1 Million+ Step-by-step solutions

1. Treating year as the independent variable and the winning value as the dependent variable, find linear equations relating these variables (separately for men and women) using the data for the years 1992 and 1996. Compare the equations and comment on any similarities or differences.

2. Interpret the slopes in your equations from part 1. Do the y-intercepts have a reasonable interpretation? Why or why not?

3. Use your equations to predict the winning time in the 2004 Olympics. Compare your predictions to the actual results (2.18 hours for men and 2.44 hours for women). How well did your equations do in predicting the winning times?

4. Repeat parts 1 to 3 using the data for the years 1996 and 2000. How do your results compare?

5. Would your equations be useful in predicting the winning marathon times in the 2104 Summer Olympics? Why or why not?

**Internet-based Project**

Predicting Olympic Performance Measurements of human performance over time sometimes follow a strong linear relationship for reasonably short periods. In 2004 the Summer Olympic Games returned to Greece, the home of both the ancient Olympics and the first modern Olympics. The following data represent the winning times (in hours) for men and women in the Olympic marathon.

Solve the equation 2(x + 3) - 1 = -7.

To complete the square of x^{2} + 10x, you would (add /subtract) the number ________.

In problem, find the following for each pair of points:

(a) The distance between the points

(b) The midpoint of the line segment connecting the points

(c) The slope of the line containing the points

(d) Interpret the slope found in part (c)

(0, 0); (4, 2)

Use the Square Root Method to solve the equation (x – 2)^{2} = 9.

True or False

Every equation of the form x^{2} + y^{2} + ax + by + c = 0 has a circle as its graph.

In problem, find the following for each pair of points:

(a) The distance between the points

(b) The midpoint of the line segment connecting the points

(c) The slope of the line containing the points

(d) Interpret the slope found in part (c)

(0, 0); (-4, 6)

Solve the equation x^{2} - 9 = 0.

For the line 2x + 3y, the x-intercept is _______ and the y-intercept is ________.

In problem, use P_{1} = (-1, 3) and P_{2} = (5, -1).

Find the midpoint of the line segment joining P_{1} and P_{2}.

The slope of a vertical line is__________ ; the slope of a horizontal line is _________ .

The points, if any, at which a graph crosses or touches the coordinate axes are called ____________.

A horizontal line is given by an equation of the form _________, where b is the __________.

In problem, use P_{1} = (-1, 3) and P_{2} = (5, -1).

(a) Find the slope of the line containing P_{1} and P_{2}.

(b) Interpret this slope.

In problem, find the following for each pair of points:

(a) The distance between the points

(b) The midpoint of the line segment connecting the points

(c) The slope of the line containing the points

(d) Interpret the slope found in part (c)

(1, -1); (-2, 3)

Graph y = x^{2} – 9 by plotting points.

True or False

Vertical lines have an undefined slope.

The x-intercepts of the graph of an equation are those x-values for which ___________.

For a circle, the ____________ is the distance from the center to any point on the circle.

In problem, find the following for each pair of points:

(a) The distance between the points

(b) The midpoint of the line segment connecting the points

(c) The slope of the line containing the points

(d) Interpret the slope found in part (c)

(-2, 2); (1, 4)

Use the converse of the Pythagorean Theorem to show that a triangle whose sides are of lengths 11, 60, and 61 is a right triangle.

Sketch the graph of y^{2} = x.

True or False

The slope of the line 2y = 3x + 5 is 3.

If for every point (x, y) on the graph of an equation the point (-x, y) is also on the graph, then the graph is symmetric with respect to the __________.

In problem, find the following for each pair of points:

(a) The distance between the points

(b) The midpoint of the line segment connecting the points

(c) The slope of the line containing the points

(d) Interpret the slope found in part (c)

(4, -4); (4, 8)

The area A of a triangle whose base is b and whose altitude is h is A = _______ .

List the intercepts and test for symmetry: x^{2} + y = 9.

True or False

The point (1, 2) is on the line 2x + y = 4.

If the graph of an equation is symmetric with respect to the y-axis and -4 is an x-intercept of this graph, then _________ is also an x-intercept.

