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# ALGEBRA & TRIG FOR CALCULUS MA 110

UK

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This 7 page Class Notes was uploaded by Kennith Herman on Friday October 23, 2015. The Class Notes belongs to MA 110 at University of Kentucky taught by Jakayla Robbins in Fall. Since its upload, it has received 10 views. For similar materials see /class/228143/ma-110-university-of-kentucky in Mathematics (M) at University of Kentucky.

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Date Created: 10/23/15

NAME GRAPHING BY HAND MA 110 I ll Cartesian Coordinate System A Your personal review Axes7 Quadrants7 Points 0 Locate points on a Cartesian Coordinate System 0 Discuss the four quadrants of a Cartesian Coordinate System 0 Identify the z 7 axis7 the y 7 ans7 and the origin on a Cartesian Coordinate System B An Example Example 1 Circle the correct response Every point one the z accls has 1 an x coordlnate that equals 0 2 a y coordlnate that equals 0 3 Both 1 and 2 Distance Example 2 Use the Pythagorean Theorem to nd the distance between the points 2 5 and 73 72 Use the Pythagorean Theorem to nd the distance between the points zhyl and 27112 ll gt lt Theorem 1 The Distance Formula The distance between the points zhyl and 27112 is 1 2 91 132 The Distance Formula is just the Pythagorean Theorem in disguise Midpoints ls the distance from 2 0 to 05 equal to the distance from 72 0 to 05 ls 05 the midpoint of the line segment from 2 0 to 720 To prove that e f is the midpoint of the line segment between ab and cd we need to show A The distance between a b and e f equals the distance between C d and e f AND B e f is on the line segment between ab and cd We can show the e f is on the line segment between a b and e d by showing that the distance between ab and e f plus the distance between ed and e f equals the distance between ab and cd Think about triangles Theorem 2 The midpoint of the line segment from zhyl t0 zgy2 is 12 y1y2 2 i 2 PROOF lV Graphs The graph of an equation is the set of all solutions of the equation ls 23 on the graph ofy 3 5 ls 18 on the graph ofy 3x 5 V Circles Example 3 Find an equation for the circle with center 725 and radius 4 Example 4 Find an equation for the circle with eenterz1y1 and radius r Example 5 Find an equation for the circle with center 2 71 that passes through the point 4 76 Example 6 A diameter of a circle has endpoints 2 5 arid 4 71 Firid ari equatiori for the circle Example 7 Is the graph of 2 10x y2 7 6y 32 a circle If so rid its eeriter arid radius Example 8 Is the graph of 2 10x y2 7 6y 34 a circle If so rid its eeriter arid radius Example 9 Is the graph of 2 10x y2 7 6y 40 a circle If so rid its eeriter arid radius Vl Lines Consider the mountain below ls it equally steep everywhere What kind of mountain would be equally steep everywhere The steepness of a line is determined by comparing the change in the vertical distance to the change in the horizontal distance The ratio of the change in vertical distance to the change in horizontal distance is constant for a line For a line7 this ratio is called the slope of the line De nition 1 If 1 31 2 then the slope of the line through the points phyl and 27112 is Au 7 22 7 in Ax 2 7 1 The slope of a line is a rate of change If the line is in terms of the variables x and y7 then the slope of the line is the rate of Change of y with respect to x This means that the slope is the ratio of the Change in y to the Change in x The word per77 is often associated with rates of Change Think about speed Speed can be measured in miles per39 hour Speed is a rate of Change of distance with respect to time Example 10 Rate of Change A particle is traveling along a straight line Its po sition s at time t seconds is given by s 60t where s is measured in feet A Is this a linear equation B What is the slope of this equation 0 What are the units on the slope of the equation D Express the slope of the equation as a rate of change Example 11 Rate of Change If the radius r of a circle is measured in meters then the circumference C of the circle in meters is given by C 27H A Is this a linear equation B What is the slope of this equation 0 What are the units on the slope of the equation D Express the slope of the equation as a rate of change Example 12 Find an equation for the line that passes through 43 and has slope Important Things About Lines 0 Slope intercept form is NOT the only way to write the equation of a line Some times it is not even the most useful way to write the equation of a line 0 Point slope form is often more useful than slope intercept forrn Learn it 0 Parallel lines have the same slope o The product of the slopes of perpendicular lines is 71 o The slope of vertical lines x a is unde ned o The slope of horizontal lines y a is zero To nd the equation of a line7 you need the slope and a point Example 13 Find an equation for the line that is perpendicular to 2n By 7 5 0 and passes through the point 2 71

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