Algebra 2: Functions and Linear Systems

1.5 Graphing Linear Functions and Inequalities

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1.5 Graphing Linear Functions and Inequalities

Page 1 · Lines 1–29

1o Linear Tables
2-> Solve for y before building a table.
3-> x + 3y - 6 = 0 gives 3y = -x + 6.
4-> Divide by 3: y = -x/3 + 2.
5-> x values: -6, -3, 0, 3, 6.
6-> y values: 4, 3, 2, 1, 0, respectively.
7-> Plot the ordered pairs; connect with a straight line.
8-> Check: 6x + 2y = 10 gives y = -3x + 5.
9-> Pick x values that make the division easy, like 0 and 6.
10-> Equal x steps of 3 give equal y steps of -1: slope -1/3.
11o Intercepts and Slope
12-> For 3x - 2y = -12, set y = 0 to find x.
13-> 3x = -12 gives the x-intercept (-4, 0).
14-> Set x = 0: -2y = -12 gives (0, 6).
15-> Draw the line through these two intercepts.
16-> In y = (3/2)x - 4, slope is 3/2.
17-> Start at (0, -4); move right 2 and up 3.
18-> Slope means change in y divided by change in x.
19-> The intercepts are (-4, 0) and (0, 6); no other crossings.
20-> Slope from the intercepts is (6 - 0)/(0 - (-4)) = 3/2.
21o Inequality Boundaries
22-> Replace the inequality sign with = for the boundary.
23-> Use a solid line for <= or >=; equality is included.
24-> Use a dashed line for < or >; equality is excluded.
25-> For y < mx + b, shade below the boundary.
26-> For y > mx + b, shade above the boundary.
27-> Multiplying or dividing by a negative reverses the sign.
28-> For x < c, the vertical boundary is x = c; shade left.
29-> For x > c, use a dashed vertical line and shade right.

1.5 Graphing Linear Functions and Inequalities

Page 2 · Lines 30–58

30o Rearrange and Test
31-> 12 - 4y > x gives -4y > x - 12.
32-> Divide by -4: y < -x/4 + 3; dashed, shade below.
33-> 9 + 3y <= 8x gives y <= (8/3)x - 3.
34-> Use a solid boundary and shade below it.
35-> At (3, 5), 9 + 3(5) = 24 and 8(3) = 24.
36-> Equality holds, so (3, 5) is included on the boundary.
37-> Test a point off the line to confirm the shaded side.
38-> Test (0, 0): 9 <= 0 is false, so the shade is below.
39-> The boundary y = -x/4 + 3 passes (0, 3), (4, 2), (8, 1).
40o Weighted Grade Constraints
41-> Let x = test grade and y = homework grade.
42-> With 0 <= x, y <= 100, target: 0.7x + 0.3y >= 90.
43-> Multiply by 10: 7x + 3y >= 900.
44-> Solve: y >= 300 - (7/3)x.
45-> If y = 100, then x >= 600/7, about 85.71.
46-> If x = 100, then y >= 200/3, about 66.67.
47-> Shade above the solid line within the grade square.
48-> y = 300 - (7/3)x has intercept (0, 300), above the square.
49-> On the boundary, raising x by 3 lowers required y by 7.
50o Regions and Functions
51-> A system's solution satisfies every inequality.
52-> Keep only the overlap of all allowed regions.
53-> The grade region includes (100, 100), not (0, 0).
54-> Its corners are (600/7, 100), (100, 100), (100, 200/3).
55-> A function assigns exactly one output to each input.
56-> An inequality region generally allows many y values per x.
57-> At x = 100, both y = 80 and y = 100 satisfy the target.
58-> At x = 600/7 (about 85.7), y = 100 is the cutoff.

1.6.1 Piecewise Functions

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1.6.1 Piecewise Functions

Page 1 · Lines 1–29

1o Choosing a Branch
2-> A piecewise function uses different rules on input intervals.
3-> Choose the branch by the input, not by the output.
4-> First compare x with each branch's interval condition.
5-> Substitute into only the rule whose condition holds.
6-> An included input must have exactly one output.
7-> Check boundary inputs carefully for < versus <=.
8-> An input outside all listed intervals is outside the domain.
9-> A branch condition may test x, or it may name an interval.
10-> With x >= 2 listed, every larger x is already covered.
11o Worked Piecewise Example
12-> Illustrative example using three distinct input intervals.
13-> Define f(x) = 2x for x < 0.
14-> Define f(x) = 3 for 0 <= x < 2.
15-> Define f(x) = -x + 5 for x >= 2.
16-> f(-2) = 2(-2) = -4 using the first branch.
17-> f(1) = 3 using the middle branch.
18-> f(4) = -4 + 5 = 1 using the last branch.
19-> f(0) uses the middle branch because 0 satisfies 0 <= 0.
20-> f(2) uses the last branch because 2 satisfies x >= 2.
21o Restricted Graphs
22-> Graph each rule only on its assigned input interval.
23-> For the example, draw y = 2x only to the left of x = 0.
24-> Mark (0, 0) open; mark (0, 3) closed.
25-> Draw y = 3 from x = 0 up to, but not including, x = 2.
26-> The middle branch has an open endpoint at (2, 3).
27-> The last branch includes (2, 3); the final point is closed.
28-> Thus f(0) = 3 and f(2) = 3, each with one output.
29-> f is increasing, flat, then decreasing across its domain.

