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