quadratic relations and conic sections
DESCRIPTION
Quadratic Relations and Conic Sections. Algebra 2 Chapter 9. This Slideshow was developed to accompany the textbook Larson Algebra 2 By Larson , R., Boswell, L., Kanold , T. D., & Stiff, L. 2011 Holt McDougal Some examples and diagrams are taken from the textbook. Slides created by - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: Quadratic Relations and Conic Sections](https://reader036.vdocument.in/reader036/viewer/2022081508/5681327e550346895d991997/html5/thumbnails/1.jpg)
QUADRATIC RELATIONS AND CONIC SECTIONSAlgebra 2
Chapter 9
![Page 2: Quadratic Relations and Conic Sections](https://reader036.vdocument.in/reader036/viewer/2022081508/5681327e550346895d991997/html5/thumbnails/2.jpg)
This Slideshow was developed to accompany the textbookLarson Algebra 2By Larson, R., Boswell, L., Kanold, T. D., & Stiff,
L. 2011 Holt McDougal
Some examples and diagrams are taken from the textbook. Slides created by
Richard Wright, Andrews Academy [email protected]
![Page 3: Quadratic Relations and Conic Sections](https://reader036.vdocument.in/reader036/viewer/2022081508/5681327e550346895d991997/html5/thumbnails/3.jpg)
9.1 APPLY THE DISTANCE AND MIDPOINT FORMULAS
Distance Formula d2 = AC2 + BC2
d2 = (x2 – x1)2 + (y2 – y1)2
A (x1, y1) C (x2, y1)
B (x2, y2)
![Page 4: Quadratic Relations and Conic Sections](https://reader036.vdocument.in/reader036/viewer/2022081508/5681327e550346895d991997/html5/thumbnails/4.jpg)
9.1 APPLY THE DISTANCE AND MIDPOINT FORMULAS
Find the distance between(1, -3) and (2, 5)
What type of triangle is ∆RST if R(2, -2), S(4, 2), T(6, 0)?
![Page 5: Quadratic Relations and Conic Sections](https://reader036.vdocument.in/reader036/viewer/2022081508/5681327e550346895d991997/html5/thumbnails/5.jpg)
9.1 APPLY THE DISTANCE AND MIDPOINT FORMULAS
Midpoint formula
Find the midpoint of (1, -3) and (-2, 5)
![Page 6: Quadratic Relations and Conic Sections](https://reader036.vdocument.in/reader036/viewer/2022081508/5681327e550346895d991997/html5/thumbnails/6.jpg)
9.1 APPLY THE DISTANCE AND MIDPOINT FORMULAS
Find the equation of a perpendicular bisector1. Find the midpoint2. Find the slope3. Write the equation of the line using the
midpoint and the negative reciprocal of the slope
![Page 7: Quadratic Relations and Conic Sections](https://reader036.vdocument.in/reader036/viewer/2022081508/5681327e550346895d991997/html5/thumbnails/7.jpg)
9.1 APPLY THE DISTANCE AND MIDPOINT FORMULAS Find the perpendicular bisector of segment AB if A(-2, 1)
and B(1, 4).