True or False

The center of the circle

(x + 3)^{2} + (y - 2)^{ 2} = 13 is (3, -2)

In problem, find the following for each pair of points:

(a) The distance between the points

(b) The midpoint of the line segment connecting the points

(c) The slope of the line containing the points

(d) Interpret the slope found in part (c)

(-3, 4); (2, 4)

Write the slope–intercept form of the line with slope -2 containing the point (3, -4). Graph the line.

If the graph of an equation is symmetric with respect to the origin and (3, -4) is a point on the graph, then is also a point on the graph.

True or False

To find the y-intercepts of the graph of an equation, let x = 0 and solve for y.

In problem, find the center and radius of each circle. Write the standard form of the equation.

Graph y = x^{2} + 4 by plotting points.

If (x, y) are the coordinates of a point P in the xy-plane, then x is called the ___________ of P and y is the ___________ of P.

Write the general form of the circle with center (4, -3) and radius 5.

The lines y = 2x + 3 and y = ax + 5 are parallel if a = ______.

In problem, find the center and radius of each circle. Write the standard form of the equation..

List the intercepts of the graph below.

Find the center and radius of the circle x^{2} + y^{2} + 4x – 2y – 4 = 0. Graph this circle.

The lines y = 2x – 1 and y = ax + 2 are perpendicular if a =.

True or False

The y-coordinate of a point at which the graph crosses or touches the x axis is an x-intercept.

In problem, find the center and radius of each circle. Write the standard form of the equation.

In problem, list the intercepts and test for symmetry with respect to the x-axis, the y-axis, and the origin.

2x = 3y^{2}

If three distinct points P, Q, and R all lie on a line and if d(P, Q) = d(Q, R), then Q is called the ___________ of the line segment from P to R.

For the line 2x + 3y = 6, find a line parallel to it containing the point (1, -1). Also find a line perpendicular to it containing the point (0, 3).

True or False

Perpendicular lines have slopes that are reciprocals of one another.

True or False

If a graph is symmetric with respect to the x-axis, then it cannot be symmetric with respect to the y-axis.

In problem, find the center and radius of each circle. Write the standard form of the equation.

In problem, list the intercepts and test for symmetry with respect to the x-axis, the y-axis, and the origin.

y = 5x

True or False

The distance between two points is sometimes a negative number.

In problem, write the standard form of the equation and the general form of the equation of each circle of radius r and center (h, k). Graph each circle.

r = 1/2; (h, k) = (1/2, 0)

In problem, find the intercepts and graph each equation by plotting points. Be sure to label the intercepts.

y = 3x - 9

In problem, write the standard form of the equation and the general form of the equation of each circle of radius r and center (h, k). Graph each circle.

r = 1/2; (h, k) = (0, 1/2)

In problem, plot each pair of points and determine the slope of the line containing them. Graph the line.

(-1, 2); (-1, -2)

In problem, find the intercepts and graph each equation by plotting points. Be sure to label the intercepts.

y = x^{2} - 1

In problem,

(a) Find the center (h, k) and radius r of each circle;

(b) Graph each circle;

(c) Find the intercepts, if any.

x_{2} + y_{2} = 4

In problem, find the center and radius of each circle. Graph each circle. Find the intercepts, if any, of each circle.

x^{2} + (y - 1)^{2} = 4

In problem, plot each pair of points and determine the slope of the line containing them. Graph the line.

(2, 0); (2, 2)

In problem, find the intercepts and graph each equation by plotting points. Be sure to label the intercepts.

y = x^{2} - 9

In problem,

(a) Find the center (h, k) and radius r of each circle;

(b) Graph each circle;

(c) Find the intercepts, if any.

x^{2} + (y - 1)2^{2 }= 1

In problem, find the center and radius of each circle. Graph each circle. Find the intercepts, if any, of each circle.

(x + 2)^{2} + y^{2} = 9

In problem, graph the line containing the point P and having slope m.

P = (1, 2); m = 3

y = -x^{2} + 4

In problem,

(a) Find the center (h, k) and radius r of each circle;

(b) Graph each circle;

(c) Find the intercepts, if any.