1.6.1 Piecewise Functions

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30o Jersey Total Cost
31-> Let n be the number of jerseys; C(n) is total cost.
32-> The domain is positive integers, not all real numbers.
33-> C(n) = 88n for 1 <= n <= 5.
34-> C(n) = 75n for 6 <= n <= 11.
35-> C(n) = 65n for 12 <= n <= 29.
36-> C(n) = 56n for n >= 30.
37-> These are all-units discounts: one rate for the whole order.
38-> The tiers leave no gap: 1-5, 6-11, 12-29, then 30 and up.
39-> Reading the domain first stops you from using 0 jerseys.
40o Jersey Calculations
41-> For 11 jerseys, choose the 6 through 11 tier.
42-> C(11) = 75(11) = 825 dollars.
43-> For 25 jerseys, choose the 12 through 29 tier.
44-> C(25) = 65(25) = 1625 dollars, not 1885.
45-> C(5) = 440, while C(6) = 450.
46-> C(11) = 825, while C(12) = 780: total cost can fall.
47-> C(29) = 1885, while C(30) = 1680.
48-> Rates step down: 88, 75, 65, then 56 per jersey.
49-> At n = 30 the total drops the most: 205 dollars.
50o Rates and Model Checks
51-> An all-units price changes for every item at a tier boundary.
52-> A progressive rate applies each rate only to its own band.
53-> Do not add earlier tier costs in this all-units model.
54-> Unit price is constant within tiers and jumps between them.
55-> Total cost rises within tiers; it is not a step function.
56-> Plot separate integer points; do not fill gaps in the domain.
57-> Tier prices jump; the integer-domain total has tier changes.
58-> Compare the two models at n = 12 before choosing one.

1.6.2 Step Functions

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1.6.2 Step Functions

Page 1 · Lines 1–29

1o Step Function Basics
2-> A step function is constant on each input interval.
3-> Its graph consists of horizontal segments or rays.
4-> Output changes at boundaries between steps.
5-> Outputs form a discrete set in the usual step models.
6-> Inputs may vary continuously across real intervals.
7-> A context may instead allow only discrete inputs.
8-> State the domain from the model, not just the picture.
9-> Inside one step, rise over run is 0, so the slope is 0.
10-> Jumps happen only at interval endpoints, never inside.
11o Greatest Integer Rule
12-> floor(x) is the greatest integer less than or equal to x.
13-> It rounds down toward negative infinity.
14-> floor(10.7) = 10 because 10 <= 10.7 < 11.
15-> floor(-6.35) = -7 because -7 <= -6.35 < -6.
16-> floor(5) = 5; an integer stays unchanged.
17-> floor(-0.2) = -1, not 0.
18-> Truncation toward zero is not floor for negative nonintegers.
19-> floor(2.999) = 2; the value never reaches the next integer.
20-> Writing x = 5 + r with 0 <= r < 1 gives floor(x) = 5.
21o Floor Intervals
22-> For an integer k, floor(x) = k when k <= x < k + 1.
23-> The left endpoint is closed; the right endpoint is open.
24-> y = 0 on 0 <= x < 1; y = 1 on 1 <= x < 2.
25-> y = -2 on -2 <= x < -1.
26-> At x = -1, use the next step: floor(-1) = -1.
27-> The domain of floor(x) is all real numbers.
28-> The range of floor(x) is all integers.
29-> floor(x) = -1 on -1 <= x < 0, not -1 < x < 0.

1.6.2 Step Functions

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30o Transformed Floor Rule
31-> Let f(x) = floor(2x - 1); let k be an integer.
32-> f(x) = k exactly when k <= 2x - 1 < k + 1.
33-> Add 1: k + 1 <= 2x < k + 2.
34-> Divide by 2: (k + 1)/2 <= x < (k + 2)/2.
35-> Each step has width 1/2, closed left and open right.
36-> The domain is all real numbers; the range is all integers.
37-> A positive input coefficient preserves endpoint order.
38-> A coefficient a > 0 makes each step width 1/a.
39-> With 2x - 1 inside, steps break where 2x - 1 is an integer.
40o Transformed Examples
41-> For k = 0, f(x) = 0 on 1/2 <= x < 1.
42-> f(0.75) = floor(0.5) = 0.
43-> At x = 1, f(1) = floor(1) = 1, not 0.
44-> For k = -1, f(x) = -1 on 0 <= x < 1/2.
45-> f(0.25) = floor(-0.5) = -1.
46-> For k = -2, f(x) = -2 on -1/2 <= x < 0.
47-> f(-0.25) = floor(-1.5) = -2.
48-> For k = 1, f(x) = 1 on the next step to the right.
49-> f(0.6) = floor(0.2) = 0, so the input lies in 1/2 to 1.
50o Context and Endpoints
51-> Parking fees can stay constant within each time bracket.
52-> Read each interval and its included endpoints from the source.
53-> Open circles exclude points; closed circles include them.
54-> Two included outputs at one input violate the function rule.
55-> Included branches with the same output still give one value.
56-> Two open endpoints at an input may leave it out of the domain.
57-> Check boundary values before using a step model for a charge.
58-> Two brackets meeting at one time must agree or the rule fails.