617 #3-55 every other odd + 6 = 20
![Page 8: Quadratic Relations and Conic Sections](https://reader036.vdocument.in/reader036/viewer/2022081508/5681327e550346895d991997/html5/thumbnails/8.jpg)
QUIZ 9.1 Homework Quiz
![Page 9: Quadratic Relations and Conic Sections](https://reader036.vdocument.in/reader036/viewer/2022081508/5681327e550346895d991997/html5/thumbnails/9.jpg)
9.2 GRAPH AND WRITE EQUATIONS OF PARABOLAS Parabola
Shape of the graph of a quadratic equation All the points so that the distance to the focus and to the
directrix is equal
Vertex
Axis of Symmetry
![Page 10: Quadratic Relations and Conic Sections](https://reader036.vdocument.in/reader036/viewer/2022081508/5681327e550346895d991997/html5/thumbnails/10.jpg)
9.2 GRAPH AND WRITE EQUATIONS OF PARABOLAS
Standard Equation of a Parabola (vertex at origin)
Equation Focus Directrix Axis Opens x2 = 4py (0, p) y = -p x = 0 up y2 = 4px (p, 0) x = -p y = 0 right
![Page 11: Quadratic Relations and Conic Sections](https://reader036.vdocument.in/reader036/viewer/2022081508/5681327e550346895d991997/html5/thumbnails/11.jpg)
9.2 GRAPH AND WRITE EQUATIONS OF PARABOLAS Identify the focus, directrix, and
graph x = 1/8 y2
Solve for squared termy2 = 8 x
Coefficient of non-squared term = 4p
8 = 4pp = 2
Plot the directrix and focusx = -2, (2, 0)
Plot other points from a table of values
x y
2 -4, 4
1 -2√2, 2√2
![Page 12: Quadratic Relations and Conic Sections](https://reader036.vdocument.in/reader036/viewer/2022081508/5681327e550346895d991997/html5/thumbnails/12.jpg)
9.2 GRAPH AND WRITE EQUATIONS OF PARABOLAS
Write the equation for the parabola.
-1 0 1 2 3 4 5
-10
-6
-2
2
6
10
F
![Page 13: Quadratic Relations and Conic Sections](https://reader036.vdocument.in/reader036/viewer/2022081508/5681327e550346895d991997/html5/thumbnails/13.jpg)
9.2 GRAPH AND WRITE EQUATIONS OF PARABOLAS
623 #3-47 every other odd, 53, 55, 57 + 5 = 20
![Page 14: Quadratic Relations and Conic Sections](https://reader036.vdocument.in/reader036/viewer/2022081508/5681327e550346895d991997/html5/thumbnails/14.jpg)
QUIZ 9.2 Homework Quiz
![Page 15: Quadratic Relations and Conic Sections](https://reader036.vdocument.in/reader036/viewer/2022081508/5681327e550346895d991997/html5/thumbnails/15.jpg)
9.3 GRAPH AND WRITE EQUATIONS OF CIRCLES
Circle Set of points a fixed distance (radius) from the center
Derivation of equation (center at origin) r = distance from center
r2 = x2 + y2
x2 + y2 = r2
![Page 16: Quadratic Relations and Conic Sections](https://reader036.vdocument.in/reader036/viewer/2022081508/5681327e550346895d991997/html5/thumbnails/16.jpg)
9.3 GRAPH AND WRITE EQUATIONS OF CIRCLES
To graphFind the radiusPlot the center (0, 0)Move up, down, left,
and right from the center the distance of the radius
Draw a good circle Graph x2 + y2 = 16
![Page 17: Quadratic Relations and Conic Sections](https://reader036.vdocument.in/reader036/viewer/2022081508/5681327e550346895d991997/html5/thumbnails/17.jpg)
9.3 GRAPH AND WRITE EQUATIONS OF CIRCLES
Write the equation of a circle with center at the origin and goes through point (-3, 5)
![Page 18: Quadratic Relations and Conic Sections](https://reader036.vdocument.in/reader036/viewer/2022081508/5681327e550346895d991997/html5/thumbnails/18.jpg)
9.3 GRAPH AND WRITE EQUATIONS OF CIRCLES Finding a tangent line to a circle
Tangent lines are perpendicular to the radiusFind the slope of the radius to the point of
intersectionUse the negative reciprocal of the slope as the
slope of the tangent lineUse the slope and the point of intersection to
write the equation of the line
![Page 19: Quadratic Relations and Conic Sections](https://reader036.vdocument.in/reader036/viewer/2022081508/5681327e550346895d991997/html5/thumbnails/19.jpg)
9.3 GRAPH AND WRITE EQUATIONS OF CIRCLES Find the equation of the tangent line at (1, 5) to x2 + y2 =
26
