o Parent Functions
-> A parent is a basic function used to build a family.
-> R means all real numbers; inf means infinity.
-> Domain lists inputs; range lists outputs.
-> Square brackets include an endpoint; parentheses do not.
-> A function assigns each input exactly one output.
-> A vertical line meets a function graph at most once.
-> Compare shape, intercepts, domain, and range.
-> The x-intercept is where y = 0; the y-intercept is x = 0.
-> Rates of change separate constant, linear, and curved parents.
o Constant Parent
-> f(x) = a gives the same output for every input.
-> Its graph is the horizontal line y = a.
-> Domain: R. Range: {a}, a single value.
-> For x = -2, -1, 0, 1, 2, outputs are a, a, a, a, a.
-> Example: f(x) = 3 has range {3}.
-> Its y-intercept is (0,a); if a != 0, it has no zero.
-> If a = 0, every real input is a zero.
-> Slope is 0, so every secant line is horizontal.
-> Average rate of change is 0 between any two distinct inputs.
o Identity Parent
-> f(x) = x returns the input unchanged.
-> Its graph is a line through (0,0) with slope 1.
-> Domain: R. Range: R.
-> Table inputs: x = -2, -1, 0, 1, 2.
-> Table outputs: y = -2, -1, 0, 1, 2.
-> Example: f(-2) = -2 and f(2) = 2.
-> Both intercepts are (0,0); the function increases.
-> Average rate of change is 1 on every interval.
o Absolute Value Parent
-> f(x) = |x| measures distance from zero.
-> Domain: R. Range: [0,inf).
-> Table inputs: x = -2, -1, 0, 1, 2.
-> Table outputs: y = 2, 1, 0, 1, 2.
-> The V has vertex (0,0) and y-axis symmetry.
-> For x < 0, |x| = -x; for x >= 0, |x| = x.
-> The minimum is 0; opposite inputs give equal outputs.
-> Rate of change: -1 on (-inf,0) and 1 on (0,inf).
-> It is not linear: slopes jump from -1 to 1 at x = 0.
o Quadratic Parent
-> f(x) = x^2 squares each input.
-> Domain: R. Range: [0,inf).
-> Table inputs: x = -2, -1, 0, 1, 2.
-> Table outputs: y = 4, 1, 0, 1, 4.
-> The parabola opens up with vertex (0,0).
-> It has y-axis symmetry and a minimum of 0.
-> Unlike |x|, it curves; both have nonnegative outputs.
-> From 0 to h != 0, average rate of change is h^2/h = h.
-> Equal input steps give unequal changes in squared outputs.
o Choosing Models
-> Constant: temperature held fixed over a time interval.
-> Linear: wages W(t) = r*t at a fixed hourly rate r.
-> Identity is the linear parent; wages need scaling.
-> Quadratic: hypothetical projectile height H(t).
-> H(t) = -t^2 + 4*t; 0 <= t <= 4; range: [0,4].
-> Absolute value: |s-c| is distance from reference c.
-> s is signed position, not total distance traveled.
-> Wages at $12 per hour: W(30) = 12*30 = 360 dollars.