1.6.3 Absolute Value Functions

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1.6.3 Absolute Value Functions

Page 1 · Lines 1–29

1o Absolute Value Rule
2-> Absolute value measures distance from zero, never negative.
3-> |x| = x if x >= 0.
4-> |x| = -x if x < 0.
5-> For example, |-5| = -(-5) = 5 and |5| = 5.
6-> The parent graph is a V with vertex (0, 0).
7-> Its domain is all real numbers; its range is y >= 0.
8-> Opposite inputs have equal absolute values.
9-> |x - c| is the distance between the input and the point c.
10-> Factoring: |2x - 6| = 2|x - 3| leaves the zeros unchanged.
11o General Graph Form
12-> Write y = a|x - h| + k, with a != 0.
13-> The vertex is (h, k); the symmetry line is x = h.
14-> The domain is all real numbers.
15-> If a > 0, the V opens upward and the range is y >= k.
16-> If a < 0, the V opens downward and the range is y <= k.
17-> The left branch slope is -a; the right branch slope is a.
18-> Larger |a| makes a steeper V; smaller |a| makes it flatter.
19-> The V meets the x-axis exactly when a*k <= 0.
20-> Points the same distance from h give the same output y.
21o Upward Worked Example
22-> For y = |x - 1| + 3, a = 1, h = 1, and k = 3.
23-> The vertex is (1, 3); the symmetry line is x = 1.
24-> The minimum output is 3; the range is y >= 3.
25-> At x = 0 or x = 2, y = 4.
26-> At x = -1 or x = 3, y = 5.
27-> For x < 1, y = -(x - 1) + 3 = -x + 4.
28-> For x >= 1, y = x - 1 + 3 = x + 2.
29-> x-intercepts: |x - 1| = -3 has none, so no crossings.

1.6.3 Absolute Value Functions

Page 2 · Lines 30–58

30o Downward Worked Example
31-> For y = -(2/3)|x + 3| + 1, h = -3, not 3.
32-> The vertex is (-3, 1); the symmetry line is x = -3.
33-> Since a = -2/3, the maximum output is 1.
34-> The domain is all real numbers; the range is y <= 1.
35-> The left slope is 2/3; the right slope is -2/3.
36-> At x = -6 or x = 0, y = -(2/3)(3) + 1 = -1.
37-> Plot these symmetric points and draw the two rays.
38-> At x = -3 the absolute term vanishes, leaving y = 1.
39-> The y-intercept is (0, -1), one of the symmetric points.
40o Equation Strategy
41-> Isolate the absolute value before splitting into cases.
42-> For |u| = q with q < 0, there are no real solutions.
43-> For q = 0, solve u = 0; for nonconstant linear u, one root.
44-> For q > 0, solve u = q or u = -q.
45-> If u is linear with nonzero slope, q > 0 gives two roots.
46-> For nonlinear u, the number of roots can differ.
47-> Check roots in the original equation for extraneous roots.
48-> Graph view: |u| = q cuts the curve y = |u| at height q.
49-> After splitting, keep each branch with its case condition.
50o Solve and Check
51-> Solve 2|x - 1| + 3 = 11: first get |x - 1| = 4.
52-> x - 1 = 4 or x - 1 = -4 gives x = 5 or x = -3.
53-> Check both: 2(4) + 3 = 11, so both are solutions.
54-> |x + 2| = -3 has no solutions: distance cannot be negative.
55-> |x + 2| = 0 gives x = -2, exactly one solution.
56-> Squaring |x + 2| = -3 would falsely suggest x = 1 or -5.
57-> Both fail the original equation; reject those extra roots.
58-> |x - 2| = 5 gives x = 7 and x = -3, both checked.

1.7.1 Parent Functions

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1.7.1 Parent Functions

Page 1 · Lines 1–29

1o Parent Functions
2-> A parent is a basic function used to build a family.
3-> R means all real numbers; inf means infinity.
4-> Domain lists inputs; range lists outputs.
5-> Square brackets include an endpoint; parentheses do not.
6-> A function assigns each input exactly one output.
7-> A vertical line meets a function graph at most once.
8-> Compare shape, intercepts, domain, and range.
9-> The x-intercept is where y = 0; the y-intercept is x = 0.
10-> Rates of change separate constant, linear, and curved parents.
11o Constant Parent
12-> f(x) = a gives the same output for every input.
13-> Its graph is the horizontal line y = a.
14-> Domain: R. Range: {a}, a single value.
15-> For x = -2, -1, 0, 1, 2, outputs are a, a, a, a, a.
16-> Example: f(x) = 3 has range {3}.
17-> Its y-intercept is (0,a); if a != 0, it has no zero.
18-> If a = 0, every real input is a zero.
19-> Slope is 0, so every secant line is horizontal.
20-> Average rate of change is 0 between any two distinct inputs.
21o Identity Parent
22-> f(x) = x returns the input unchanged.
23-> Its graph is a line through (0,0) with slope 1.
24-> Domain: R. Range: R.
25-> Table inputs: x = -2, -1, 0, 1, 2.
26-> Table outputs: y = -2, -1, 0, 1, 2.
27-> Example: f(-2) = -2 and f(2) = 2.
28-> Both intercepts are (0,0); the function increases.
29-> Average rate of change is 1 on every interval.

1.7.1 Parent Functions

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30o Absolute Value Parent
31-> f(x) = |x| measures distance from zero.
32-> Domain: R. Range: [0,inf).
33-> Table inputs: x = -2, -1, 0, 1, 2.
34-> Table outputs: y = 2, 1, 0, 1, 2.
35-> The V has vertex (0,0) and y-axis symmetry.
36-> For x < 0, |x| = -x; for x >= 0, |x| = x.
37-> The minimum is 0; opposite inputs give equal outputs.
38-> Rate of change: -1 on (-inf,0) and 1 on (0,inf).
39-> It is not linear: slopes jump from -1 to 1 at x = 0.
40o Quadratic Parent
41-> f(x) = x^2 squares each input.
42-> Domain: R. Range: [0,inf).
43-> Table inputs: x = -2, -1, 0, 1, 2.
44-> Table outputs: y = 4, 1, 0, 1, 4.
45-> The parabola opens up with vertex (0,0).
46-> It has y-axis symmetry and a minimum of 0.
47-> Unlike |x|, it curves; both have nonnegative outputs.
48-> From 0 to h != 0, average rate of change is h^2/h = h.
49-> Equal input steps give unequal changes in squared outputs.
50o Choosing Models
51-> Constant: temperature held fixed over a time interval.
52-> Linear: wages W(t) = r*t at a fixed hourly rate r.
53-> Identity is the linear parent; wages need scaling.
54-> Quadratic: hypothetical projectile height H(t).
55-> H(t) = -t^2 + 4*t; 0 <= t <= 4; range: [0,4].
56-> Absolute value: |s-c| is distance from reference c.
57-> s is signed position, not total distance traveled.
58-> Wages at $12 per hour: W(30) = 12*30 = 360 dollars.

