Math 1310 Section 1.1 Points, Regions, Distance and Midpoints


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1 Math 1310 Section 1.1 Points, Regions, Distance and Midpoints In this section, we ll review plotting points in the coordinate plane, then graph vertical lines, horizontal lines and some inequalities. We ll also review the Pythagorean Theorem, develop a formula for finding the distance between two points in the coordinate plane and one for finding the midpoint of a line segment with given endpoints. Here s the coordinate plane: Graphing Points It consists of two number lines that are perpendicular to one another. The horizontal number line is called the x axis and the vertical number line is called the y axis. The point where the two number lines intersect is called the origin. The number lines divide the plane into four regions, called quadrants. Math 1310 Class Notes Section 1.1, Page 1 of 1
2 The pair, ( x, y), is called an ordered pair. It corresponds to a single unique point in the coordinate plane. The first number is called the x coordinate, and the second number is called the y coordinate. The ordered pair (0, 0) names to the origin. The x coordinate tells us the horizontal distance a point is from the origin. The y coordinate tells us the vertical distance a point is from the origin. You ll move right or up for positive coordinates and left or down for negative coordinates. So, the point (3, 6) is three units to the right of the origin, and then 6 units up. The point (, 8) is two units to the left of the origin, and then 8 units down. The point (5, 4) is five units to the right of the origin, and then 4 units down. Example 1: State the ordered pairs associated with each point labeled on the graph: Math 1310 Class Notes Section 1.1, Page of 1
3 Example : Plot the following points in the coordinate plane: A (3, 5) B (6, 0) C (7, 1) D (0, 8) E (5, 3) F (8, ) Math 1310 Class Notes Section 1.1, Page 3 of 1
4 Graphing Regions in the Coordinate Plane The set of all points in the coordinate plane with y coordinate k is the horizontal line y = k. The set of all points in the coordinate plane with x coordinate k is the vertical line x = k. Example 3: Graph y = 3 in the coordinate plane. Example 4: Graph x = 7 in the coordinate plane. Math 1310 Class Notes Section 1.1, Page 4 of 1
5 The region y < k is the part of the coordinate plane below the line y = k, not including the line itself. The region y k is the part of the coordinate plane below the line y = k, including the line itself. The region y > k is the part of the coordinate plane above the line y = k, not including the line itself. The region y k is the part of the coordinate plane above the line y = k, including the line itself. The region x < k is the part of the coordinate plane to the left of the line x = k, not including the line itself. The region x k is the part of the coordinate plane to the left of the line x = k, including the line itself. The region x > k is the part of the coordinate plane to the right of the line x = k, not including the line itself. The region x k is the part of the coordinate plane to the right of the line x = k, including the line itself. For y > k or y < k, the line y = k is drawn as a dotted line. For x < k or x > k, the line x = k is drawn as a dotted line. For y k or y k, the line y = k is drawn as a solid line. For x k or x k, the line x = k is drawn as a solid line. In the exercises, you will see these written using set notation, e.g., ( x y) Example 4: Graph in the coordinate plane: y. { y k},. Math 1310 Class Notes Section 1.1, Page 5 of 1
6 Example 5: Graph in the coordinate plane: x < 3. Example 6: Graph in the coordinate plane: < x 5 Math 1310 Class Notes Section 1.1, Page 6 of 1
7 Example 7: Graph ( y) {, 3 y 8 } x. Example 8: Graph ( y) {, x < 4 and y 1 } x. Math 1310 Class Notes Section 1.1, Page 7 of 1
8 The Pythagorean Theorem In a previous course, you probably learned to work with the Pythagorean Theorem. This theorem states that, in a right triangle, with legs measuring a and b and with hypoteneuse measuring c, c = a + b. Given any two of these values, you can solve the equation to find the third. Example 9: In right triangle ABC, with right angle C, a = 0 and b = 15. Find c. Example 10: Find the length of the missing side of the right triangle Example 11: Find the length of the missing side of the right triangle Math 1310 Class Notes Section 1.1, Page 8 of 1
9 The Distance Formula We can use the Pythagorean Theorem to develop a formula for finding the distance between any two points in the coordinate plane. Suppose you have two ordered pairs, ( x, y ) and ( x y ). We can find a point C, so that we can draw a right triangle in the coordinate plane. Then we can find the lengths of the legs of the right triangle and use the Pythagorean Theorem to find the length of the hypoteneuse. This will give us the distance between the two points: 1 1, This gives us a formula that we can use to find the distance between any two points in the coordinate plane. Suppose (, y ) and ( x y ) x are points in the coordinate plane. Then the distance between the 1 1, two points is ( ) ( ) d = x. We call this the Distance Formula. x1 + y y1 Math 1310 Class Notes Section 1.1, Page 9 of 1
