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Criswell PreCalculus
Linear Forms and Parabolas
| Question | Answer |
|---|---|
| Linear Forms | 1) y = mx + b or y = (∆y / ∆x) x + b. "slope intercept" 2) Ax + By = C "standard" 3) (y–k) = m (x–h) "point slope" |
| Criteria for Standard Form Linear Equations | 1) Must be in "Ax + By = C" form 2) where A, B, & C are integers 3) and "A" must be a positive value m = –A / B or first number over the second and "change the sign". |
| Parallel Slopes | The slopes of parallel " // "lines are the SAME! Examples: m = 4/5, m // = 4/5 m = –2/3, m // = –2/3 |
| Perpendicular Slopes | The slopes of perpendicular "⊥" lines are OPPOSITE RECIPROCALS! Examples: m = 2/7, m ⊥ = –7/2 m = –3/5, m ⊥ = 5/3 |
| Characteristics to Identify on Parabolas | 1) Vertex 2) Max / Min points 3) Opens up/ down 4) Opening (obtuse, acute, standard) 5) x - intercepts 6) y - intercepts 7) axis of symmetry 8) Domain & Range 9) Focus & Directrix |
| Completing the Square on Quadratic Equations | 1) go from "standard" form to "vertex" form 2) to find roots |
| Graphing Parabolas with Lattice Points | When considering y = a (x –h)^2 +k, the denominator for "a" is the ∆x that should be used in a t chart centered around the vertex |
| Discriminant | = b^2 – 4(a)(c) if result is negative, then no solutions. if result is zero, then one rational root "double root". if result is positive non-square value, then two irrational roots. if result is positive square value, then two rational roots. |
| Quadratic Formula | x = (–b ± √ [ b^2–4(a)(c)] ) / (2a) x = (–b ± √ [ discriminant] ) / (2a) |
| Write Equation of Parabola in "Vertex Form" from 3 points on the curve | Two phase problem. 1) Write equations in standard form for each point where Ax^2 +Bx+C = y 2) Put the three equations into the matrix editor 3) Write the standard form parabola with values for A,B, & C 4) Complete the square to get the Vertex Form |
| Movements for graphing lattice points on "Standard" parabolas | When a =1 from the vertex move, right /left 1 unit, then up 1 right /left 2 unit, then up 4 right /left 3 unit, then up 9 |
| Movements for graphing lattice points on "acute" parabolas | When a >1 but still an integer, from the vertex move, right /left 1 unit, then up 1(a) right /left 2 unit, then up 4(a) right /left 3 unit, then up 9(a) |
| Movements for graphing lattice points on "obtuse" parabolas | When a <1 but the reciprocal of an integer ex "1/7", from the vertex move, right /left 1(a) unit, then up 1(a) right /left 2(a) unit, then up 4(a) right /left 3(a) unit, then up 9(a) |
| Movements for graphing lattice points on "blended" parabolas | When "a" is some some type of rational in the form of m/n like "5/3" or "2/11", from the vertex move, right /left 1(n) unit, then up 1(m*n) right /left 2(n) unit, then up 4(m*n) right /left 3(n) unit, then up 9(m*n) |
| Inverse graphs of parabolas | Be prepared to write and graph equations in the form x = a (y–k)^ + h and be able to identify related characteristics. |
| Focus of parabola | The Focus is always located in the interior of a parabola and is p units away from the vertex. |
| Directrix of a parabola | The Directrix is always located on the exterior of a parabola and is p units away from the vertex. |
| Locus of a parabola | The essence of a parabola is found with the relationship between the focus and the directrix. The collection of points that are equal distance from the focus to (x , y) and then perpendicular to the directrix is what creates the parabolic shape. |
| x-intercept(s) to a parabola | If these points exist, they can be found by using the quadratic formula. An important side note: When the MIDPOINT of the intercepts is found, it leads to the x coordinate of the VERTEX. The y coordinate is found through substitution. |
Created by:
Troy.Criswell
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