1.7.2 Translations of Functions

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1.7.2 Translations of Functions

Page 1 · Lines 1–29

1o Translation Rule
2-> A translation moves a graph without changing its shape.
3-> Write g(x) = f(x-h) + k for parent function f.
4-> h is the horizontal shift; k is the vertical shift.
5-> h > 0 moves right; h < 0 moves left.
6-> k > 0 moves up; k < 0 moves down.
7-> Inside signs look opposite; outside signs act the same.
8-> A parent point (u,v) becomes (u+h,v+k).
9-> Translations preserve the size of the domain and range.
10-> Every point shifts the same way; the graph never flips.
11o Reading Shifts
12-> g(x) = (x+2)^2 - 4 uses parent f(x) = x^2.
13-> x+2 = x-(-2), so h = -2 and k = -4.
14-> Shift left 2 and down 4; vertex: (-2,-4).
15-> Parent (1,1) becomes (-1,-3).
16-> g(x) = |x-7| + 1 shifts right 7 and up 1.
17-> Its vertex is (7,1); (1,1) becomes (8,2).
18-> Check: g(8) = |8-7| + 1 = 2.
19-> For g(x) = |x-7| + 1, the range is [1,inf).
20-> The parent range [0,inf) shifts to [-4,inf).
21o Writing Equations
22-> Begin with the parent, then insert the requested shifts.
23-> Quadratic right 10 and up 4: h = 10, k = 4.
24-> The equation is g(x) = (x-10)^2 + 4.
25-> The vertex moves from (0,0) to (10,4).
26-> Parent (2,4) moves to (12,8).
27-> Check: g(12) = (12-10)^2 + 4 = 8.
28-> Shifts preserve the parabola's width and opening.
29-> Left 10 and up 4 gives h = -10 and k = 4.

1.7.2 Translations of Functions

Page 2 · Lines 30–58

30o Quadratic Worked Points
31-> For g(x) = (x+5)^2 - 3, h = -5 and k = -3.
32-> Translate the parent left 5 and down 3.
33-> (0,0) becomes vertex (-5,-3).
34-> (1,1) becomes (-4,-2).
35-> (2,4) becomes (-3,1).
36-> Check: g(-3) = 2^2 - 3 = 1.
37-> Domain: R. Range: [-3,inf); axis: x = -5.
38-> Symmetry pairs: (-6,-2) and (-4,-2) share output -2.
39-> The x-intercept: (x+5)^2 = 3, so x = -5 +/- sqrt(3).
40o Absolute Value Worked Points
41-> For g(x) = |x+8| - 5, h = -8 and k = -5.
42-> Translate left 8 and down 5; vertex: (-8,-5).
43-> Parent (-1,1) becomes (-9,-4).
44-> Parent (1,1) becomes (-7,-4).
45-> Domain: R. Range: [-5,inf).
46-> The axis of symmetry is x = -8.
47-> Check: g(-7) = |-7+8| - 5 = -4.
48-> Each pair equidistant from x = -8 gives one output.
49-> x-intercepts: |x+8| = 5 gives x = -3 and x = -13.
50o Factoring and Checks
51-> Read h from a fully factored inside expression.
52-> With a scale, use b(x-h), not the raw constant.
53-> Example: 2x-6 = 2(x-3), so h = 3, not 6.
54-> This combines horizontal scaling with a shift.
55-> For pure translations, the inside coefficient is 1.
56-> Shift domain horizontally and range vertically.
57-> Verify a vertex and a second point by substitution.
58-> Factored check: x^2 - 6x + 9 = (x-3)^2, so h = 3, k = 0.

1.7.3 Dilations and Reflections

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1.7.3 Dilations and Reflections

Page 1 · Lines 1–29

1o Vertical Dilations
2-> In g(x) = a*f(x), multiply each output by a.
3-> Use a != 0 for the dilation rules here.
4-> |a| > 1 stretches vertically by |a|.
5-> 0 < |a| < 1 compresses vertically by |a|.
6-> a < 0 also reflects across the x-axis.
7-> A parent point (u,v) becomes (u,a*v).
8-> x stays fixed; a negative a reverses output signs.
9-> Zeros persist: f(c) = 0 gives g(c) = a*f(c) = 0.
10-> The y-intercept is not fixed: it becomes a*f(0) at x = 0.
11o Horizontal Dilations
12-> In g(x) = f(b*x), use b != 0.
13-> The horizontal factor is 1/|b|, not |b|.
14-> |b| > 1 compresses horizontally by 1/|b|.
15-> 0 < |b| < 1 stretches horizontally by 1/|b|.
16-> b < 0 also reflects across the y-axis.
17-> A parent point (u,v) becomes (u/b,v).
18-> Solve b*x = u to locate the new input x = u/b.
19-> Outputs never move; only the input coordinate is rescaled.
20-> With b = 1/2, stretch by 2; with b = 2, compress by 1/2.
21o Linear Worked Examples
22-> Use identity f(x) = x as the parent.
23-> g(x) = -(2/5)*x has a = -2/5.
24-> Compress vertically by 2/5 and reflect across x-axis.
25-> (5,5) becomes (5,-2); check: -(2/5)*5 = -2.
26-> g(x) = -3*x has a = -3.
27-> Stretch vertically by 3 and reflect across x-axis.
28-> (1,1) becomes (1,-3); both graphs pass through (0,0).
29-> The slope of the image line equals a times the parent slope.