10 Example 1: Use the distance formula to find the distance between the points (3, ) and (4, 1). Example 13: Use the distance formula to find the distance between the points , and,. 6 4 Example 14: Use the distance formula to find the distance between the points, 3 and, 3 3. ( ) ( ) Math 1310 Class Notes Section 1.1, Page 10 of 1
11 The Midpoint Formula x 1 1, and ask for the coordinates of the point that is halfway between the two given points. We call this point the midpoint. Suppose we look at two points in the coordinate plane (, y ) and ( x y ) So the midpoint of a segment with two given endpoints, ( x ) and ( y ) x + x y1 + M =, 1 y 1, y 1. This is the Midpoint Formula. x is Example 15: Find the midpoint of the segment with endpoints at (4, 7) and (3, 1)., Math 1310 Class Notes Section 1.1, Page 11 of 1
12 Example 16: Find the midpoint of the segment with endpoints at , and,. 6 4 Example 17: Find the midpoint of the segment with endpoints at (.3, 0.7) and (5.8, .9). Example 18: The midpoint of a line segment is (3, 8). One endpoint is (4, ). Find the other endpoint. Math 1310 Class Notes Section 1.1, Page 1 of 1
13 Math 1310 Section 1. Lines Topics in this section include finding the slope of a line, writing an equation of the line, graphing a line in the coordinate plane, finding x and y intercepts, and writing an equation of a line that is parallel or perpendicular to a given line through a stated point. Slope The slope of a line represents the rate of change of the line. A steep line that rises from left to right will have a large positive slope, while a line that falls gradually from left to right will have a small negative slope. Example 1: Determine whether each of these lines has a positive slope, negative slope, slope 0 or undefined slope. Math 1310 Class Notes Section 1., Page 1 of 8
14 You can find the slope of a line that contains two given points ( x, y ) and ( x y ) y y1 using the slope formula: m =. x x 1 1 1, Example : Find the slope of the line containing the points (4, 3) and (, 1). by Example 3: Find the slope of the line containing the points 1 5, and, 3 4. Example 4: Find the slope of the line containing the points (7, 4) and (3, 4). Example 5: Find the slope of the line containing the points (, 5) and (, 7). Math 1310 Class Notes Section 1., Page of 8
15 You can also find the slope of a line from its graph. Example 6: Find the slope of the line: Example 8: Find the slope of the line: Math 1310 Class Notes Section 1., Page 3 of 8
16 Writing Equations of Lines One of the main objectives of this section is to write an equation of a line. You ll need to know the slope of the line and a point that lies on the line in order to write an equation of the line. You can be given this information in a number of ways. You can write an equation of the line using either the point slope form of the equation of the line, y y1 = m( x x1), or the slope intercept form of the equation of the line, y = mx + b. In both forms, m represents the slope. In the point slope form, ( x 1, y1) is a point on the line. In the slope intercept form, b represents the y intercept, the point where the graph of the line crosses the y axis.. Example 9: Write an equation of the line: Example 10: Write an equation of the line: Math 1310 Class Notes Section 1., Page 4 of 8
17 Example 11: Write an equation of the line that has slope 3 and y intercept 7. Example 1: Write an equation of the line that has slope 5 1 and passes through the point (10, 4). Example 13: Write an equation of the line that passes through the points (3, 8) and (4, 1). Example 14: Write an equation of the line that passes through the points (4, 0) and (4, 7). Math 1310 Class Notes Section 1., Page 5 of 8
18 Sometimes you will be given the x intercept and the y intercept. The x intercept is the point where the graph of the line crosses the x axis. Example 15: Write an equation of the line with x intercept 6 and y intercept 1. The form Ax + By = C is called the standard form of the equation of the line. You will need to be able to rewrite an equation given in standard form in slope intercept form. Example 16: Write the equation in slope intercept form. Then identify the slope and the y intercept: 4 x + y = 3 Graphing Lines You ll need to be able to graph lines in the coordinate plane. You can use the slope and the y intercept, or any two points, such as the x and y intercepts. Math 1310 Class Notes Section 1., Page 6 of 8
19 Example 17: Graph the line y = x + 1 using the slope and the y intercept. Example 18: Graph the line x 5y = 10 using the intercepts. Math 1310 Class Notes Section 1., Page 7 of 8
20 Example 19: Graph the line x + 3y = 1. Math 1310 Class Notes Section 1., Page 8 of 8
The slope m of the line passes through the points (x 1,y 1 ) and (x 2,y 2 ) e) (1, 3) and (4, 6) = 1 2. f) (3, 6) and (1, 6) m= 6 6
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