5Two cities are located on a map using a coordinate system Two cities are located on a map using a coordinate system. Your house is exactly half-way between the two cities. If city #1 is located at (-12, 2) and your house is at (-7.75, -4.5). What is the grid location of city #2?
6A circle has diameter 𝐴𝐵 A circle has diameter 𝐴𝐵 . If A is at (-3,-5) and the center of the circle is at (2, 3), find the coordinates of B. Then find the circumference and area of the circle.
7Find the perimeter of a triangle with vertices of A(4, 1), B(-3, -2), and C(-1, -4).
9Conic section: Any figure that can be obtained by slicing a double cone Focus: the point that is the same distance from all points in a parabolaDirectrix: a given line that is the same distance from all points in a parabolaLatus rectum: the line segment through the focus of a parabola and perpendicular to the axis of symmetry
10Upward if a>0, downward if a<0 Parabolas-y = a(x – h)2 + kx = a(y – k)2 + hVertex(h, k)Axis of symmetryx = hy = kFocus(h, k 𝑎 )(h 𝑎 , k)Directrixy = k 𝑎x = h 𝑎Direction of OpeningUpward if a>0, downward if a<0Right if a>0,Left if a<0Length of Latus rectum1 𝑎 units
11Write the equation in standard form Write the equation in standard form. Identify the coordinates of the vertex and focus, the equations of the axis of symmetry and directrix, and the direction of opening.y = x2 – 6x + 11
12Write the equation in standard form Write the equation in standard form. Identify the coordinates of the vertex and focus, the equations of the axis of symmetry and directrix, and the direction of opening.x = 3y2 + 5y - 9
13Vertex (8, 6) focus (2, 6)Vertex (3, 4)axis of symmetry x = 3, measure of latus rectum 4, a>0
18Equation of a circle (x – h)2 + (y – k)2 = r2 Circle: the set of all points in a plane that are equidistant from a given point in the planeCenter: the point that all points in a circle are equidistant fromEquation of a circle (x – h)2 + (y – k)2 = r2h = x value of centerk = y value of centerr = radius length .
23Write the equation in standard form then graph. x2 + y2 – 4x + 8y – 5 = 0
24Write the equation in standard form then graph. x2 + y2 + 4x - 10y – 7 = 0
25Write the equation for the circle described. Center (-1,-5) radius 2 unitsEndpoints of a diameter at(-4, 1) and (4, -5)
26A plan for a park puts the center of a circular pond of radius 0 A plan for a park puts the center of a circular pond of radius 0.6mi, 2.5mi east and 3.8mi south of the park headquarters. Use the headquarters as the origin and write an equation to represent the situation.
28Ellipse: the set of all points in a plane such that the sum of the distance from two fixed points is constantFoci: the two fixed points of an ellipseMajor axis: the longer line segment that forms an axis of symmetry for an ellipseMinor axis: the shorter line segment that forms an axis of symmetry for an ellipseCenter: the intersection of the axes of symmetry for an ellipse
29Direction of major axis (𝒙−𝒉) 𝟐 𝒂 𝟐 + (𝒚−𝒌) 𝟐 𝒃 𝟐 =𝟏(𝒚 −𝒌) 𝟐 𝒂 𝟐 + (𝒙 −𝒉) 𝟐 𝒃 𝟐 =𝟏Direction of major axishorizontalverticalFoci(h + c, k) and (h - c, k)(h, k + c) and (h, k - c)Length of major axis2a unitsLength of minor axis2b unitsCenter is (h , k) andNOTE:
30State the center, the direction of the major axis, the length of the major and minor axis, the value of c, and the foci.𝑥 𝑦 2 4 =1𝑥 𝑦 =1
39Hyperbola: the set of all points in a plane such that the absolute value of the differences of the distances from two fixed points is constantCenter: intersection of transverse and conjugate axesTransverse axis: axis of symmetry whose endpoints are the vertices of the hyperbolaConjugate axis: axis of symmetry perpendicular to the transverse axis
40𝒙 − 𝒉 𝟐 𝒂 𝟐 − 𝒚 −𝒌 𝟐 𝒃 𝟐 =𝟏 𝒚 −𝒌 𝟐 𝒂 𝟐 − 𝒙 −𝒉 𝟐 𝒃 𝟐 =𝟏 𝒙 − 𝒉 𝟐 𝒂 𝟐 − 𝒚 −𝒌 𝟐 𝒃 𝟐 =𝟏𝒚 −𝒌 𝟐 𝒂 𝟐 − 𝒙 −𝒉 𝟐 𝒃 𝟐 =𝟏Direction of transverse axishorizontalverticalFoci(h ± c, k)(h, k ± c)Vertices(h ± a, k)(h, k ± a)Length of transverse axis2a unitsLength of conjugate axis2b unitsAsymptotesy – k = ± 𝑏 𝑎 (x – h)y – k = ± 𝑎 𝑏 (x – h)*Note: Center is (h , k) and c2 = a2 + b2
51A and C have the same sign and A ≠ C Hyperbola Ax2 + Bxy + Cy2 + Dx + Ey + F = 0ParabolaA = 0 or C = o but not bothCircleA = CEllipseA and C have the same sign and A ≠ CHyperbolaA and C have opposite signs
52Without writing the equation in standard form, state whether the graph of each equation is a parabola, circle, ellipse, or hyperbola.y2 – x – 10y + 34 = 0y2 – 2x2 - 4x – 4y – 4 = 03x2 + 2y x – 28y = 04x2 + 4y x – 12y + 30 = 0
53A military jet performs for an air show A military jet performs for an air show. The path of the plane during one trick can be modeled by a conic section with equation 24x y – 31,680x – 45,600 = 0. Distances are represented in feet.Identify the shape of the curved path of the jet. Write the equation in standard form.
54If the jet begins its path upward or ascent at (0, 0), what is the horizontal distance traveled by the jet from the beginning of the ascent to the end of the descent?
57Review how to solve a system Elimination:If two coefficients are the same add or subtract to cancel that variableIf needed, multiply to get like coefficients and then add or subtractSubstitution:Solve one of the equations for a variable and then replace that variable in the other equation to solve.