1.7.3 Dilations and Reflections

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30o Even Parent Examples
31-> g(x) = (-2.5*x)^2 uses f(x) = x^2, b = -2.5.
32-> Compress horizontally by 2/5 and reflect across y-axis.
33-> Since x^2 is even, that reflection is invisible.
34-> (2,4) becomes (-0.8,4); g(x) = 6.25*x^2.
35-> g(x) = (2/3)*|-x| reflects across y-axis.
36-> It also compresses vertically by 2/3.
37-> Since |-x| = |x|, this reflection is also invisible.
38-> The same trick works for |x| and every other even parent.
39-> Even symmetry gives f(-b*x) = f(b*x), not f(b*x) = f(x).
40o Combining Two Dilations
41-> In g(x) = a*f(b*x), use nonzero a and b.
42-> Map (u,v) to (u/b,a*v); divide input, scale output.
43-> Reflect x, compress horizontally 1/2, stretch vertically 8.
44-> For x^2, choose a = -8 and b = 2.
45-> g(x) = -8*(2*x)^2 = -32*x^2.
46-> For g(x) = -4*|2*x|, (-2,2) becomes (-1,-8).
47-> The other parent point (2,2) becomes (1,-8).
48-> For a power parent, the coefficient becomes a*b^k.
49-> The sign of b is free for even parents: -2 works like 2.
50o Equivalent Descriptions
51-> g(x) = (-3*x)^2 = 9*x^2.
52-> Inside form: compress horizontally 1/3; reflect y-axis.
53-> Outside form: stretch vertically by 9, with no reflection.
54-> Both descriptions produce the same graph.
55-> Even parents satisfy f(-x) = f(x).
56-> Their y-axis reflections can be invisible.
57-> Parameter representations need not be unique.
58-> Simplify first: (-3*x)^2 = 9*x^2 keeps a positive a.

1.7.3 Combined Transformations

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1.7.3 Combined Transformations

Page 1 · Lines 1–29

1o Combined Rule
2-> Use g(x) = a*f(b*(x-h)) + k with a,b != 0.
3-> Map parent (u,v) to (u/b+h,a*v+k).
4-> Divide by b; do not multiply the input by b.
5-> Dilation and reflection come before the shifts.
6-> a controls vertical scale and x-axis reflection.
7-> b controls reciprocal horizontal scale and y-reflection.
8-> h shifts horizontally; k shifts vertically.
9-> Test the rule with the parent vertex or any known point.
10-> Domain and range come from the parent; k moves the range.
11o Factor Before Reading
12-> Factor the entire inside to find b and h.
13-> 0.4*x+2 = 0.4*(x+5), so b = 0.4, h = -5.
14-> The shift is left 5, not left 2.
15-> g(x) = -(0.4*x+2) uses identity f(x) = x.
16-> Here a = -1, b = 0.4, h = -5, k = 0.
17-> Stretch horizontally by 2.5; reflect x; shift left 5.
18-> Equivalent: g(x) = -0.4*x-2; (0,0) maps to (-5,0).
19-> Always factor out b first, then read the leftover constant.
20-> Confirm with a value: g(0) = -2 matches the expanded form.
21o Absolute Value Combined
22-> g(x) = -(2/3)*|x+3| + 1 uses a = -2/3, b = 1.
23-> Reflect x; compress vertically 2/3; left 3; up 1.
24-> h = -3, k = 1, so the vertex is (-3,1).
25-> (3,3) maps to (0,-1); check: g(0) = -2+1 = -1.
26-> The V opens down; its maximum output is 1.
27-> Domain: R. Range: (-inf,1].
28-> With b = 1 only, the mapping is (u+h,a*v+k).
29-> Its ray slopes are 2/3 on the left and -2/3 on the right.

1.7.3 Combined Transformations

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30o Quadratic Combined
31-> Start with f(x) = x^2.
32-> Open down, stretch vertically 4, right 2, down 5.
33-> Choose a = -4, b = 1, h = 2, k = -5.
34-> The equation is g(x) = -4*(x-2)^2 - 5.
35-> Vertex: (2,-5); axis: x = 2; maximum: -5.
36-> (1,1) maps to (3,-9); check: g(3) = -4-5 = -9.
37-> Domain: R. Range: (-inf,-5]; there are no real zeros.
38-> Expand to confirm: g(x) = -4*x^2 + 16*x - 21.
39-> The y-intercept (0,-21) lies 16 units below the maximum.
40o Dolphin Model
41-> g(x) = -0.125*(x-12)^2 + 18 models height.
42-> Vertex: (12,18); maximum height: 18 distance units.
43-> Zero height: 0 = -0.125*(x-12)^2 + 18.
44-> Thus (x-12)^2 = 144; x-12 = +/-12.
45-> Zeros: x = 0 and x = 24; symmetry axis: x = 12.
46-> Context domain: [0,24]; context range: [0,18].
47-> The packet does not specify distance units.
48-> Spot check the middle: g(12) = 18, the peak of the arc.
49-> At x = 6 the height is 13.5, three quarters of 18.
50o Reflection and Point Check
51-> g(x) = -2*|-(x-5)| + 4 has a = -2, b = -1.
52-> Reflect y then x; stretch vertically 2; right 5; up 4.
53-> h = 5, k = 4; vertex: (5,4); range: (-inf,4].
54-> (0,0) maps to (5,4); domain: R.
55-> (-1,1) maps to (6,2); (1,1) maps to (4,2).
56-> Absolute value hides the y-axis reflection.
57-> Check each mapped point by substituting into g(x).
58-> Verify the far ray: g(9) = -2*4 + 4 = -4.

