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Analytic geometry

Conic Sections, Rational Exponents, and Radical Functions Review

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Algebra
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graph quadratics in vertex form

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Algebra
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Geometry of roots of real polynomials

3 H 5 Vt4Section 4.7: Graph of Quadratic Functions in Vertex or Intercept rm. J *Reca ll the transformations from Section 2.7 we learned f(x) = a I x - hi + Ic for absolute value functions. Vertex Form of a Quadratic f(x) =' (x - h)2 + ? Vertex: -Kit itc'1eS-- 0 r 2f4oO&1 Axis of Symmetry: In 0 *l.L tot,&) J'&H- ? Opens up/down if: Example 1: Given f(x) = x2, write it in vertex form and graph. totAJe-k- polv1+ on hne con1-aiw\k -' 4?LiOOlO?' j x ' - t 03 I \ z_ (oo) h t. Example 2: Given f(x) = (x - 3)2 - 5, state the vertex and axis of symmetry, describe the translations, and graph. (h,-) Vertex: (y) Axis of Symm: 3 Translations: 110flZ0iT1t'A 1 (C\\* 3 \rc&k C1$SIO(\ Vi (extjj \- CoLJJfl .iej~ 5o~ I ~~ k q I

Calculus 1 Exam 3 3of4

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14) Determine whether the equation represents an ellipse, a parabola, a hyperbola, or a degenerate conic. 15) Find the focus, directrix, and focal diameter of the parabola, and sketch its graph. a) b) Vertex: Focus: Directrix:

Limits

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Limits Number value that f(x) approaches as the values of x approach a specific numberIf f(x) is a polynomial or a rational function and a is in the domain of f(x), then Limits that fail to exist 1. If f(x) approaches a different number from the right than from the left: 2. If f(x) increases or decreases without bound as it approaches a number: 3. If f(x) oscillates between 2 fixed value One-sided limits If only approaching from the right If only approaching from the left Limit Existence Theorem = L iff = L = Vertical Asymptote If = or = Then x=a is a vertical asymptote Horizontal Asymptote If or Then y=b is a horizontal asymptote

math study guide 4 grade

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Math study guide Discrete a point is a dot T geometry a line is a set of dots in a row T a line has no thickness F 2 crossing lines intersect in a point F Synthetic a point is a physical dot F (exact location) Geometry a line is a set of points going in both directions going the shortest paths T A line has no thickness T A line extends 2 directions T Plane coordinate a point is an ordered pair of numbers not a # T Geometry a line is a set of ordered pairs that go with ax+by=c when they are both not 0 T

Point Slope Formula

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Point-slope refers to a method for graphing a linear equation on an x-y axis. When graphing a linear equation, the whole idea is to take pairs of x's and y's and plot them on the graph. While you could plot several points by just plugging in values of x, the point-slope form makes the whole process simpler. Point-slope form is also used to take a graph and find the equation of that particular line. Point slope form gets its name because it uses a single point on the graph and the slope of the line. Think about it this way: You have a starting point on a map, and you are given a direction to point. You have all the information you need to draw a single line on the map. The standard point-slope equation looks like this:

Cross Product

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Cross Product

Vector Algebra ? Vector Addition ? Scalar Multiplication ? How about the product of two vectors? 1. Dot Product ?v ? ?u 2. Cross Product ?v ? ?u Before the geometry, Determinant ? Determinant of a 2? 2 matrix, ? ? ? ? a b c d ? ? ? ? = ad ? bc ? Determinant of a 3? 3 matrix, ? ? ? ? ? ? a 1 a 2 a 3 b 1 b 2 b 3 c 1 c 2 c 3 ? ? ? ? ? ? = a 1 ? ? ? ? b 2 b 3 c 2 c 3 ? ? ? ? ? a 2 ? ? ? ? b 1 b 3 c 1 c 3 ? ? ? ? + a 3 ? ? ? ? b 1 b 2 c 1 c 2 ? ? ? ? =a 1 (b 2 c 3 ? b 3 c 2 )? a 2 (b 1 c 3 ? b 3 c 1 ) + a 3 (b 1 c 2 ? b 2 c 1 ) Example 1 1. ? ? ? ? ?1 2 3 5 ? ? ? ? = ?5? 6 = ?11 2. ? ? ? ? ? ? 2 4 6 ?1 3 5 7 2 6 ? ? ? ? ? ? = 2 ? ? ? ? 3 5 2 6 ? ? ? ? ? 4 ? ?

Cross Product

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Calculus
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Cross Product

Vector Algebra ? Vector Addition ? Scalar Multiplication ? How about the product of two vectors? 1. Dot Product ?v ? ?u 2. Cross Product ?v ? ?u Before the geometry, Determinant ? Determinant of a 2? 2 matrix, ? ? ? ? a b c d ? ? ? ? = ad ? bc ? Determinant of a 3? 3 matrix, ? ? ? ? ? ? a 1 a 2 a 3 b 1 b 2 b 3 c 1 c 2 c 3 ? ? ? ? ? ? = a 1 ? ? ? ? b 2 b 3 c 2 c 3 ? ? ? ? ? a 2 ? ? ? ? b 1 b 3 c 1 c 3 ? ? ? ? + a 3 ? ? ? ? b 1 b 2 c 1 c 2 ? ? ? ? =a 1 (b 2 c 3 ? b 3 c 2 )? a 2 (b 1 c 3 ? b 3 c 1 ) + a 3 (b 1 c 2 ? b 2 c 1 ) Example 1 1. ? ? ? ? ?1 2 3 5 ? ? ? ? = ?5? 6 = ?11 2. ? ? ? ? ? ? 2 4 6 ?1 3 5 7 2 6 ? ? ? ? ? ? = 2 ? ? ? ? 3 5 2 6 ? ? ? ? ? 4 ? ?

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