4 The Cartesian Coordinate System Pictures of Equations


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1 The Cartesian Coordinate Sstem Pictures of Equations Concepts: The Cartesian Coordinate Sstem Graphs of Equations in Two Variables intercepts and intercepts Distance in Two Dimensions and the Pthagorean Theorem Equations of Circles The Distance Formula and the Standard Form for an Equation of a Circle. Writing Equations of Circles Identifing Equations of Circles Midpoints Finding Midpoints Verifing that a Point Is the Midpoint of a Line Segment Steepness Rates of Change Lines The Slope The Slope as a Rate of Change Linear Equations PointSlope Form Vertical Lines Horizontal Lines Parallel Lines and Perpendicular Lines (Sections..) In this section, we leap from the onedimensional number line to the twodimensional Cartesian Coordinate Sstem. This leap will allow us to see pictures of equations that contain two variables ( and, for eample). We will also see how these pictures can be used to help us approimate solutions of equations that contain onl one variable.
2 . Your Personal Review Undoubtedl, ou have seen the Cartesian Coordinate Sstem in previous classes. We will not review the basics of plotting points during lecture, but ou should read section. in our tetbook for a quick review. In particular, ou need to be able to: Locate points on a Cartesian Coordinate Sstem. Discuss the four quadrants of a Cartesian Coordinate Sstem. Identif the ais, the ais, and the origin on a Cartesian Coordinate Sstem. Eample. (A Review of the Cartesian Coordinate Sstem) Ever point on the ais has: A. an coordinate that equals. B. a coordinate that equals. C. Both (A) and (B).. Graphs of Equations with Two Variables Definition. The graph of an equation in the variables and is the set of all points (a,b) such that = a, = b is a solution to the equation. If a point is on the graph of an equation, then the point is a solution of the equation. If a point is a solution of an equation, then the point is on the graph of the equation. If a point is not on the graph of an equation, the the point is not a solution of the equation. If a point is not a solution of an equation, then the point is not on the graph of the equation. Eample. Is (,) on the graph of = +?? = ()+? = 6+ Since (,) is not a solution to the equation = +, then it is not on its graph.
3 Is (,8) on the graph of = +? 8 8? = ()+? = + 8 = 8 Since (,8) is a solution to the equation = +, then it is on its graph. Eample. Eplain wh the graph below is not the graph of = Notice that the point (,) is a solution to = ++. However, (,) isnot a point on the given graph. Therefore this is not the graph of = ++. Eample. Sketch the graph of = b making a table of values Definition.6 (intercepts and intercepts) If a graph intersects the ais at the point (a,), then a is called an intercept of the graph. If a graph intersects the ais at the point (,b), then b is called a intercept of the graph. Notice that the value equals zero at an intercept and the value equals zero at a  intercept.
4 Eample.7 Find the intercepts of the graph of = + intercepts occur when =, = +() = The intercept is: (, ). The intercepts occur when =, = + = ( +)( ) + = or = = = The intercepts are: (, ),(, ). Distance When we were working with the onedimensional number line, we saw the relationship between distance and the absolute value. In the twodimensional Cartesian Coordinate Sstem, we will use the Pthagorean Theorem to discuss distance. We will also see the relationship between the distance formula and the equations for circles. Eample.8 Use the Pthagorean Theorem to find the distance between the points (,) and (, ). 6 distance B Pthagorean s Theorem, distance = + = +8 = 9+6 = 7 distance = Eample.9 Use the Pthagorean Theorem to find the distance between the points (6, ) and (b,). Note that b could be an number. Since is the distance between b and 6, then = b 6. Note that 6 = ( 6). Then, b Pthagorean s Theorem, distance distance = + = (b 6) +7 = (b 6) +9 distance = (b 6) +9
5 Theorem. (The Distance Formula) The distance between the points (, ) and (, ) is ( ) +( ) The Distance Formula is just the Pthagorean Theorem in disguise. Eample. Find the perimeter of the triangle with vertices A(, ), B(,7), and C(,). perimeter = (Distance from A to B) + (Dist. from B to C) + (Dist. from A to C) = ( ( )) +(7 ( )) + ( ( )) +( (7)) + ( ( )) +( ( )) = Equations of Circles Recall the definition of a circle. Definition. A circle is the set of all points that are a fied distance r from a specified point called the center of the circle. The distance r is called the radius of the circle. radius center Since the definition of a circle is based on a distance, we can use the distance formula to find the equation of a circle. Eample. Find an equation for the circle with center (,) and radius. Let (,) b an point on the circle. Then, the distance from (,) to (,) is. B the distance formula, we have ( ( )) +( ) = (+) +( ) = 6
