Algebra II. Midpoint and Distance Formula. Slide 1 / 181 Slide 2 / 181. Slide 3 / 181. Slide 4 / 181. Slide 6 / 181. Slide 5 / 181.

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1 Slide 1 / 181 Slide 2 / 181 lgebra II onic Sections Slide 3 / 181 Slide 4 / 181 Table of ontents click on the topic to go to that section Review of Midpoint and istance Formulas Introduction to onic Sections Parabolas ircles Midpoint and istance Formula Ellipses Hyperbolas Recognizing onic Sections from General Form Return to Table of ontents Slide 5 / 181 What is the midpoint of segment? Slide 6 / 181 The Midpoint Formula Given points (x1,y1) and (x2,y2), the midpoint of is (-3, 6) (x,y) (5, 2) Look at this segment - the midpoint is halfway. To find the coordinates of the midpoint, find the average of the x-values and the average of the y-values. Examples: Find the midpoint of the segment with the given endpoints.

2 Slide 7 / 181 Slide 8 / Find the midpoint of K(1,8) & L(5,2). (2,3) (3,5) (-2,-3) (-3,-5) Slide 9 / Find the midpoint of H(-4, 8) & L(6, 10). (5,9) (-1,9) (1,9) (5,1) Slide 10 / If the midpoint of a segment is (4,9) and one endpoint is (-3,10), find the other endpoint. (-10,8) (11,8) (-10,11) (.5,9.5) Slide 11 / 181 Slide 12 / 181 (1, 4) (1,-2) (7, -2) How far apart are points and? The istance Formula is derived from the Pythagorean Theorem, a 2 + b 2 = c 2. In this example, = = 2 72 = 2 = =

3 Slide 13 / 181 Slide 14 / What is the distance between (2, 4) and (-1, 8)? Slide 15 / What is the distance between (0, 7) and (5, -5)? Slide 16 / Given ( 4, 5) and (x, 1) and =5, find all of the possible values of x E 0 F 1 G 3 H 5 I 7 J 9 Slide 17 / 181 Slide 18 / If the distance between (4,5) and (x,-2) is 10, what are the possible values of x? Introduction to onic Sections Return to Table of ontents

4 Slide 19 / 181 onic Sections are created by intersecting a set of double cones with a plane. Slide 20 / 181 More Info bout onics lick the link below for a YouTube video that demonstrates the cutting of the cones. SalMathGuy onics Video lick on the title below to take you to a webpage for more background information about conic sections: "The Occurrence of the onics", by r. Jill ritton iscussion Question: Which conic sections are functions? Slide 21 / 181 Slide 22 / 181 ircle comes from cutting parallel to the "base". The ircle The term base is misleading because like lines and planes, conic sections continue on forever. The Ellipse n Ellipse comes from cutting skew to the "base". The Parabola Parabola comes from intersecting the cone with a plane that is parallel to a side of the cone. Slide 23 / 181 The Hyperbola Hyperbola comes from cutting the cones perpendicular to the "bases". This is the only cross section that intersects both cones. Slide 24 / 181

5 Slide 25 / 181 Slide 26 / 181 This is the graph of y = x 2. omplete the table below: Parabolas x y iscuss the patterns that you observe. Return to Table of ontents graph that has this shape is called a parabola. y = x 2 is the "parent function". Slide 27 / 181 The equation of a parabola can be written in two forms: y = ax 2 + bx + c (the General Form) Slide 28 / 181 Match each equation to its parabola - drag the number of the graph to its equation. y = a(x - h) 2 + k (Standard Form) where (h,k) is the vertex. This is also called Vertex Form. Example: Name the vertex of each equation: ) y= -3(x - 4) f(x) = (x - 3)² - 2 g(x) = -2(x + 1)² h(x) = 2 / 3 (x + 5)² - 7 ) y= 2(x + 7) ) y= (x -3) 2 Slide 29 / 181 Slide 30 / What is the vertex of? (3, 2) (-3, -2) (2, 3) (-2, -3)

6 Slide 31 / 181 Slide 32 / What is the vertex of? (2, -3) (-3, -2) (2, 3) (-2, -3) Slide 33 / 181 Slide 34 / What is the vertex of? (-3, 2) (-3, -2) (2, 3) (-2, -3) Slide 35 / 181 Slide 36 / What is the vertex of? (3, 2) (-3, -2) (2, 3) (-2, -3)

7 Slide 37 / 181 onverting from General Form to Standard Form Slide 38 / 181 onverting from General Form to Standard Form Fill in the blank to complete the square: Half of 6 is 3, 3 2 = 9 y = x 2-8x + 5 y = (x 2-8x + ) What number completes the square in the parenthesis above? Slide 39 / 181 Slide 40 / 181 Slide 41 / 181 Slide 42 / 181

8 Slide 43 / 181 Slide 44 / What is the vertex of x = y 2-10y + 29? (4, 5) (-4, 5) (-5, 4) (5, 4) Slide 45 / 181 Slide 46 / What is the vertex of y= x 2-8x +21? (4, 5) (-4, 5) (-5, 4) (5, 4) Slide 47 / 181 onverting from General Form to Standard Form Slide 48 / What should be factored out of x = (4y 2-8y + )+ 9 -?