2(x - 3)^{2} + 2y^{2 }= 8

In problem, graph the line containing the point P and having slope m.

P = (2, 1); m = 4

y = -x^{2} + 1

In problem,

(a) Find the center (h, k) and radius r of each circle;

(b) Graph each circle;

(c) Find the intercepts, if any.

3(x + 1)^{2} + 3(y - 1)^{2} = 6

In problem, find the center and radius of each circle. Graph each circle. Find the intercepts, if any, of each circle.

x^{2} + y^{2} + 4x - 4y - 1 = 0

In problem, graph the line containing the point P and having slope m.

P = (2, 4); m = - 3/4

2x + 3y = 6

In problem,

(a) Find the center (h, k) and radius r of each circle;

(b) Graph each circle;

(c) Find the intercepts, if any.

x^{2} + y^{2} - 2x - 4y - 4 = 0

3x^{2} + 3y^{2} - 6x + 12y = 0

In problem, graph the line containing the point P and having slope m.

P = (1, 3); m = - 2/5

5x + 2y = 10

In problem,

(a) Find the center (h, k) and radius r of each circle;

(b) Graph each circle;

(c) Find the intercepts, if any.

x^{2} + y^{2} + 4x + 2y - 20 = 0

2x^{2} + 2y^{2} - 4x = 0

In problem, graph the line containing the point P and having slope m.

P = (-1, 3); m = 0

9x^{2} + 4y = 36

In problem,

(a) Find the center (h, k) and radius r of each circle;

(b) Graph each circle;

(c) Find the intercepts, if any.

x^{2} + y^{2} + 4x - 4y - 1 = 0

In problem, graph the line containing the point P and having slope m.

P = (2, -4); m = 0

4x^{2} + y = 4

In problem,

(a) Find the center (h, k) and radius r of each circle;

(b) Graph each circle;

(c) Find the intercepts, if any.

x^{2} + y^{2} - 6x + 2y + 9 = 0

In problem, find an equation of the line having the given characteristics. Express your answer using either the general form or the slope–intercept form of the equation of a line, whichever you prefer.

Slope = 0; containing the point (-5, 4)

In problem, plot each point and form the triangle ABC. Verify that the triangle is a right triangle. Find its area.

A = (-2, 5); B = (1, 3); C = (-1, 0)

In problem, graph the line containing the point P and having slope m.

P = (0, 3); slope undefined

In problem,

(a) Find the center (h, k) and radius r of each circle;

(b) Graph each circle;

(c) Find the intercepts, if any.

x^{2} + y^{2} - x + 2y + 1 = 0

In problem, find an equation of the line having the given characteristics. Express your answer using either the general form or the slope–intercept form of the equation of a line, whichever you prefer.

Vertical; containing the point (-3, 4)

In problem, plot each point and form the triangle ABC. Verify that the triangle is a right triangle. Find its area.

A = (-2, 5); B = (12, 3); C = (10, -11)

In problem, graph the line containing the point P and having slope m.

P = (-2, 0); slope undefined

In problem,

(a) Find the center (h, k) and radius r of each circle;

(b) Graph each circle;

(c) Find the intercepts, if any.

x^{2} + y^{2} + x + y - 1/2 = 0

In problem, find an equation of the line having the given characteristics. Express your answer using either the general form or the slope–intercept form of the equation of a line, whichever you prefer.

x-intercept = 2; containing the point (4, -5)

In problem, the slope and a point on a line are given. Use this information to locate three additional points on the line.

Slope 4; point (1, 2)

In problem,

(a) Find the center (h, k) and radius r of each circle;

(b) Graph each circle;

(c) Find the intercepts, if any.

2x^{2} + 2y^{2} - 12x + 8y - 24 = 0

y-intercept = -2; containing the point (5, -3)

In problem, plot each point and form the triangle ABC. Verify that the triangle is a right triangle. Find its area.

A = (-6, 3); B = (3, -5); C = (-1, 5)

In problem, the slope and a point on a line are given. Use this information to locate three additional points on the line.

Slope 2; point (-2, 3)

Containing the points (3, -4) and (2, 1)

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