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ACT · Math · Preparing for Higher Math

Ranges, holes, asymptotes and changing rates

Connect equations to output sets and graph features, including gaps, inverse proportion and periodic patterns.

What you will tackle

  • Derive a polynomial range
  • Distinguish a rational hole from a vertical asymptote
  • Build inverse-proportion models
  • Compare changing graphical slopes
  • Recover a sine equation from a graph

Features include what is excluded

For f(x) = (x − 3)² − 4 over all real x, the square can take every nonnegative value, so the range is [−4,∞). A denominator zero always excludes an input. In 1/(x − 1), values become unbounded near 1: x = 1 is a vertical asymptote. In (x² − 1)/(x − 1), cancellation gives x + 1 only for x ≠ 1; the graph has a removable hole at (1,2), not a vertical asymptote there.

Recover relationships from displays

Inverse proportion y = k/x has constant product xy. Recover k from a known pair and state the meaningful domain. On a graph, equal horizontal steps with increasingly large rises indicate an increasing rate; a straight line has constant rate. For a sine curve, amplitude is half the vertical range, midline is its vertical centre, and period T gives angular factor 2π/T. The initial value and direction help distinguish sine from cosine and phase shifts.

Curves U and VU: (0, 0); (1, 3); (2, 6); (3, 9). V: (0, 0); (0.1, 0.010000000000000002); (0.2, 0.04000000000000001); (0.3, 0.09); (0.4, 0.16000000000000003); (0.5, 0.25); (0.6, 0.36); (0.7, 0.48999999999999994); (0.8, 0.6400000000000001); (0.9, 0.81); (1, 1); (1.1, 1.2100000000000002); (1.2, 1.44); (1.3, 1.6900000000000002); (1.4, 1.9599999999999997); (1.5, 2.25); (1.6, 2.5600000000000005); (1.7, 2.8899999999999997); (1.8, 3.24); (1.9, 3.61); (2, 4); (2.1, 4.41); (2.2, 4.840000000000001); (2.3, 5.289999999999999); (2.4, 5.76); (2.5, 6.25); (2.6, 6.760000000000001); (2.7, 7.290000000000001); (2.8, 7.839999999999999); (2.9, 8.41); (3, 9). Horizontal axis: x; vertical axis: Output. Invented teaching data.002.250.754.51.56.752.2593xOutput U V
Curves U and V · original teaching data; the table/description supplies exact coordinates.
Periodic curve TT: (0, 1); (0.1308996938995747, 1.2610523844401031); (0.2617993877991494, 1.5176380902050415); (0.39269908169872414, 1.7653668647301797); (0.5235987755982988, 2); (0.6544984694978736, 2.2175228580174413); (0.7853981633974483, 2.414213562373095); (0.916297857297023, 2.5867066805824703); (1.0471975511965976, 2.732050807568877); (1.1780972450961724, 2.8477590650225735); (1.3089969389957472, 2.9318516525781364); (1.4398966328953218, 2.982889722747621); (1.5707963267948966, 3); (1.7016960206944713, 2.982889722747621); (1.832595714594046, 2.9318516525781364); (1.9634954084936205, 2.847759065022574); (2.0943951023931953, 2.7320508075688776); (2.2252947962927703, 2.5867066805824703); (2.356194490192345, 2.414213562373095); (2.4870941840919194, 2.2175228580174418); (2.6179938779914944, 2); (2.748893571891069, 1.7653668647301797); (2.8797932657906435, 1.517638090205042); (3.010692959690218, 1.261052384440104); (3.141592653589793, 1.0000000000000002); (3.272492347489368, 0.7389476155598964); (3.4033920413889427, 0.4823619097949584); (3.5342917352885173, 0.23463313526982066); (3.665191429188092, 5.551115123125783e-16); (3.796091123087667, -0.2175228580174413); (3.926990816987241, -0.41421356237309426); (4.057890510886816, -0.5867066805824699); (4.1887902047863905, -0.7320508075688767); (4.319689898685966, -0.8477590650225737); (4.4505895925855405, -0.9318516525781366); (4.581489286485115, -0.9828897227476208); (4.71238898038469, -1); (4.843288674284264, -0.982889722747621); (4.974188368183839, -0.9318516525781368); (5.105088062083413, -0.8477590650225739); (5.235987755982989, -0.7320508075688772); (5.3668874498825625, -0.5867066805824714); (5.497787143782138, -0.41421356237309537); (5.6286868376817125, -0.21752285801744176); (5.759586531581287, -8.881784197001252e-16); (5.8904862254808625, 0.23463313526982088); (6.021385919380436, 0.48236190979495686); (6.152285613280012, 0.7389476155598966); (6.283185307179586, 0.9999999999999996). Horizontal axis: x (radians); vertical axis: T(x). Invented teaching data.-1001.5713.1424.7136.28x (radians)T(x) T
Periodic curve T · original teaching data; the table/description supplies exact coordinates.

