Algebra 2 Lap 7: Conic figures (parabola, circle, ellipse, hyperbola)
What is the general form equation for ALL conics?
Ax²+Cy²+Dx+Ey+F=0
diameter of a circle
a chord that passes through a center and the other endpoints is on the circle (d = 2r OR d= circumference / pi)
What is a circle?
a circle is the set of all points (x, y) that are equidistant from a fixed point, called the center of the circle
What is an ellipse?
an ellipse is the set of all points P such that the sum of the distances between P and two distinct fixed points, called the foci, is a constant
What factoring rule is used in completing the square?
a²±2ab+b² = (a±b)²
c of an ellipse
c is the distance from each focus to the center
c in a hyperbola
c is the distance from the center to each foci (for hyperbolas, c will always be greater than a)
What is the equation for finding c in an ellipse?
c²=a²-b² (because a>b) (note: for hyperbolas, its + not -)
All points on a parabola (including the vertex) are equidistant between the
directrix and the focus
Equations for the parts of a hyperbola:
horizontal hyperbola: - center = (h, k) - vertexes = (h±a, k) - unnamed points = (h, k±b) - foci = (h±c, k) - slope of asymptotes = ±b/a Vertical hyperbola: - center = (h, k) - vertexes = (h, k±a) - unnamed points = (h±b, k) - foci = (h, k±c) - slope of asymptotes = ±a/b
In a hyperbola, a is always with the....
positive piece
In a hyperbola, the piece that is positive determines...
the direction of the hyperbola (x is + → horizontal, y is + → vertical)
Radius of a circle
the distance r between the center and any point (x, y) on the circle (r= circumference / (2)pi )
What is the standard form of the equation of a circle with a center at (0, 0)?
x²+y²=r² (r is the radius)
What is the new standard form of a quadratic equation?
y= (1/4p)(x-h)²+k OR x= (1/4p)(y-k)²+h
What is p in a parabola?
→ |p| is the distance between the vertex and the focus → |p| is the distance between the vertex and the directrix → |2p| is the distance between the focus and the directrix → |4p| is the length of the latus rectum
What is the standard form of an equation of a circle with a center at (h, k)?
(x-h)²+(y-k)²=r²
recap of conic equations
*** the parabola equation is not correct in image
What is the standard form equation for a hyperbola?
- (h, k) is the center - for a hyperbola, one piece MUST be positive, and the other MUST be negative (in the horizontal example, the x piece is positive while the y piece is negative) - Note: the equation MAY appear to have a negative sign in front of the first piece and then a plus sign which is STILL a hyperbola, NOT an ellipse - must =1
What is a quadratic system?
A system of equations that includes one or more equations of conics
What is the standard form equation for an ellipse?
- (h, k) is the center of the ellipse - a is the length of the semimajor axis - b is the length of the semiminor axis - if the ellipse is horizontal, then a goes with x and b goes with y - if the ellipse is vertical, then b goes with x and a goes with y -Note: for the equation of a ellipse, the x piece and the y piece are either BOTH positive or BOTH negative - THE EQUATION FOR AN ELLIPSE ALWAYS MUST EQUAL 1 (if it does not, then divide both sides by that number to get it to equal 1)
focus of a parabola
- a point on the axis of symmetry -always located p units away from the vertex on the INSIDE of the bowl of the parabola
What is the difference between the equation of a circle and the inequality of a circle?
- equation is the edge of the circle only -an inequality less than the radius is everything inside the circle -an inequality more than the radius is everything outside the circle
how to tell if a hyperbola is horizontal or vertical:
- if the hyperbola opens left and right, it is horizontal, if the hyperbola opens up and down it is vertical -if the transverse axis is horizontal, then it is horizontal; if the transverse axis is vertical, then it is vertical -if the x piece of the equation is the positive piece, then it is horizontal; if the y piece of the equation is positive, then it is vertical -if a is with x, then it is horizontal; if a is with y, then it is vertical
What is a quadratic?
-A function in which the input variable is squared -The shape of quadratic functions is called a parabola
Latus Rectum of a parabola
-A line parallel to the directrix that passes through the focus -its entire length is |4p| (a distance of |2p| on either side of the focus) -The latus rectus ends when it hits the parabola
directrix of a parabola
-A line perpendicular to the axis of symmetry that is |p| units away from the vertex -always located OUTSIDE the bowl of the parabola -denoted by a SOLID line
axis of symmetry of a parabola
-A line perpendicular to the directrix that runs through the focus and the vertex -The axis of symmetry will always = -b/2a (same as the x value of the vertex) -splits the parabola into two symmetrical halves
Vertex of a parabola
-The point (x,y) of a parabola where it crosses the axis of symmetry -furthest point in some direction of the parabola (i.e. if the parabola opens up, then the vertex is the lowest point) -The vertex is equidistant from both the focus and directrix -located at (h, k)
what is the relationship between ellipses and circles?