6 Theorem. (Circle EquationsStandard Form) An equation for a circle with center (h,k) and radius r is equivalent to ( h) +( k) = r. Eample. Find an equation for the circle with center (, ) that passes through the point (, 6). The radius of the circle is The equation for the desired circle is r = ( ) +( ( 6)) = + = 9 ( ) +( +) = 9 Eample.6 (Do ou understand circle equations?) What is the center of the circle whose equation in standard form is ( ) + =? (a) (,) (b) (,) (c) (,) (d) (,) 6 8 (e) The graph of this equation is not a circle. Sketch the circle defined b ( ) + =. Eample.7 Is the graph of = a circle? If so, find its center and radius. B completing the square, we can tr to rewrite this equation into the standard form for a circle = ( ++ ) + ( 6 +9 ) 9+ = (+) +( ) = (+) +( ) = This is the standard form for a circle centered at (,) with radius Eample.8 Is the graph of ++ + = a circle? If so, find its center and radius. Completing the square ields, = ( ++ ) + ( + ) + = (+) ( ) + = (+) ( ) = Since the right hand side is negative, it can t possibl be r. This is not the equation for a circle. 6
7 .. Midpoints Definition.9 (Midpoint) The midpoint of the line segment AB is the point on the line segment that is equidistant from A and B. Eample. Suppose that the distance from A to B equals the distance from B to C. Is it necessaril true that B is the midpoint of AC? Wh or wh not? Not necessaril. Think about the vertices of an isosceles triangle. What etra piece of information do ou need? The distance from A to C equals the distance from A to B plus the distance from B to C. A B Eample. Consider the points A(,), B(,), and C(,). Is B the midpoint of AC? C B No, B is not the midpoint of AC. C A Theorem. The midpoint of the line segment from (, ) to (, ) is ( +, ) + Proof. This proof is left as an eercise. Eample. (Circles and Diameters) A diameter of a circle has endpoints (,) and (, ). Find an equation for the circle. To write down an equation for a circle in standard form, ou need the center and radius of the circle. Note that the center is midpoint of the endpoints of a diameter of the circle and the radius is the distance from the center to an point on the circle. ( + center:, +( ) ) = (,) radius: r = ( ) +( ) = ( ) +( ) = 7
8 . Steepness, Lines, and Rates of Change The distance formula allows us to measure the length of a line segment in two dimensions. Distance or length is just one of the useful quantities we can measure in two dimensions. The steepness of a line segment is another useful measurement in two dimensions. How fast is a line segment rising or falling? Eample. Consider the curve shown below. Discuss the steepness of the curve The curve gets less steep then starts going down then gets ver steep. Eample. Consider the curve shown below. Discuss the steepness of the curve It s hard to talk about the steepness of this curve as it changes graduall. (The steepness of curves such as this is an important topic in calculus.) Eample.6 Suppose ou want to draw a curve for which the steepness does not var. What tpe of curve could ou draw? a line 8
9 We have been discussing steepness in ver vague qualitative terms. If we want to be able to make more concrete comparisons, we need to be able to quantif, or measure, the steepness of a curve. You will need Calculus in order to measure the steepness of general curves like the curve shown in Eample.. Toda we will quantif the steepness of lines because the are the simplest curves. Moreover, this measurement will la a foundation for what ou will do in Calculus... Quantifing the Steepness of a Line  Slope The steepness of a line is determined b 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 line, this ratio is called the slope of the line. Definition.7 If, then the slope of the line through the points (, ) and (, ) is Eample.8 Find the slope of the line shown below. = Notice that (,) and (, ) are points on the line. So the slope of the line is, m = ( ) = 6 8 = Eample.9 Find the slope of the line that passes through (6,) and (6,). m = 6 6 = undefined Note that = for ever vertical line. Therefore, ever vertical line has a slope that is undefined. Eample. Find the slope of the line that passes through (,) and (,). m = ( ) = 6 = Note that = for ever horizontal line. Therefore, ever horizontal line has a slope of. 9
10 .. Rates of Change The slope of a line is a rate of change. If the line is in terms of the variables and, then the slope of the line is the rate of change of with respect to. This means that the slope is the ratio of the change in to the change in. The word per is often associated with rates of change. Think about speed. Speed can be measured in miles per hour. Speed is a rate of change of distance with respect to time. Eample. (Rate of Change) A particle is traveling along a straight line. Its position, s, at time t seconds is given b s = 6t where s is measured in feet. t = t = t = t =. t =. The twodimensional s graph of s = 6t is shown below. This graph is linear. t =.8 s t. What is the slope of this line? (,) and (,6) are points on the line so m = 6 = 6. What are the units of the slope? feet per second. Epress the slope of the line as a rate of change. 6 feet per second. What does the slope of this line tell us about the particle? The particle s position changes 6 feet ever second.
