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Cubic and reciprocal graphs GCSE lesson: shapes, roots and asymptotes

Name: ____________________ Date: ____________________

Show your working. Check every root by substitution and label your sketches.

Cubic and reciprocal graphs: practice

Show your working: for each bracket, write the x that makes it zero. The stretch is optional.

Diagnostic question: choose and explain

  1. Where does y = (x − 2)(x + 1)(x + 3) cross the axes?
    1. The x-axis at 2, −1 and −3; the y-axis at −6
    2. The x-axis at −2, 1 and 3; the y-axis at 6
    3. The x-axis at 2, −1 and −3; the y-axis at 6
    4. The x-axis at −2, 1 and 3; the y-axis at −6

Practice questions

  1. On y = x³, when x = −3, y = ___
  2. On y = 12/x, when x = 4, y = ___, and when x = −2, y = ___
  3. y = (x + 4)(x − 1)(x − 5) crosses the x-axis at the smallest x = ___, then x = ___, then the largest x = ___
  4. Where does y = (x + 4)(x − 1)(x − 5) cross the y-axis? Give the y-value.
  5. Can the graph of a cubic miss the x-axis altogether?
    1. No: one end is far below the axis and the other far above it
    2. Yes, like y = x² + 1 does
    3. Yes, if it has no brackets
  6. Which is true of the graph of y = 1/x?
    1. It never meets either axis
    2. It goes through (0, 0)
    3. It crosses the x-axis at x = 1
  7. y = −6/x. Where are the two branches of its graph?
    1. Top left and bottom right
    2. Top right and bottom left
    3. Top left and top right
    4. One branch, through the origin
  8. Complete the table for y = x³ − 2x: when x = −1, y = ___, and when x = 2, y = ___
  9. Stretch: y = (x − 2)²(x + 1) touches the x-axis at x = ___ and crosses it at x = ___

Support: For question 3: a bracket such as (x + 7) is 0 when x = −7, so each root has the opposite sign to the number in its bracket. For question 4, put x = 0 into every bracket and multiply, signs and all.

Cubic and reciprocal graphs: exit

A sketch with the crossings marked, and one sentence.

  1. Sketch y = (x + 1)(x − 2)(x − 3). Mark where it crosses the x-axis and the y-axis, and say in one sentence how you know which way its ends go.

Exit check

One sketch shows who reads roots, the y-intercept and the ends correctly.

Sketch y = (x + 1)(x − 2)(x − 3) and mark where it crosses both axes.

Cubic and reciprocal graphs: tables, roots and sketches

Show your working. Draw each curve smoothly, never with straight lines between the points, and label where it crosses the axes.

1. Complete a table for y = x³ − 3x for x = −2, −1, 0, 1, 2, then plot the graph.

2. Where does y = (x + 3)(x − 1)(x − 2) cross the x-axis? Where does it cross the y-axis?

3. Complete a table for y = 8/x for x = −8, −4, −2, −1, 1, 2, 4, 8, then plot the graph.

4. Explain why the graph of y = 1/x never crosses the x-axis, and why it has no point on the y-axis.

5. Sketch y = x³ and y = −x³ on one pair of axes. Where do they meet?

6. A cubic crosses the x-axis at −2, 1 and 3, and starts low on the left. Write its equation in brackets, and find where it crosses the y-axis.

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