-a circle is a special type of ellipse in which the foci and center are all at one point -if an ellipse's a and b are the same length, then they are the radii of a circle
What is a hyperbola?
-a hyperbola is the set of all points P such that the difference of the distances between P and two fixed points called the foci is a constant -a hyperbola is also the shape that rational functions make (indirect relationhips)
a in a hyperbola
-a is the distance from the center to each vertex - half the length of the transverse axis - located with the positive piece of the equation
a of an ellipse
-a is the length of the semiMAJOR axis -a is the distance from each vertex to the center -a is always larger than (or equal to) b
b in a hyperbola
-b is the distance from the center to each unnamed point - half the length of the conjugate axis - located with the negative piece of the equation
b of an ellipse
-b is the length of the semiMINOR axis -b is the distance from each covertex to the center -b is always smaller than (or equal to) a
y= (1/4p)(x-h)²+k
-because x is the part that is squared, this equation is a VERTICAL parabola -vertical parabolas open up (if 1/4p is positive) or down (if 1/4p is negative)
x= (1/4p)(y-k)²+h
-because y is the part that is squared, this equation is a HORIZONTAL parabola -horizontal parabolas open left ((if 1/4p is negative) or right (if 1/4p is positive) -horizontal parabolas are NOT functions
What are types of quadratic systems?
-between a line and a conic section (0, 1, or 2 points of intersection) -between 2 distinct conic sections (0, 1, 2, 3, or 4 points of intersection)
What are conics?
-conics are curves that are formed by the intersection of a plane with a right circular cone -there are four types of conics: parabolas, circles, ellipse, and hyperbolas
The box of a hyperbola
-determined by the vertices (endpoints of the transverse axis) and the unnamed points (endpoints of the conjugate axis) -signified by a dashed line (used for graphing purposes only, NOT actually part of the real graph) -determines the asymptotes of the graph
vertexes of an ellipse
-endpoints of the MAJOR AXIS (always on the longest axis of the ellipse) -located a units away from the center of the ellipse (on the major axis)
covertexes of an ellipse
-endpoints of the MINOR AXIS (always on the shortest axis of the ellipse) -located b units from the center of an ellipse (on the minor axis)
vertexes of a hyperbola
-endpoints of the transverse axis -located ON the branches of the hyperbola -located a units from the center, always on the transverse axis
branches of a hyperbola
-every hyperbola has two arms -the branches never enter the box of the hyperbola -the branches start at each vertex and reach to the asymptotes, curving around the foci
asymptotes of a hyperbola
-every hyperbola has two diagonal asymptotes drawn through the corners of the box -each asymptote passes through the center -the equation of each asymptote can be found by finding the slope of the asymptote, and using this slope and the center point to form a point-slope form linear equation -For a vertical hyperbola, the slope of the asymptotes is ±a/b, for a horizontal hyperbola, the slope of the asymptotes is ±b/a (can also be found by using rise/run) -the branches of the graph MUST show asymptotic behavior (they must reach TOWARDS the asymptotes)
In the equation for an ellipse, the a always goes with the variable that controls the direction that the ellipse is in
-ex: x controls horizontal, so if the ellipse is horizontal, then a goes with x -ex: y controls vertical, so if the ellipse is vertical, then a goes with y
what are the parts of an ellipse?
-foci -center -major axis -minor axis -vertices -covertices -a -b -c
How to classify conics based on the general form equation:
-if only one variable is squared, then it is a parabola -if two variables are squared and they have opposite signs, then it is a hyperbola -if two variables are squared and the have the same sign, then it is an ellipse -if two variables are squared and they have the same coefficients, then its a circle
horizontal vs vertical ellipse
-if the major axis is parallel to the y axis, then it is a vertical ellipse -if the major axis is parallel to the x axis, then it is a horizontal ellipse
center of an ellipse
-midpoint between the two foci -midpoint of the major axis -midpoint of the minor axis -located at the intersection of the major and minor axes
center of a hyperbola
-midpoint between the two foci -midpoint of the transverse and conjugate axes -point of intersection between the two axes -located at (h, k)
What are the parts of a circle?