629 #3-55 every other odd, 63, 65 + 4 = 20
![Page 20: Quadratic Relations and Conic Sections](https://reader036.vdocument.in/reader036/viewer/2022081508/5681327e550346895d991997/html5/thumbnails/20.jpg)
QUIZ 9.3 Homework Quiz
![Page 21: Quadratic Relations and Conic Sections](https://reader036.vdocument.in/reader036/viewer/2022081508/5681327e550346895d991997/html5/thumbnails/21.jpg)
9.4 GRAPH AND WRITE EQUATIONS OF ELLIPSES
Set of points so that the sum of the distances to the 2 foci is constant
Co-vertex (0, -b)
Co-vertex (0, b)
![Page 22: Quadratic Relations and Conic Sections](https://reader036.vdocument.in/reader036/viewer/2022081508/5681327e550346895d991997/html5/thumbnails/22.jpg)
9.4 GRAPH AND WRITE EQUATIONS OF ELLIPSES
Horizontal Ellipse.Center at origin
a > bc2 = a2 – b2
c
![Page 23: Quadratic Relations and Conic Sections](https://reader036.vdocument.in/reader036/viewer/2022081508/5681327e550346895d991997/html5/thumbnails/23.jpg)
9.4 GRAPH AND WRITE EQUATIONS OF ELLIPSES
Vertical Ellipse.Center at origin
a > bc2 = a2 – b2
![Page 24: Quadratic Relations and Conic Sections](https://reader036.vdocument.in/reader036/viewer/2022081508/5681327e550346895d991997/html5/thumbnails/24.jpg)
9.4 GRAPH AND WRITE EQUATIONS OF ELLIPSES
Graph EllipseWrite in standard
form (find a and b)Plot vertices and co-
verticesDraw ellipse
Graph 4x2 + 25y2 = 100and find foci
![Page 25: Quadratic Relations and Conic Sections](https://reader036.vdocument.in/reader036/viewer/2022081508/5681327e550346895d991997/html5/thumbnails/25.jpg)
9.4 GRAPH AND WRITE EQUATIONS OF ELLIPSES
Write the equation for an ellipse with center at (0, 0) and …a vertex at (0, 5), and a co-vertex at (4, 0)
![Page 26: Quadratic Relations and Conic Sections](https://reader036.vdocument.in/reader036/viewer/2022081508/5681327e550346895d991997/html5/thumbnails/26.jpg)
9.4 GRAPH AND WRITE EQUATIONS OF ELLIPSES
Write the equation for an ellipse with center at (0, 0) and …A vertex at (-6, 0) and a focus at (3, 0)
637 #3-35 every other odd, 37, 39, 41, 43, 49, 51 + 5 = 20
![Page 27: Quadratic Relations and Conic Sections](https://reader036.vdocument.in/reader036/viewer/2022081508/5681327e550346895d991997/html5/thumbnails/27.jpg)
QUIZ 9.4 Homework Quiz
![Page 28: Quadratic Relations and Conic Sections](https://reader036.vdocument.in/reader036/viewer/2022081508/5681327e550346895d991997/html5/thumbnails/28.jpg)
9.5 GRAPH AND WRITE EQUATIONS OF HYPERBOLAS
Set of all points so the difference of the distances between a point and the two foci is constant
![Page 29: Quadratic Relations and Conic Sections](https://reader036.vdocument.in/reader036/viewer/2022081508/5681327e550346895d991997/html5/thumbnails/29.jpg)
9.5 GRAPH AND WRITE EQUATIONS OF HYPERBOLAS
Horizontal transverse axis
Asymptotes
![Page 30: Quadratic Relations and Conic Sections](https://reader036.vdocument.in/reader036/viewer/2022081508/5681327e550346895d991997/html5/thumbnails/30.jpg)
9.5 GRAPH AND WRITE EQUATIONS OF HYPERBOLAS
Vertical transverse axis
Asymptotes
![Page 31: Quadratic Relations and Conic Sections](https://reader036.vdocument.in/reader036/viewer/2022081508/5681327e550346895d991997/html5/thumbnails/31.jpg)
9.5 GRAPH AND WRITE EQUATIONS OF HYPERBOLAS
Graphing HyperbolasPlot the vertices and “co-vertices”Draw the “box”Draw the asymptotesDraw the hyperbola
![Page 32: Quadratic Relations and Conic Sections](https://reader036.vdocument.in/reader036/viewer/2022081508/5681327e550346895d991997/html5/thumbnails/32.jpg)
9.5 GRAPH AND WRITE EQUATIONS OF HYPERBOLAS
Graph 9x2 – 16y2 = 144
![Page 33: Quadratic Relations and Conic Sections](https://reader036.vdocument.in/reader036/viewer/2022081508/5681327e550346895d991997/html5/thumbnails/33.jpg)
9.5 GRAPH AND WRITE EQUATIONS OF HYPERBOLAS Write the equation of hyperbola with foci (0, -5) and (0, 5)
and vertices at (0, -3) and (0, 3).
