Algebra II Unit ❹⓸𝟜
The Equation of a circle
- (x-h)^2 + (y-k)^2 = r^2 - see distance formula
midpoint formula
- (x1 + x2 / 2, y1 + y2 / 2) - endpoints (x1, y1) + (x2, y2)
the distance formula
- d = √[( x₂ - x₁)² + (y₂ - y₁)²] - distance between any two points (x1, y1) + (x2, y2) - derived from Pythagorean theorem
why can't functions have x axis symmetry?
- doesn't pass vertical line test
vertical translation/shift
- effect of adding/subtracting a constant from f(x) in y = f(x) => shift of the graph f(x) *up/down* - y-coordinate changes - b > 0: up - b < 0: down - up: y = f(x) + b - down: y = f(x) - b - change on the *outside*
Horizontal translation/shift
- effect of adding/subtracting a constant from x-value in y = f(x) => shift of the graph f(x) *right/left* - slide left/right; the x-coordinate changes - d > 0: left - d < 0: right - left: y = f(x + d) - right: y = f(x - d) - change on the *inside*
reciprocal parent function
- f(x)=1/x - gets infinitely close to zero but never touches zero - ODD FUNCTION!! - for every point (x,y) reflect across origin to get (-x, -y) - to obtain quadrant 1 upper values, plug in fractions for x (e.g. 0.5, 2)
Algabraic procedure for determining even/odd functions
- given f(x) 1. find f(-x) and simplify: if f(x) = f(-x) then f is *even* - replace all x's with -x 2. find -f(x) and simplify: if -f(x) = f(-x) then f is *odd* - make the whole function neg. (f(-x) = -(2x + 1)
odd function
- graph of f is symmetric with respect to *origin* - f(-x) = -f(x)
even function
- graph of f is symmetric with respect to the *y-axis* - f(x) = f(-x)
Reflections
- graph of y = -f(x) is *reflection* of graph of y = f(x) across x axis - graph of y = f(-x) is *reflection* of graph of y = f(x) across y axis - if point (x, y) is on graph of y = f(x), then (x, -y) is on graph of y = -f(x) and (-x, y) is on graph of y = f(-x)
Symmetry with respect to the origin
- if for any point (x, y) on graph and point (-x, -y) also on graph - origin graph: rotate graph 180º (90 then 90) about the origin => should result in original graph
Symmetry with respect to the y-axis
- if for any point (x, y) on graph and point (-x, y) also on graph - fold line: fold graph on y axis
Symmetry with respect to the x-axis
- if for any point (x, y) on graph, point (x, -y) also on graph - fold line: fold graph on x axis
Algebraic tests of symmetry: origin
- if replacing x and y with -x and -y produces an equivalent equation, then the graph is symmetrical with respect to the origin - e.g. -y = 2(-x)^3 -y = -2x^3 y = 2x^3
Algebraic tests of symmetry: y axis
- if replacing x with -x produces an equivalent equation, then the graph is symmetrical with respect to the y-axis - y = 2(-x)^3 - y = -2x^3
Algebraic tests of symmetry: x axis
- if replacing y with -y produces an equivalent equation, then the graph is symmetrical with respect to the x-axis - *multiply by -1* - -y = 2x^3 y = -2x^3
Reflections across the x-axis
- points have same x-value, opposite y-values - (x, y) => (x, -y)
Reflections across the y-axis
- points have same y-value, opp. x-values - (x, y) => (-x, y)
terminology
- vertical/horizontal compression by a factor of... - reflection with respect to the x/y axis - horizontal/vertical shift right/left ___units or up/down ___ units
how to find a point symmetric to the given point
- x axis: (x, -y) - y axis: (-x, y) - origin: (-x, -y)
Transformations of Parent Functions
Parent functions can be transformed to create other members in a family of graphs
circle
The set of all points in a plane that are the same distance (radius) (x, y) from a given point called the center(h, k)
Graphical procedure for determining even/odd functions
compare graphs to definitions
parent functions
constant/identity, linear, squaring, cubic, cube root, reciprocal, absolute value
neither even nor odd
graph of f is not symmetric to neither y - axis or origin
Vertical Stretching and Shrinking
the graph of y = af(x) can be obtained from graph of y = f(x) by... - *stretching* vertically for |a| > 1 - *shrinking* (compressing) vertically for 0 < |a| < 1 - for a < 0, graph is also reflected across x-axis - y coordinates obtained by *multiplying* y coordinates of y = f(x) by a - change on outside = vertical shift (y-values)
Horizontal Stretching and Shrinking
the graph of y = f(cx) can be obtained from graph of y = f(x) by... - *shrinking* (compressing) horizontally for |c| > 1 - *stretching* horizontally for 0 < |c| < 1 - for c < 0, graph is also reflected across y-axis - x coordinates obtained by *dividing* x coordinates by c - change on inside = horizontal shift (x-values) - has to match formula EXACTLY (coefficient of 1 and + sign)