2.3 Equations of Linear Functions

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2.3 Equations of Linear Functions

Page 1 · Lines 1–29

1o Standard Form
2-> Standard form is Ax + By = C with A, B not both zero.
3-> Convention: A >= 0; A, B, C integers; gcd 1.
4-> gcd is greatest common factor; clear fractions when rational.
5-> Example: y = (2/5)x + 14 becomes 2x - 5y = -70.
6-> Subtract (2/5)x: -(2/5)x + y = 14.
7-> Multiply by -5: 2x - 5y = -70.
8-> Here A = 2, B = -5, and C = -70.
9-> Check: -2x + 2y = 2 is not standard; divide by 2.
10-> It reduces to -x + y = 1; multiply by -1: x - y = -1.
11o Slope Intercept
12-> Slope-intercept form is y = mx + b.
13-> m is the slope; b is the y-intercept, not standard B.
14-> Solve the equation for y before reading either value.
15-> Example: 12x - 4y = 24.
16-> Subtract 12x: -4y = -12x + 24.
17-> Divide by -4: y = 3x - 6, so m = 3 and b = -6.
18-> Check 4x = -2y + 22.
19-> Add 2y: 4x + 2y = 22, then 2y = -4x + 22.
20-> Thus y = -2x + 11; a positive 2 is wrong.
21o Point Slope
22-> Point-slope form is y - y1 = m(x - x1).
23-> m is the slope and (x1, y1) is one point on the line.
24-> Example: slope 7 through (2, -3).
25-> Substitute: y - (-3) = 7(x - 2).
26-> Simplify the left side: y + 3 = 7(x - 2).
27-> Expand and solve for y to reach slope-intercept form.
28-> y + 3 = 7x - 14 gives y = 7x - 17.
29-> Check (2, -3): 7(2) - 17 = -3, as required.

2.3 Equations of Linear Functions

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30o Two Points
31-> Given two points, first compute the slope.
32-> m = (y2 - y1)/(x2 - x1), provided x2 != x1.
33-> Example: (1, 4) and (5, 16).
34-> m = (16 - 4)/(5 - 1) = 12/4 = 3.
35-> With (1, 4): y - 4 = 3(x - 1).
36-> With (5, 16): y - 16 = 3(x - 5).
37-> Both simplify to y = 3x + 1.
38-> Check (1, 4): 3 + 1 = 4.
39-> Check (5, 16): 15 + 1 = 16.
40o Context Models
41-> Define the input and output with units before writing.
42-> Example: a gym charges $40 to join and $25 per month.
43-> Let x be months and y be total cost in dollars.
44-> The fixed fee is the y-intercept: b = 40.
45-> The monthly rate is the slope: m = 25.
46-> Model: y = 25x + 40.
47-> After 6 months, y = 25(6) + 40 = 190 dollars.
48-> The point (6, 190) satisfies the model.
49-> Use the model only while the rate and fee stay fixed.
50o Model Limits
51-> A linear model assumes a constant rate of change.
52-> Real prices, populations, and demand often change rates.
53-> Extrapolation far beyond the data is a claim, not a fact.
54-> Rounded slopes make predictions approximate, not exact.
55-> Example: slope 4/3 rounded to 1.33 loses precision.
56-> From (0, 2), x = 30 gives (4/3)(30) + 2 = 42 exactly.
57-> The rounded model gives 1.33(30) + 2 = 41.9.
58-> State the domain and the rounding when reporting a result.

2.4 Solving Systems of Equations Graphically

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2.4 Solving Systems of Equations Graphically

Page 1 · Lines 1–29

1o Graphing Basics
2-> A system of equations shares the same two variables.
3-> Graph both related lines on one coordinate plane.
4-> An intersection point satisfies both equations.
5-> Plot intercepts or use slope from the y-intercept.
6-> Example: y = 2x - 1 and y = -x + 5.
7-> First line: (0, -1), then up 2 and right 1.
8-> Second line: (0, 5), then down 1 and right 1.
9-> The lines meet at (2, 3).
10-> Check: 2(2) - 1 = 3 and -(2) + 5 = 3.
11o Classification
12-> One intersection means one solution.
13-> The system is consistent and independent.
14-> Identical lines mean infinitely many solutions.
15-> The system is consistent and dependent.
16-> Parallel distinct lines mean no solution.
17-> The system is inconsistent.
18-> Different slopes: one solution.
19-> Same slope and same intercept: infinitely many.
20-> Same slope and different intercepts: none.
21o One Solution
22-> Example: 3x + y = 10 and x - y = 2.
23-> Solve each for y: y = -3x + 10 and y = x - 2.
24-> Graph or equate: -3x + 10 = x - 2.
25-> Add 3x: 10 = 4x - 2; add 2: 12 = 4x.
26-> So x = 3 and y = 3 - 2 = 1.
27-> The solution is (3, 1).
28-> Check both: 3(3) + 1 = 10 and 3 - 1 = 2.
29-> Always verify in both original equations, not just one.