645 #3-11 odd, 15-33 odd, 41, 43 + 3 = 20
![Page 34: Quadratic Relations and Conic Sections](https://reader036.vdocument.in/reader036/viewer/2022081508/5681327e550346895d991997/html5/thumbnails/34.jpg)
QUIZ 9.5 Homework Quiz
![Page 35: Quadratic Relations and Conic Sections](https://reader036.vdocument.in/reader036/viewer/2022081508/5681327e550346895d991997/html5/thumbnails/35.jpg)
9.6 TRANSLATE AND CLASSIFY CONIC SECTIONS
Remember when we studied quadratics and absolute value equations?
y = a(x – h)2 + k
h is how far the graph moved right k is how far the graph moved up
We can apply this concept for conics, too.
![Page 36: Quadratic Relations and Conic Sections](https://reader036.vdocument.in/reader036/viewer/2022081508/5681327e550346895d991997/html5/thumbnails/36.jpg)
9.6 TRANSLATE AND CLASSIFY CONIC SECTIONS
Parabola:
Ellipse:
Hyperbola:
Vertical Axis
Circle:
Horizontal Axis
![Page 37: Quadratic Relations and Conic Sections](https://reader036.vdocument.in/reader036/viewer/2022081508/5681327e550346895d991997/html5/thumbnails/37.jpg)
9.6 TRANSLATE AND CLASSIFY CONIC SECTIONS
How to graphFind the center/vertex (h, k)Graph the rest as before
![Page 38: Quadratic Relations and Conic Sections](https://reader036.vdocument.in/reader036/viewer/2022081508/5681327e550346895d991997/html5/thumbnails/38.jpg)
9.6 TRANSLATE AND CLASSIFY CONIC SECTIONS
Graph
![Page 39: Quadratic Relations and Conic Sections](https://reader036.vdocument.in/reader036/viewer/2022081508/5681327e550346895d991997/html5/thumbnails/39.jpg)
9.6 TRANSLATE AND CLASSIFY CONIC SECTIONS
Write equations of a translated conicGraph known points to determine horizontal or
vertical axisFind the center/vertex to give (h, k)Use the known points to find a and b (or p)
![Page 40: Quadratic Relations and Conic Sections](https://reader036.vdocument.in/reader036/viewer/2022081508/5681327e550346895d991997/html5/thumbnails/40.jpg)
9.6 TRANSLATE AND CLASSIFY CONIC SECTIONS
Write an equation of a parabola with vertex (3, -1) and focus at (3, 2).
Write an equation of a hyperbola with vertices (-7, 3) and (-1, 3) and foci (-9, 3) and (1, 3).