9 Slide 49 / What value completes the square of x = 4(y 2-2y + )+ 9 -? Slide 50 / What value should follow "+ 9" in x = 4(y 2-2y + ) + 9? Slide 51 / Which is the correct standard form of x = (4y 2-8y + )+ 9 -? Slide 52 / What should be factored out of y = (-5x 2-20x + )+ 7 -? x = 4(y - 1) x = 4(y + 1) x = 4(y - 1) x = 4(y + 1) Slide 53 / What value completes the square of click to reveal y = -5(x 2 + 4x + )+ 7 -? Slide 54 / What value should follow "+7" in click y to = reveal -5(x 2 + 4x + )+ 7?

10 Slide 55 / Which is the correct standard form of y = (-5x 2-20x + )+ 7 -? y = -5(x - 2) y = -5(x + 2) y = -5(x - 2) 2-13 y = -5(x - 2) Slide 56 / 181 Geometric efinition parabola is a locus* of points equidistant from a fixed point, the focus, and a fixed line, the directrix. *locus is just a fancy word for set. Every parabola is symmetric with respect to a line through the focus and perpendicular to the directrix. The vertex of the parabola is the "turning point" and is on the axis of symmetry. Slide 57 / 181 Focus and irectrix of a Parabola Every point on the parabola is the same distance from the directrix and the focus. Slide 58 / 181 Eccentricity of a Parabola L 1=L 2 L 1=L 2 L 1 L 1 L 2 Focus xis of Symmetry L 2 Focus irectrix irectrix ll parabolas have an eccentricity of 1. Slide 59 / 181 Parts of a Parabola Parts are the same for all parabolas, regardless of the direction in which they open. Slide 60 / 181 ompare the graphs below: What makes the graph more "narrow" or "wide"? y = 2x 2 y=ax 2 +bx+c x=ay 2 +by+c y = x 2 Focus Vertex irectrix xis of Symmetry Focus Vertex irectrix y =.5x 2 xis of Symmetry

11 Slide 61 / Which of the parabolas below are narrower than their parent functions? Slide 62 / 181 Focal istance To calculate: focal distance = The distance from the vertex to the focus is 1. The distance from the vertex to the directrix is 1. Slide 63 / 181 Slide 64 / 181 Parabola Summary General Form y= ax 2 + bx + c x= ay 2 +by + c Standard Form y= a(x - h) 2 +k x= a(y - k) 2 + h Opens a>0 opens up a<0 opens down a>0 opens to the right a<0 opens to the left xis of Symmetry x = h y = k Vertex (h, k) (h, k) Focal istance irectrix Focus Eccentricity 1 1 Slide 65 / 181 Graph the equation from the last example. Slide 66 / 181 Identify the vertex and the focus, the equations for the axis of symmetry and the directrix, and the direction of the opening of the parabola with the given equation. What is the parabola's eccentricity? irectrix Focus xis of Symmetry

12 Graph Slide 67 / 181 Slide 68 / 181 Identify the vertex and the focus, the equations for the axis of symmetry and the directrix, and the direction of the opening of the parabola with the given equation. What is the parabola's eccentricity? Step 1: onvert the equation from general to standard form. Slide 69 / 181 Step 2: Identify the vertex and the focus, the equations for the axis of symmetry and the directrix, and the direction of the opening of the parabola with the given equation. What is the parabola's eccentricity? Graph Slide 70 / 181 Slide 71 / 181 Slide 72 / Given the following equation, which direction does it open? 30 How does the following equation compare to the parent function Up own Left Right Is narrower Is wider Is the same width

13 Slide 73 / Where is the vertex for the following equation? (-3, 4) (3, 4) (4, 3) (4, -3) Slide 74 / What is the equation of the axis of symmetry for the following equation? y = 3 y = -3 x = 4 x = -4 Slide 75 / What is the focal distance in the following equation? Slide 76 / What is the equation of the directrix for the following equation? y = 2 y = -4 x = 3 x = -5 Slide 77 / Where is the focus for the following equation? Slide 78 / What is the eccentricity of the following conic section? (-3, 5) (3, 5) (5, 3) (5, -3)