2.4 Solving Systems of Equations Graphically

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30o No Solution
31-> Example: y = 4x + 1 and y = 4x - 3.
32-> The slopes are equal but the intercepts differ.
33-> The lines are parallel and never meet.
34-> Equating gives 4x + 1 = 4x - 3.
35-> Subtract 4x: 1 = -3, which is false.
36-> A false statement signals no solution.
37-> Do not invent an intersection from a rough sketch.
38-> A true statement such as 0 = 0 signals dependent lines.
39-> Then every point on the shared line solves the system.
40o Technology
41-> Solve each equation for y before entering it.
42-> Example: 2.5y - 4x = 7.5 and y + 0.5x = 6.
43-> First: y = 1.6x + 3.
44-> Second: y = -0.5x + 6.
45-> Enter both in a graphing tool and find the intersection.
46-> The tool reports approximately (1.43, 5.29).
47-> Check with exact fractions: x = 10/7.
48-> Then y = -5/7 + 6 = 37/7.
49-> The decimal display is rounded, not the exact solution.
50o Equation as System
51-> A single equation can be split at the equals sign.
52-> Set each side equal to y to form a system.
53-> Example: 5x - 2 = 13 - 3x.
54-> System: y = 5x - 2 and y = 13 - 3x.
55-> Equate: 5x - 2 = 13 - 3x.
56-> Add 3x and 2: 8x = 15, so x = 15/8.
57-> Check: 5(15/8) - 2 = 75/8 - 16/8 = 59/8.
58-> Right side: 13 - 45/8 = 104/8 - 45/8 = 59/8.

2.5 Solving Systems of Equations Algebraically

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2.5 Solving Systems of Equations Algebraically

Page 1 · Lines 1–29

1o Substitution
2-> Solve one equation for one variable.
3-> Substitute that expression into the other equation.
4-> Solve the resulting one-variable equation.
5-> Back-substitute to find the other variable.
6-> Example: x + 3y = 7 and 2x - y = 0.
7-> From the second equation, y = 2x.
8-> Substitute: x + 3(2x) = 7.
9-> Then 7x = 7, so x = 1 and y = 2.
10-> Check: 1 + 3(2) = 7 and 2(1) - 2 = 0.
11o Elimination
12-> Line up like terms in both equations.
13-> Add or subtract to remove one variable.
14-> Example: 3x + 4y = 11 and 3x - 2y = 5.
15-> Subtract: 6y = 6, so y = 1.
16-> Substitute into the first: 3x + 4 = 11.
17-> Then 3x = 7 and x = 7/3.
18-> The solution is (7/3, 1).
19-> Check the second: 3(7/3) - 2 = 7 - 2 = 5.
20-> Opposite terms add away; equal terms subtract away.
21o Scaling
22-> Multiply equations by nonzero constants to match terms.
23-> Example: 2x + 3y = 8 and 5x - 2y = 1.
24-> Multiply by 2 and 3: 4x + 6y = 16.
25-> The second becomes 15x - 6y = 3.
26-> Add: 19x = 19, so x = 1.
27-> Then 2(1) + 3y = 8 gives y = 2.
28-> Check: 5(1) - 2(2) = 1.
29-> Scaling changes the form, not the solution set.

2.5 Solving Systems of Equations Algebraically

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30o Special Outcomes
31-> Example: 2x + 4y = 6 and x + 2y = 3.
32-> The second is the first divided by 2.
33-> Same line: infinitely many solutions.
34-> Multiply the second by -2 and add: 0 = 0.
35-> Example: 2x + 4y = 6 and x + 2y = 4.
36-> Proportional left sides, different constants.
37-> Multiply the second by -2 and add: 0 = -2.
38-> This is false, so there is no solution.
39-> A true identity is dependent; a contradiction is inconsistent.
40o Mixture Model
41-> Track total amount and the active ingredient.
42-> Example: blend 12 L of 20% juice with 8 L of 45% juice.
43-> Total 20 L; juice: 0.20(12) + 0.45(8) = 6 L.
44-> Concentration: 6/20 = 0.30, or 30%.
45-> Reverse problem: how much 45% juice makes 20 L at 30%?
46-> Let x be liters of 45% juice; then 20 - x is 20% juice.
47-> 0.45x + 0.20(20 - x) = 6.
48-> 0.25x + 4 = 6 gives x = 8 L of 45% juice.
49-> Then 12 L of 20% juice; check 0.45(8) + 0.20(12) = 6.
50o Checks
51-> Substitute into both original equations; one is not enough.
52-> Check units and whether the answer is viable.
53-> Example: (3, -1) in 2x + y = 5 and x - y = 4.
54-> First: 2(3) - 1 = 5, true.
55-> Second: 3 - (-1) = 4, true.
56-> Negative amounts can satisfy algebra but fail context.
57-> For whole items, check that an integer solution exists.
58-> Report the solution with the variable meanings.

2.6 Solving Systems of Inequalities

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2.6 Solving Systems of Inequalities

Page 1 · Lines 1–29

1o Boundary Lines
2-> Replace each inequality sign with = to get a boundary.
3-> Solid boundary: <= or >= includes the line.
4-> Dashed boundary: < or > excludes the line.
5-> Shade the half-plane that satisfies the inequality.
6-> Test a point not on the boundary.
7-> Example: y > 2x - 3.
8-> Test (0, 0): 0 > -3 is true, so shade above.
9-> Example: x < 2 uses a dashed vertical boundary at x = 2.
10-> Points on x = 2 are excluded; shade left.
11o Sign Reversal
12-> Multiplying or dividing by a negative reverses the sign.
13-> Example: -3y <= 12 becomes y >= -4.
14-> Dividing by -3 flips <= to >=.
15-> Example: 2y > -4x + 8.
16-> Divide by 2: y > -2x + 4, with the sign unchanged.
17-> Rewrite 5 - 2y < x by isolating y carefully.
18-> -2y < x - 5, then y > -0.5x + 2.5.
19-> The division by -2 flips < to >.
20-> Test (0, 3): 5 - 6 = -1 < 0 is true.
21o Overlap
22-> The feasible region satisfies every inequality at once.
23-> Shade each half-plane lightly, then keep the common part.
24-> Example: y >= x and y <= 4.
25-> The overlap is above y = x and below y = 4.
26-> Point (1, 3) satisfies both: 3 >= 1 and 3 <= 4.
27-> Point (5, 2) fails y >= x.
28-> Example: y > x and y < x - 2 has no overlap.
29-> Empty overlap means the system has no solution.