![Page 41: Quadratic Relations and Conic Sections](https://reader036.vdocument.in/reader036/viewer/2022081508/5681327e550346895d991997/html5/thumbnails/41.jpg)
9.6 TRANSLATE AND CLASSIFY CONIC SECTIONS
Identify lines of symmetry Conics are symmetric along their axes which go
through their center/vertex
![Page 42: Quadratic Relations and Conic Sections](https://reader036.vdocument.in/reader036/viewer/2022081508/5681327e550346895d991997/html5/thumbnails/42.jpg)
9.6 TRANSLATE AND CLASSIFY CONIC SECTIONS Classifying Conics from general equations Ax2 + Bxy + Cy2 + Dx + Ey + F = 0
Discriminant: B2 – 4AC
B2 – 4AC < 0, B = 0 and A = C Circle B2 – 4AC < 0, B ≠ 0 or A ≠ C Ellipse B2 – 4AC = 0 Parabola B2 – 4AC > 0 Hyperbola
If B = 0, the axes are horizontal or vertical. If B ≠ 0, the axes are rotated
![Page 43: Quadratic Relations and Conic Sections](https://reader036.vdocument.in/reader036/viewer/2022081508/5681327e550346895d991997/html5/thumbnails/43.jpg)
9.6 TRANSLATE AND CLASSIFY CONIC SECTIONS
An asteroid's path is modeled by where x and y are in astronomical units from the sun. Classify the path and write its equation in standard form.
655 #3, 7, 11-19 odd, 23-43 odd, 49 + 1 = 20
![Page 44: Quadratic Relations and Conic Sections](https://reader036.vdocument.in/reader036/viewer/2022081508/5681327e550346895d991997/html5/thumbnails/44.jpg)
QUIZ 9.6 Homework Quiz
![Page 45: Quadratic Relations and Conic Sections](https://reader036.vdocument.in/reader036/viewer/2022081508/5681327e550346895d991997/html5/thumbnails/45.jpg)
9.7 SOLVE QUADRATIC SYSTEMS You have already learned how to solve systems
usingGraphingSubstitutionElimination
You can use all three methods to solve quadratic systems.
![Page 46: Quadratic Relations and Conic Sections](https://reader036.vdocument.in/reader036/viewer/2022081508/5681327e550346895d991997/html5/thumbnails/46.jpg)
9.7 SOLVE QUADRATIC SYSTEMS Quadratic systems of two equations can have up to four
solutions.
![Page 47: Quadratic Relations and Conic Sections](https://reader036.vdocument.in/reader036/viewer/2022081508/5681327e550346895d991997/html5/thumbnails/47.jpg)
9.7 SOLVE QUADRATIC SYSTEMS Solve using substitution
![Page 48: Quadratic Relations and Conic Sections](https://reader036.vdocument.in/reader036/viewer/2022081508/5681327e550346895d991997/html5/thumbnails/48.jpg)
9.7 SOLVE QUADRATIC SYSTEMS Solve using elimination
![Page 49: Quadratic Relations and Conic Sections](https://reader036.vdocument.in/reader036/viewer/2022081508/5681327e550346895d991997/html5/thumbnails/49.jpg)
9.7 SOLVE QUADRATIC SYSTEMS Solve by graphing calculator
Graph both equations You will have to solve for y. If you have a ± sign, then you will have to graph one equation for the +
and one for the -- On TI-83/84
Push Choose “intersect” Push enter for the first curve Push enter for the second curve (you may have to use the up/down arrows
to choose the right curve) Use the left and right arrows to move the cursor to an intersection and
push enter. Repeat for the rest of the intersections
![Page 50: Quadratic Relations and Conic Sections](https://reader036.vdocument.in/reader036/viewer/2022081508/5681327e550346895d991997/html5/thumbnails/50.jpg)
9.7 SOLVE QUADRATIC SYSTEMS Solve using a graphing calculator
661 #3, 5, 9-35 odd, 41 + 3 = 20
![Page 51: Quadratic Relations and Conic Sections](https://reader036.vdocument.in/reader036/viewer/2022081508/5681327e550346895d991997/html5/thumbnails/51.jpg)
QUIZ 9.7 Homework Quiz
![Page 52: Quadratic Relations and Conic Sections](https://reader036.vdocument.in/reader036/viewer/2022081508/5681327e550346895d991997/html5/thumbnails/52.jpg)
9.REVIEW 673 choose 20