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Graphing Solutions on Slope Fields worksheets

Graph one valid antiderivative curve on a supplied slope field for polynomial, rational, and trigonometric differential equations.

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Sketch one valid solution curve on a connected interval of each supplied slope field.

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Follow the local slopes

Every short field segment gives the derivative at its x-coordinate. A valid solution curve stays tangent to those directions throughout one connected domain interval.

Learning objective

What this activity builds

Connect dy/dx = F(x) to local tangent directions and sketch a domain-safe solution curve.

Worked example

If y'=2x, what curves follow the field?

Every curve y=x²+C follows the field.

Changing C translates the curve vertically without changing its derivative.

Teaching notes

Make the practice count

  • Use the field segments as tangent guides.
  • Do not cross an excluded x-value or pole.
  • Different connected domain intervals may use different constants.

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Choose 4 or 8 free-response tasks. Each task supplies the differential equation and slope field; students sketch one valid solution on a connected interval.

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Math SheetNameScoreTeacherDateGraphing Solutions on Slope FieldsSketch one valid solution curve that follows each supplied slope field on one connected interval.Infinitely many vertical translations are valid; constants may differ across separate domain intervals.1.dydx=3·sin-2-3·x-3-3-2-2-1-1112233xyDomain: all real x.2.dydx=-3-2·x-3-3-3-2-2-1-1112233xyDomain: x is not -3/2; solve on one connected side of the pole.Orange lines mark excluded x-values.3.dydx=2·x2-3-3-2-2-1-1112233xyDomain: all real x.4.dydx=23·sec-3+2·x·tan-3+2·x-3-3-2-2-1-1112233xyDomain: cos(2x-3) is not 0; use one connected interval betweenpoles. Orange lines mark excluded x-values.math-sheet.comPage 1 of 1

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