11 .. Equations of Lines As we said before, the slope of a nonvertical line is constant. A line can be totall described bitsslopeandapoint ontheline. Whdo weneed apoint? Whistheslopenot sufficient? You must place the line somewhere. Just as we used the distance formula to construct an equation for a circle, we will now use the slope formula to construct an equation for a nonvertical line. Eample. Find an equation for the line that passes through (,) and has slope . Let (,) be an point on the line other than (,). B the slope formula, = Ever point on the line satisfies this equation ecept (,). So, we can manipulate it to get = ( ) Eample. Find an equation for the line that passes through (a,b) and has slope . As before, we can use the slope formula to obtain, or b a = b = ( a) Theorem. (Equations of LinesPoint Slope Formula) The graph of b = m( a) is a line that passes through (a,b) and has slope m. Theorem. (Equations of LinesHorizontal and Vertical Lines) The graph of = b is a horizontal line that passes through (a,b). The slope of a horizontal line is. The graph of = a is a vertical line that passes through (a,b). The slope of a vertical line is undefined. Definition.6 (Linear Equations) A linear equation in and is an equation that is equivalent to an equation of the form where A, B, and C are constants. A+B +C =
12 Eample.7 Verif that b = m( a) is a linear equation. What are the values of A, B, and C in the definition of a linear equation. }{{} m A + }{{} B +ma b }{{} C b = m( a) b = m ma = Thus, this is a linear equation with A = m, B =, and C = ma b. Eample.8 (Concept Check) Which of the following are linear equations? A. + = B. = ( ) C. = + E. = F. ( ) +( +) = D. =.. Miscellaneous Information About Lines The following facts about lines are IMPORTANT: Slopeintercept form ( = m + b) is NOT the onl wa to write an equation for a line. Sometimes it is not even the most useful wa to write an equation for a line. Pointslope form is often more useful than slopeintercept form. Learn it! Slopeintercept form is useful if ou need to determine the slope of a line or if ou ou are checking to see if two lines are the same. Parallel lines have the same slope. The product of the slopes of perpendicular lines is. The slope of vertical lines ( = a) is undefined. The slope of horizontal lines ( = a) is zero. To find the equation of a line, ou need the slope and a point.
13 Eample.9 Find an equation for the line that passes through ( 9,) and is parallel to the line whose equation is = 7 (+). Line parallel: = 7 (+) Line I want: goes through: ( 9, ) slope = 7 slope = 7 = 7 (+9) Eample. Find an equation for the line that is perpendicular to + = and passes through the point (, ). Line perpendicular: + = = + slope = Line I want: goes through: (, ) slope = + = ( ) Eample. Find the slope of the line given b the equation + =? What is its intercept? Graph the line. Rewrite the equation in slope intercept form: + = = + = = + The slope of the equation is. The intercept occurs when = : = ()+ = The intercept is (, ). Note that = ()+ = + = So, (, ) is another point on the line.
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