph the circle.
SOLUTION
Warm-Up ExercisesGUIDED PRACTICE for Examples 4, and 5.
6. The equation of a circle is (x + 8)2 + (y + 5)2 = 121. Graph the circle.
SOLUTION
Warm-Up ExercisesGUIDED PRACTICE for Examples 4, and 5.
7. 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 ExercisesDaily Homework Quiz
1. Write the standard equation of a circle withcenter (–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 ExercisesDaily 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)