Section 1.1 The Distance and Midpoint Formulas 1
y axis origin x axis 2
Plot the points: ( 3, 5), (0,7), ( 6,0), (6,4) 3
Distance Formula y x 4
Finding the Distance Between Two Points Find the distance d between the points (2, 4) and ( 1, 3). 5
Midpoint Formula y (x 2, y 2 ) y M (x 1, y 1 ) x x 6
Find the midpoint of the line segment from P 1 = (4, 2) to P 2 = (2, 5). Plot the points and their midpoint. 7
Find the endpoint of the line segment P 2 given P 1 = ( 4, 5) and M = ( 6, 1). Plot the points and their midpoint. 8
Section 1.2 Graphs of Equations In Two Variables; Intercepts; Symmetry 9
Examples of Equations in Two Variables What rate of return can you expect with a 17% level of risk? What percentage of foreign vs. US investments minimizes risk? What percentage of foreign vs. US investments maximizes return? What is y when x = 4? What is x when y = 3? 10
Graph of An Equation The graph of an equation is a drawing that represents all its solutions. Solve: 2x + 3y = 6 11
Determining Whether a Point is on the Graph of an Equation Determine if the following points are on the graph of the equation 3x +y = 6 (a) (0, 4) (b) ( 2, 0) (c) ( 1, 3) 12
Graphing an Equation by Plotting Points Graph the equation: y = 2x + 5 13
Graphing an Equation by Plotting Points Graph the equation: y = x 2 14
Intercepts from a Graph 15
Find the intercepts of the graph. Finding Intercepts from a Graph 5 4 3 2 1 0 1 2 3 4 5 16
Find Intercepts from an Equation Finding Intercepts To find the x intercepts of the the graph of an equation: To find the y intercepts of the graph of an equation: 17
Finding Intercepts from an Equation Find the x intercept(s) and the y intercept(s) of the graph of then graph by plotting points. y = x 4 18
Finding Intercepts from an Equation Find the x intercept(s) and the y intercept(s) of the graph of then graph by plotting points. y = x 2 + x 6 19
Symmetry and Graphs A graph is said to be symmetric with respect to the x axis if for every point (x, y) on the graph, the point (, ) is also on the graph. 20
Points Symmetric with Respect to the x Axis If a graph is symmetric with respect to the x axis and the point (3,2) is on the graph, what other point is also on the graph? 21
Symmetry and Graphs A graph is said to be symmetric with respect to the y axis if for every point (x, y) on the graph, the point (, ) is also on the graph. 22
Points Symmetric with Respect to the y Axis If a graph is symmetric with respect to the y axis and the point (3,2) is on the graph, what other point is also on the graph? 23
Symmetry and Graphs A graph is said to be symmetric with respect to the origin if for every point (x, y) on the graph, the point (, ) is also on the graph. 24
Points Symmetric with Respect to the Origin If a graph is symmetric with respect to the origin and the point (3,2) is on the graph, what other point is also on the graph? 25
Tests for Symmetry To test the graph of an equation for symmetry with respect to the: x axis: y axis: Origin: 26
Testing an Equation for Symmetry 27
Know How to Graph Key Equations y = x y = x 2 y = x 3 y = 1 x y = x x = y 2 28
Graphing the Equation: y = x 29
Graphing the Equation: y = x 2 30
Graphing the Equation: y = x 3 31
Graphing the Equation: y = x 1 32
Graphing the Equation: y = x 33
Graphing the Equation: x = y 2 34
YOUR TURN!! Testing an Equation for Symmetry Find the intercepts and test for symmetry. 4x 2 + y 2 = 4 35
Section 1.3 Lines 36
Linear Equations and Slope y = mx + b ax + by = c Slope of a Line (m): 1. Slope is defined by the ratio: m = 2. The slope tells how "steep" a line is and what "direction" it points 3. The slope tells how to get from one point on a line to another point on the line 37
The Sign of Slope 38
Deriving the Slope Formula m = m = m = 39
Compute the slopes of the lines containing the following pairs of points. A = (2, 3) B = ( 1, 2) C = ( 5, 2) D = (3, 2) E = ( 3, 4) F = ( 7, 6) G = (7, 5) H = (7, 2) 40
(c) 2 41
Graphs of y = mx + 2 Graphs of y = 2x + b 42
Find the slope m and y intercept b of the equation 3x 2y = 6. Graph the equation. 43
Graph the linear equation 3x + 2y = 6 by finding its intercepts. 44
Graph the equation: y = 2 45
Graph the equation: x = 2 46
Finding an Equation for a Line To find the equation of a line you need: 1. 2. 47
Find the equation of a line with slope 3 and containing the point (0, 5). 48
Find the equation of a line with slope 3 and containing the point ( 1, 4). 49
Find the equation of a horizontal line containing the point (2, 4). 50
Find an equation of the line containing the points ( 1, 4) and (3, 1). 51
Parallel Lines 52
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Find an equation for the line that contains the point ( 1, 3) and is parallel to the line 3x 4y = 12. 54
Perpendicular Lines 55
Find the slope of a line perpendicular to a line with slope. 56
Find an equation for the line that contains the point ( 1, 3) and is perpendicular to the line 3x 4y = 12. 57
YOUR TURN!! Finding an Equation for a Line Find the equation for the line passing through the points ( 3, 4) and (2, 5). 58
YOUR TURN!! Finding an Equation for a Line Find the equation for the line perpendicular to the line 2x + 3y = 6 passing through the point (2, 5). 59
Section 1.4 Circles 60
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Write the standard form of the equation of the circle with radius 12 and center at the origin. 65
Write the standard form of the equation of the circle with radius 4 and center (2, 4). 66
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Graph the equation: x 2 + y 2 = 16 68
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Finding an Equations for Circle and Tangent Line Using Distance and Midpoint. (1, 6) A C B (5, 4) 74
YOUR TURN!! Finding an Equation for Circle Find the equation for the circle with radius 7 and center ( 3, 5). 75
YOUR TURN!! More Circle Stuff Find the center of the circle, any intercepts, and graph. x 2 + y 2 6x + 2y + 9 = 0 76
YOUR TURN!! Yet More Circle Stuff Find the equation for the tangent line to the circle x 2 + y 2 = 25 at the point (3, 4) 77
Proof: "If two nonvertical lines are perpendicular, then the product of their slopes is 1." y Assume lines are perpendicular 1 x 78
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