-radius -diameter -circumference -center -chord -tangent
major axis of an ellipse
-the LONGEST axis of an ellipse (can be vertical or horizontal) -half the major axis is called the semi major axis and has a length of a -passes through the foci and the center -touches the ellipse at endpoints called vertexes -the total length of the major axis is 2a -the length of the major axis is the constant (the sums of the distances between each point on the ellipse and the foci is a constant: the length of the major axis, or 2a) -perpendicular to the minor axis
minor axis of an ellipse
-the SHORTEST axis of an ellipse (can be vertical or horizontal) -half of the minor axis is called the semi minor axis and has a length of b -the whole minor axis has a length of 2b -perpendicular to the major axis -passes through the center o the ellipse -touches the ellipse at endpoints called covertexes
transverse axis of a hyperbola
-the axis of a hyperbola that passes through the foci and the center -ends at points called vertexes -always oriented in the same direction as the hyperbola (hyperbola is vertical --> transverse axis is vertical) -the length of the transverse axis (between the two vertexes) is 2a -also the axis of symmetry
conjugate axis of a hyperbola
-the axis perpendicular to the transverse axis -always oriented in the opposite direction as the hyperbola (hyperbola is vertical --> conjugate axis is horizontal) -the length of the conjugate axis is 2b -the endpoints of the conjugate axis lie at an unnamed point
unnamed points of a hyperbola
-the unnamed points of a hyperbola are the endpoints of the conjugate axis -located b units from the center -the unnamed points, along with the vertices help determine the box of the hyperbola
How to solve systems algebraically:
-try to eliminate one of the squared variables using combination, then use plysmlt to solve for one of the variables, plug back in to solve for the other -if one of the equations doesn't have a squared variable, use that equation to solve for one of the variables and then plug in this expression into the other, use plysmlt to solve one variable, then plug this value in to solve for the other
foci of hyperbola
-two distinct points that determine a hyperbola (a hyperbola is made up of every point whose difference of the distances to the foci is a constant) -located c units from center, always on the transverse axis -located inside the bowl of the hyperbola, on the outside of the vertexes (focus, vertex, center, vertex, focus)
foci of an ellipse
-two distinct points that determine an ellipse (every point whose sum of the distances to the foci is a constant is part of the ellipse) -located c units from center, always on the major axis
What are the parts of a hyperbola?
-vertexes -center -foci -transverse axis -conjugate axis -asymptotes -branches of graph -the box -unnamed point -a -b -c
Examples of classifying conics:
1. 3x²+3y²-6x+9y-14=0, two variables are squared and they have the same coefficients →circle 2. 6x²+12x-y+15=0, one variable is squared→ parabola 3. x²+2y²+4x+2y-27=0, two variables are squared with the same sign → ellipse 4. x²-y²+3x-2y-43=0, two variables are squared with different signs → hyperbola
What are the different ways to solve quadratics?
1. Factor, zero product property, solve for x 2. isolate the squared variable, square root both sides which leads to an absolute value equation, solve for x 3. Quadratic formula NOTE: the solutions of quadratic equations are the X INTERCEPTS)
How to solve systems by graphing or by calculator:
1. Graphically- graph each equation, see where they intersect 2. calculator- solve each equation for y (this often creates an absolute value situation, so put the positive version in y1 and the negative of y1 in y1) and put each equation into y= in the calculator, graph, and use intersection tool
Complete the square for conics:
1. Group x's together and y's together, equaling the constant 2. For you x's, factor out the leading coefficient 3. Repeat with the y's 4. add blanks to the x's and y's 5. Add two blanks on the constant side as well (remember to keep the balance by putting the factored out coefficient in front of the corresponding blank) 6. For both the x's and y's Divide the "b" term by two, and then square it and add the number to the blanks 7. Reduce using the factoring rule
how to graph ellipse:
1. Put it in standard form (make sure both pieces are positive and that it =1) 2. Identify if horizontal (a is with x) or vertical (a is with y) (this will tell you which way the major and minor axis goes -- the major axis is always in the direction that the ellipse is in) 3. Find a (√a²), b (√b²) and c (√(a²-b²)) (note: no need to worry about absolute value because these are distances which are always +) 4. Graph the center (h, k) with an OPEN circle 5. graph each vertex a units from the center along the major axis 6. graph each covertex b units from the center along the minor axis 7. graph each focus c units from the center along the major axis 8. connect the vertexes and covertexes with an oval shape
How to graph hyperbolas:
1. Put the equation in standard form 2. Identify which piece is positive in order to determine the direction of the hyperbola 3. Identify a (with the positive piece, √a² ), b (with the negative piece, √b² ), and c (√(a²+b²) ) 4. graph the center at (h, k) 5. Graph each vertex a units from the center 6. graph each unnamed point b units from the center 7. draw a box connecting the vertexes and unnamed points 8. Draw each asymptotes diagonally through the corners of the box, passing through the center 9. Find the equation for each asymptote 10. Draw each foci c units from the center 11. Draw the branches stemming from the vertices, reaching towards the asymptotes
What are the parts of the graph of a quadratic?