Build your notes, then practise

Work on paper before opening an explanation. Guided tasks provide a starting point; independent tasks remove the scaffolding; application tasks ask you to justify a method or interpretation in context.

Guided practice

Guided 1

Find the range of (x − 3)² − 4. Compare the behaviour at x = 1 of 1/(x − 1) and (x² − 1)/(x − 1). In inverse proportion, y = 6 when x = 4; find a rule and y at x = 8.

Show a starting hint

A cancelled factor does not restore its excluded input.

Compare with the explained answer
  1. The square is at least zero and unbounded above, giving [−4,∞).
  2. The first function has a vertical asymptote; the second simplifies to x + 1 with a hole at (1,2).
  3. xy = 24, so y = 24/x and y(8) = 3, with x ≠ 0.

Answer / response criteria: [−4,∞); asymptote versus hole; y = 24/x and 3.

Independent practice

Independent 1

Use the displayed curves U and V at x = 0,1,2,3. Which has constant rate, and which has an increasing rate? The displayed curve T starts at 1, rises to 3 at π/2, falls to −1 at 3π/2 and repeats after 2π. Give a sine equation.

Compare with the explained answer
  1. U has values 0,3,6,9: successive changes are all 3, so its rate is constant.
  2. V has values 0,1,4,9: changes 1,3,5 increase, so its rate increases over the shown positive inputs.
  3. T has midline 1, amplitude 2, period 2π and initially rises from its midline, so T(x) = 1 + 2 sin x fits.

Answer / response criteria: U constant; V increasing rate; T(x) = 1 + 2 sin x.

Application practice

Application 1

A rectangular site has fixed area 120 m². Build the width model for positive length x; find the width when x = 15 and describe the effect of doubling x. A designer instead uses R(x) = (x² − 9)/(x − 3). Explain why the line x + 3 cannot supply R(3).

Compare with the explained answer
  1. The area equation xw = 120 gives w = 120/x with domain x > 0; w(15) = 8 m.
  2. Doubling x halves w; the product remains 120, so this is inverse proportion rather than a constant decrease.
  3. R simplifies to x + 3 only when x ≠ 3. Its original denominator is zero at 3, leaving a hole at (3,6).

Answer / response criteria: w = 120/x, x > 0; 8 m; doubling halves width; R(3) remains undefined.

Recall without your notes

What can you explain now?

A rectangular site has fixed area 120 m². Build the width model for positive length x; find the width when x = 15 and describe the effect of doubling x. A designer instead uses R(x) = (x² − 9)/(x − 3). Explain why the line x + 3 cannot supply R(3).

Compare with the explanation

w = 120/x, x > 0; 8 m; doubling halves width; R(3) remains undefined.

Explain the reasoning before comparing. This review records your self-assessment rather than an official score or mastery decision.

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Source and review status

Coverage follows the named requirements and examples in the 2026–2027 ACT test guide and subject descriptions. That guide explicitly does not list every possible topic. The broader coded College and Career Readiness Standards are also mapped to conceptual teaching and practice; this does not calibrate a learner to a score band. The separate scientific-background context list and reading-complexity rubric have their own stated limits.

Original teaching and explained self-check practice are available. Specialist review and difficulty calibration remain pending. Essays require self-check or teacher review and are not automatically marked or awarded official scores.

Official source specification / guide