By Myles, Josh, Joe, and Louis Chapter 8 By Myles, Josh, Joe, and Louis

8.1 The Distance formula The distance formula is used to find the distance between two points Formula: √((x2-x1)+(y2-y1)); where (x1, y1) is one point and (x2, y2) is another point

8.1 The midpoint Formula The midpoint formula is used to find the midpoint between two points Formula: (x1+x2)/2, (y1+y2)/2; where (x1, y1) is one point and (x2, y2) is another point

8.2 Parabolas Focus: Is a point that lies on the axis of symmetry Directrix: Is a line perpendicular to the axis of symmetry Vertex: Is a point that lies halfway between the focus and directrix

8.2 Parabolas (vertical) Equation: (x-h)2=4p(y-k) Vertex: (h,k) Axis of symmetry: x=h Focus: (h,k+p) Directrix: y=k-p The graph opens up if p>0, and opens down if p<0

8.2 Parabolas (horizontal) Equation: (y-k)2=4p(x-h) Vertex: (h,k) Axis of symmetry: y=k Focus: (h+p,k) Directrix: x=h-p The graph opens to the right is p>0, and opens left if p<0

8.3 Circles The equation of a circle is (x-h)2+(y-k)2=r2 The center is (h,k) The radius is r Given the center, and a point on the circle, you can find the radius of the circle using the distance formula

8.4 Ellipses Use the pythagorean theorem to guide you. Is it horizontal or vertical? If the larger denominator is first, it is horizontal, if it isn’t, it’s vertical. A = Distance from center to Foci B = Distance from center to Co-vertices C = Distance from center to Vertices

Ellipses (horizontal) Formula:(x-h)2/c2+(y-k)2/b2=1 Center: (h,k) Vertices: (h+-c,k) Co-vertices: (h,k+-b) Foci: (h+-a,k)

8.4 Ellipses (vertical) Formula:(x-h)2/b2+(y-k)2/c2=1 Center: (h,k) Vertices: (h,k+-c) Co-vertices: (h+-b,k) Foci: (h,k+-a) 2

8.5 hyperbolas Foci- distance between P and two fixed points (use c) Vertices- center of each parabola (use a) Transverse axis- joins the vertices Center- midpoint of hyperbola

8.5 hyperbola (vertical) Equation: (y-k)2/a2-(x-h)2/b2=1 Center: (h,k) Vertices: (h,k+-a) Foci: (h,k+-c) Slope of Transverse Axis: +-a/b

8.5 hyperbola (Horizontal) Equation: (x-h)2/a2-(y-k)2/b2=1 Center: (h,k) Vertices: (h+-a,k) Foci: (h+-c,k) Slope of Transverse Axis: +-b/a

8.6 Classifying conic sections Is there BOTH X2 and Y2? NO: Parabola YES: Go to step 2 Do the X2 and Y2 values have the same coefficient? NO: go to step 3 YES: Circle What sign is in between the fractions? (+) : Ellipse (-) : Hyperbola

8.7 Solving Systems of Quadratic Equations (Substitution) Given: two equations; asked to find where both equations intersect with each other, and there can be one, two, or no answers. Step 1: Solve one equation so that you can substitute it in for the second equation. Step 2: Plug the first equation into the second equation, and solve the second equation so that you end up with a quadratic equation. Step 3: use either the quadratic formula or completing the square to factor the quadratic equation and end up with the solution (these are the “x” values). Step 4: Plug the “x” values back into the initial solved equation to get the corresponding “y” values.

8.7 Solving Systems of Quadratic Equations (Substitution) Example: 2x-y=1 x=y2-5x+7 Step 1: 2x-y+y=1+y 2x-1=y+1-1 y=2x-1 Step 2: x=(2x-1)2-5x+7 x-x=4x2-6x+1-5x+7-x 4x2-12x+8=0 Step 3: 4(x2-3x+2)=0(4) x2-3x+2-2=0-2 x2-3x+2.25=-2+2.25 (x-1.5)2=0.25 x-1.5+1.5=0.5+1.5 x=2 Step 4: y=2(2)-1 y=3 Answer:(2,3)