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16 Slide 91 / 181 Slide 92 / 181 Slide 93 / What is the equation of the parabola with vertex (2,3) and directrix y = 4? y = 4(x - 2) y = -1/4(x - 2) x = 4(y - 2) x = 1/4(y - 2) Slide 94 / 181 hallenge Problem The St. Louis rch is 630 feet tall and 630 feet wide at the base. Write an equation to represent the shape of the arch. Slide 95 / 181 Slide 96 / 181 nswer on next page... ircles Return to Table of ontents

17 Slide 97 / 181 Slide 98 / 181 circle is a locus of points in a plane that are equidistant from a given point. Radius (x,y) enter (h,k) The distance from the center to a point on the circle is Slide 99 / 181 Slide 100 / 181 Slide 101 / Write the equation of the circle with center (5, 2) and radius 6 Slide 102 / Write the equation of the circle with center (-5,0) and radius 7

18 Slide 103 / Write the equation of the circle with center (-2,1) and radius Slide 104 / What is the center and radius of the following equation? Slide 105 / 181 Slide 106 / What is the center and radius of the following equation? Slide 107 / What is eccentricity of a circle? Slide 108 / 181 Write the equations for each part of this unfortunate snowman.

19 Slide 109 / 181 Write the equation of the circle that meets the following criteria: enter (1, -2) and passes through (4, 6) Slide 110 / 181 Write the equation of the circle that meets the following criteria: iameter with endpoints (4, 7) and (-2, -1). Since we know the center we only need to find the radius. The radius is the distance from the center to the point. The equation of the circle is: Slide 111 / 181 Slide 112 / 181 Write the equation of the circle that meets the following criteria: enter at (-5, 6) and tangent to the y-axis. "Tangent to the y-axis" means the circle only touches the y-axis at one point. Look at the graph. Slide 113 / 181 Write the equation of the circle in standard form that meets the following criteria: omplete the square for the x's. (Remember, the y-term is 0y.) Slide 114 / What is the equation of the circle that has a diameter with endpoints (0,0) and (16,12)?

20 Slide 115 / What is the equation of the circle with center (-3,5) that contains the point (1,3)? Slide 116 / What is the equation of the circle with center (7,-3) and tangent to the x-axis? Slide 117 / 181 Slide 118 / 181 Slide 119 / 181 Slide 120 / 181 hallenge Question: What is the equation of a circle that passes through the three points (2,3), (2,-2), and (5,-3)? Ellipses Remember that the distance from the radius to the circle is the same for every radius. Let (x,y) be the center and use the distance formula twice. Return to Table of ontents

21 Slide 121 / 181 n ellipse is a focus of points in a plane that are each the same total distance from 2 fixed points, called the foci (plural of focus). P 1 P 2 F 1 F 2 Slide 122 / 181 Eccentricity of an Ellipse 0 < e < 1 The eccentricity of an ellipse is a number between 0 and 1. The more elongated the ellipse the closer the eccentricity is to 1. The closer an ellipse is to being a circle, the closer the eccentricity is to 0. For example, P 1F 1 + P 1F 2 = P 2F 1 + P 2F 2 Slide 123 / 181 The major axis,, is the segment through both foci whose endpoints are on the ellipse. The minor axis,, is perpendicular to the major axis through the center, O. The vertices of an ellipse are the endpoints of the major axis, points and. The co-vertices are the endpoints of the minor axis, points and. Minor axis Major axis Vertex Slide 124 / 181 Parts of an Ellipse Horizontal ellipse Vertical ellipse o-vertex Major axis o-vertex Minor axis Vertex O Focus o-vertex Vertex Slide 125 / What letter or letters corresponds with ellipse's center? The length of the major axis is 2a. The length of the minor axis is 2b. Slide 126 / What letter or letters corresponds with ellipse's foci? E F G E F G E F G E F G

22 Slide 127 / What letter or letters corresponds with ellipse's major axis? E F G H I E I H G F Slide 128 / Which choice best describes an ellipse's eccentricity? e = 0 0< e < 1 e = 1 e > 1 Slide 129 / 181 Slide 130 / Which of the ellipses has the greater eccentricity? Slide 131 / 181 Slide 132 / 181 Finding the foci: (Note that in this case, a represents the hypotenuse of the triangle.) 69 What is the center of (9, 4)? a b a (5, 6) (-5, -6) a c (3, 2) In this ellipse, a = 5 and b = 4, so c = 3. The coordinates of the foci are (3-3,2) and (3+3,2) or (0,2) and (6,2)

23 Slide 133 / How long is the major axis of ? Slide 134 / How long is the minor axis of Name one foci of Slide 135 / Name one foci of Slide 136 / 181 Graphing an Ellipse Slide 137 / 181 Find and graph the center Find the length and direction of the major and minor axes raw the major and minor axes raw the ellipse Slide 138 / 181 The center is (4, -2) The major axis is 6 units and horizontal The minor axis is 4 units and vertical

