Optional course topic. Revisit quadratic equations, sign charts and graph transformations.
Hypothetical tariffs depend on usage x≥0. One has rate a and initial charge 3; the other rate 1 and charge b. Fix parameters a,b for one problem and solve for x. Then classify the whole family by parameter values. Here a,b≥0; money and usage have agreed arbitrary units. This is a teaching model.
Check before dividing
Equal charges give:
T_(1) = ax + 3, T_(2) = x + b, (a − 1)x = b − 3.
Dividing immediately by a−1 loses a=1. Partition all admissible parameters:
| Condition | Solutions for x≥0 |
|---|---|
| a≠1 | one if (b−3)/(a−1)≥0; otherwise none |
| a=1, b=3 | all x≥0 |
| a=1, b≠3 | none |
For the first row:
x = (b − 3)/(a − 1) (a ≠ 1), x ≥ 0.
The other two give 0·x=b−3, an identity at b=3 and a contradiction otherwise. These cases are disjoint and cover every a,b≥0. OpenStax
At a=2,b=5 we get x=2; both charges are 7. At a=1,b=5 equality is impossible; at a=1,b=3 charges always agree. At a=0.5,b=5 algebra gives −4, but negative usage is inadmissible. Boundary b=3 with a≠1 gives allowed x=0.
A parameter moves a graph
Now a different problem: x²=a over the whole real line, for any real a. Negative a is no real square; zero gives the one root 0; positive a gives two roots ±√a.
(x)^(2) = a: a < 0⇒0, a = 0⇒1, a > 0⇒2.
The numbers 0,1,2 after arrows are counts of distinct real roots, not their values.
Three parameter values
a=−1 · a=0 · a=1
The graph f_a(x)=x²−a moves down as a grows. Axis intersections appear after touching at a=0. The figure shows three parameter values; the square’s properties establish the complete classification. Vertical shifts.
Branch · DeeperOptional · may be skipped
Exponential and periodic families
From exponential equations, e^x is positive and takes each positive value exactly once:
(e)^(x) = a: a > 0⇒x = lna; a ≤ 0⇒∅.
For sin x=a on half-open 0≤x<2π there are two roots for |a|<1, one for |a|=1, and none for |a|>1. Half-openness avoids duplicate period endpoints. On the entire line |a|≤1 gives infinitely many roots through periodicity. Keep a fixed while solving for x. Trigonometric equations.
Name the domains of variable and parameter, identify zero denominators and coefficients, solve each branch and check boundaries. Asking which parameters requires a complete partition rather than a few successful examples.