2.6 Solving Systems of Inequalities

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30o Bounded Region
31-> A bounded region fits inside a finite polygon.
32-> Bounded describes extent, not whether edges are included.
33-> A dashed boundary excludes its points, so it is not closed.
34-> Example: x >= 0, y >= 0, x + y <= 6.
35-> The vertices are (0, 0), (6, 0), and (0, 6).
36-> Points on x + y = 6 are included by <=.
37-> The region is a triangle with finite area.
38-> Example with x > 0 excludes the y-axis edge.
39-> It excludes (0, 0) and (0, 6), but stays bounded.
40o Unbounded Region
41-> An unbounded region extends without limit.
42-> Example: y >= x + 1 and x >= 0.
43-> The region continues up and to the right forever.
44-> Its only vertex is (0, 1).
45-> A single vertex does not make the region bounded.
46-> Test points: (0, 2) and (10, 20) both satisfy.
47-> Boundedness concerns the whole region, not one corner.
48-> Parallel boundaries may leave a strip or no overlap.
49-> Example: y > x and y < x + 2 gives an unbounded strip.
50o Real Constraints
51-> Count variables in whole items; add nonnegative constraints.
52-> Example: x caps and y shirts, with x + y <= 40.
53-> If at least 10 caps are needed, add x >= 10.
54-> If shirts cost $8 and caps $5, $300 gives 5x + 8y <= 300.
55-> The feasible region is the overlap of all constraints.
56-> Check a candidate such as (20, 12): 20 + 12 <= 40.
57-> Cost: 5(20) + 8(12) = 196 <= 300.
58-> A fractional point may be infeasible if items are discrete.

2.7 Optimization with Linear Programming

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2.7 Optimization with Linear Programming

Page 1 · Lines 1–29

1o Objective and Constraints
2-> Define variables and state their units.
3-> Constraints are inequalities the variables must satisfy.
4-> The feasible region is the overlap of all constraints.
5-> The objective is the quantity to maximize or minimize.
6-> Example: maximize P = 3x + 2y.
7-> Constraints: x >= 0, y >= 0, x + y <= 8, x <= 5.
8-> Each vertex is an intersection of two boundary lines.
9-> A vertex must satisfy every constraint, not just two lines.
10-> Check candidate vertices in all constraints before use.
11o Bounded Case
12-> Example: maximize P = 4x + 5y.
13-> Constraints: x >= 0, y >= 0, x + 2y <= 10, 3x + y <= 15.
14-> Boundaries meet at (0, 0), (5, 0), (4, 3), and (0, 5).
15-> Check (4, 3): 4 + 6 = 10 and 12 + 3 = 15.
16-> (0, 0): P = 0.
17-> (5, 0): P = 20.
18-> (4, 3): P = 16 + 15 = 31.
19-> (0, 5): P = 25.
20-> The maximum is 31 at (4, 3).
21o Vertex Test
22-> On a nonempty closed bounded polygon, a linear objective
23-> attains both extremes at vertices; otherwise check existence.
24-> Evaluate every vertex; do not stop at the first maximum.
25-> Example: minimize C = 2x + 7y on the same region.
26-> (0, 0): C = 0.
27-> (5, 0): C = 10.
28-> (4, 3): C = 8 + 21 = 29.
29-> (0, 5): C = 35; the minimum is 0 at (0, 0).

2.7 Optimization with Linear Programming

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30o Unbounded Case
31-> An unbounded region may lack a maximum or minimum.
32-> Example: minimize C = 2x + 3y.
33-> Constraints: x >= 0, y >= 0, x + y >= 4.
34-> The vertices are (0, 4) and (4, 0).
35-> C(0, 4) = 12 and C(4, 0) = 8.
36-> At x = 0, C = 3y grows without bound, so no maximum exists.
37-> Since x + y >= 4, C = 2(x + y) + y >= 8.
38-> Thus the minimum is 8 at (4, 0).
39-> One extra test point cannot prove that no maximum exists.
40o Application
41-> Each table needs 4 board feet; each chair needs 2.
42-> Only 24 board feet are available.
43-> At least 2 tables and at least 3 chairs are required.
44-> Profit: $30 per table and $20 per chair.
45-> Maximize P = 30t + 20c.
46-> Constraints: 4t + 2c <= 24, t >= 2, c >= 3; t, c integers.
47-> Real-relaxation vertices: (2, 3), (2, 8), and (4.5, 3).
48-> Check (2, 3): 8 + 6 = 14 <= 24.
49-> P values: 120, 220, and 195.
50o Check and Interpret
51-> The relaxed maximum is 220 at integer point (2, 8).
52-> Integer points form a feasible subset of the relaxation.
53-> So no feasible integer point can exceed 220.
54-> Rounding a fractional optimal vertex is not a proof.
55-> If needed, enumerate all feasible integer points.
56-> Report the number of tables, chairs, and profit with units.
57-> State whether the region is bounded and the extreme claimed.
58-> Confirm every constraint at the chosen point.