24 Slide 139 / 181 Slide 140 / 181 Slide 141 / Given that an ellipse has foci (4,1) and (-4,1) and major axis of length 10, what is the center of the ellipse? Slide 142 / Given that an ellipse has foci (4,1) and (-4,1) and major axis of length 10, in which direction is the ellipse elongated? (8, 2) (0, 2) (0, 1) (-8, 1) horizontally vertically obliquely it is not elongated Slide 143 / Given that an ellipse has foci (4,1) and (-4,1) and major axis of length 10, how far is it from the center to an endpoint of the major axis? Slide 144 / Given that an ellipse has foci (4,1) and (-4,1) and major axis of length 10, which equation would be used to find the distance from the center to an endpoint of the minor axis?

25 Slide 145 / 181 Slide 146 / Given that an ellipse has foci (4, 1) and (-4, 1) and major axis of length 10, find b. Slide 147 / Given that an ellipse has foci (4,-4) and (4,2) and minor axis of length 8, which is the equation of the ellipse? Slide 148 / 181 Steps for onverting the ellipse from General Form to Standard Form factor the x's and y's divide by the constant complete the square for x and/or y Slide 149 / 181 Slide 150 / 181

26 Slide 151 / onvert the following ellipses to standard form. Slide 152 / onvert the following ellipses to standard form. Slide 153 / 181 Slide 154 / 181 Hyperbolas Return to Table of ontents Slide 155 / 181 Slide 156 / 181 Like the ellipse, the hyperbola is a set of points at a given distance from two foci. In the case of the hyperbola, the absolute value of the difference of the distances from a point to the foci is constant. Horizontal Hyperbola symptote b Focus a Vertex enter b symptote a Focus Vertex Equation: a b d c Vertices: move a units to the left and right of the center Foci: move c units to the left and right of the center, where F1 F2 a - b = c - d (on't worry so much about this definition - it is just to put things in perspective.) symptotes: slope = ± b/a (The asymptotes are lines that pass through the vertices of the rectangle between the vertices with length 2a and width 2b. n asymptote is a line that the graph approaches but never touches.)

27 Vertical Hyperbola Equation: Vertices: move a units up and down from the center Foci: move c units up and down from the center, where Slide 157 / 181 symptote Focus Vertex a b b enter a Vertex Focus symptote Slide 158 / 181 To graph a hyperbola in standard form: Find and graph the center Plot points a right and left of the center, and b up and down for horizontal, or b right and left, and a up and down for vertical Make a rectangle through the four points from previous step raw asymptotes that contain the diagonals of the rectangle Sketch the graph of the hyperbola enter: (-1,2) symptotes: slope = ± a/b Example: Graph Slide 159 / 181 Example: Graph Slide 160 / 181 click The center of the rectangle is ( -5, 4 ) From the center move left/right 2 From the center move up/down 3 The hyperbola opens What are the slopes of the asymptotes? How does this relate to a and b? Why? up and down nswer The center of the rectangle is ( 6, 0 ) From the center move left/right 4 From the center move up/down 5 The hyperbola opens click left and right Slide 161 / 181 Slide 162 / 181

28 Slide 163 / 181 Slide 164 / 181 Slide 165 / 181 Slide 166 / 181 Slide 167 / 181 Slide 168 / What is the equation of a hyperbola that has vertices (±6,0) and foci (±10,0)?

29 Slide 169 / 181 Slide 170 / 181 onvert to standard form: Slide 171 / 181 Slide 172 / 181 Recognizing onic Sections from General Form Return to Table of ontents Slide 173 / 181 General Form: ax 2 + bx + cy 2 + dy + e = 0 Slide 174 / 181 This form could represent any conic under the following conditions: In a parabola, either a=0 or b=0. In a circle, a=c and both a and c are positive. ax 2 + bx + dy +e =0 cy 2 + dy + bx + e=0 ax 2 + bx + cy 2 + dy + e = 0 In an ellipse, a and c are both positive, and a c. * ax 2 + bx + cy 2 + dy + e = 0 In a hyperbola, either a<0 ax 2 + bx - cy 2 + dy + e = 0 and c>0 or a>0 and c<0. cy 2 + dy - ax 2 + bx + e = 0 * circle is a special type of ellipse in which a = c.

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31 Slide 181 / 181 Eccentricity of onic Sections Ellipse 0<e<1 Parabola e = 1 ircle e=0 Hyperbola e > 1 This picture depicts the comparative eccentricity of conic sections. Eccentricity (e) is a measure of "unroundness". circle is round, so has e=0. For an ellipse, as the ellipse becomes more elongated, e increases from 0 to 1, not-including 1. parabola has e=1, and for a hyperbola e>1.