1. The Parabola 2. vertex 3. Axis of Symmetry 4. x and y intercepts 5. focus 6. directrix 7. latus rectum
How to solve quadratic systems:
1. algebraically (substitution or elimination) 2. graphically 3. calculator
How to graph parabolas in the new standard form:
1. find the vertex (h, k) 2. find p (1/4p = the a value) 3. find the focus (p units away from the vertex inside the parabola) 4. find the AOS (h, dashed line) 5. find the directrix (p units away from the vertex outside the parabola, solid line) 6. find the latus rectum (4p, goes through the focus parallel to the directrix) 7. Plot points on either end the the latus rectum and connect to the vertex 8. Domain and Range
How do you change a conic equation from general form to its graphing form?
1. group x's together and y's together, equaling the constant 2. Do "complete the square" to each variable grouping 3. Simplify 4. Make sure it fits the requirements of the shape (ex: ellipses must =1, hyperbolas must have opposite signs)
What are the four types of quadratic equations?
1. old standard form: y=ax²+bx+c → -b/2a is the x coord of vertex 2. Intercept form: y=a(x-p)(x-q) → (p,0) and (q,0) are the x intercepts → (p+q)/2 is the x coord of the vertex 3. Vertex form: y=a(x-h)²+k → (h, k) is the vertex → can be used with coordinate rule: (x,y) = (x+h, ay+k) 4. NEW standard form
Example of changing general form to graphing form by completing the square:
1.) 4x²-9y²+32x-144y-548=0 2.) 4x²+32x-9y²-144y=548 3.) 4(x²+8x)-9(y²+16y)=548 4.) 4(x²+8x+___)-9(y²+16y+___)=548+4___-9___ 5.) 4(x²+8x+16)-9(y²+16y+64)=548+4(16)-9(64) 6.) 4(x+4)²-9(y+8)²=36 **Because this is an ellipse, it must =1 so divide by 36 7.) (x+4)²/9 - (y+8)² /4 = 1
tangent to a circle
A line in the plane of a circle that intersects the circle at exactly one point (tangents are perpendicular to the radius drawn to the point of intersection)
chord of a circle
A segment whose endpoints are on a circle (if a radius is perpendicular to a chord, then it also bisects this chord)
ellipse vs hyperbola
DIFFERENCES 1. + vs - -ellipse: both pieces are either + or - -hyperbola: one piece is + and one piece is - 2. location of foci -ellipse: foci are in between the vertexes -hyperbola: foci are outside the vertexes 3. location of a in equation -ellipse: the location of a is based on the direction of the ellipse (horizontal --> with x, vertical --> with y) -hyperbola: the location of a is based on the signs of the equation (a goes with the positive piece) 4. The length of a -ellipse: a ≥ b -hyperbola: c > a 5. The equation for c -ellipse: c²=a²-b² -hyperbola: c²=a²+b² 6. How to tell if it is vertical or horizontal -ellipse: if the bigger number (the "a") is with x, then its horizontal; with y, then its vertical -hyperbola: if the x piece is positive, then its horizontal; if the y piece is positive, then its vertical 7. Shape of graph -ellipse: one oval shape -hyperbola: two branching shapes SIMILARITIES 1. both shapes have the same overall equation 2. both shapes curve around the foci 3. The major axis and the transverse axis pass through the center, vertexes, and foci 4. The meaning of a, b, and c - a is always the length from the center to the vertex - b is always the length from the center to the covertex (the unnamed point in the case of hyperbolas, but same idea) - c is always the length from the center to the foci 5. Both equations MUST =1
equations for points on an ellipse:
Horizontal ellipse: - center = (h, k) - vertexes = (h±a, k) - covertexes = (h, k±b) - foci = (h±c, k) Vertical ellipse: - center = (h, k) - vertexes = (h, k±a) - covertexes = (h±b, k) - foci = (h, k±c)