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Cambridge IGCSE0607

Sketching graphs on a calculator

International Mathematics 0607 Chapter Notes

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Sketching graphs on a calculator
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In this chapter

  1. 1Sketching a graph on the calculator
  2. 2Table of values and plotted points
  3. 3Zeros, local maxima, local minima and the vertex
  4. 4Intersection of two graphs

1. Sketching a graph on the calculator

Use a graphic display calculator to sketch the graph of a function. Type the function into the graph menu, then set the window to the range of x the question gives.

If the shape looks wrong, the window is wrong. Change the range of x and y until the whole curve and its key points are on screen.

Copy the shape onto the paper. Show the axes, and mark the points the question asks about. This includes unfamiliar functions not mentioned explicitly in the syllabus.

Examiner insight

“Many candidates produced a suitable sketch here, although the shapes seen would suggest that not all had set suitable ranges on their calculators.” “Some candidates sketched the wrong graph.”

Worked example 13 marks

Sketch the graph of y = x³ - 3x for values of x between -3 and 3. Mark the local maximum and the local minimum.

  1. 1

    Enter y = x³ - 3x. Set the window -3 ≤ x ≤ 3 and y from -20 to 20 so the whole curve shows.

  2. 2

    Cubic shape: rises, falls, then rises again, through the origin [1]

  3. 3

    Local maximum read from the calculator: (-1, 2) [1]

  4. 4

    Local minimum read from the calculator: (1, -2) [1]

2. Table of values and plotted points

A graphic display calculator can produce a table of values for a function. Set the first value of x and the step, then read off each y.

It can also plot points from a table of values or a list. Enter the x values and the y values and plot them.

When a question says sketch, do not work out points by hand and plot them. Sketch from the graph on the calculator.

Examiner insight

“Many candidates were plotting points whilst others were just guessing what the curve looked like.” “A number of candidates plotted points rather than using the graphic display calculator.”

Worked example 13 marks

Make a table of values for y = x² - 2x - 1 for x = 0, 1, 2, 3, 4. Write down the points to plot.

  1. 1

    Enter the function. Open the table with start 0 and step 1.

  2. 2

    y = -1, -2, -1, 2, 7 [2]

  3. 3

    Points: (0, -1), (1, -2), (2, -1), (3, 2), (4, 7) [1]

3. Zeros, local maxima, local minima and the vertex

Use the calculator to find zeros, local maxima and local minima. A zero is a value of x where the graph crosses the x-axis. Use the calculator's zero, maximum and minimum functions.

Do not trace along the curve. The built-in functions give values to the accuracy needed.

The vertex of a quadratic is its local maximum or its local minimum. Read off both coordinates when the question asks for a point.

zero: y = 0

line of symmetry of a quadratic: x = x-coordinate of the vertex

Examiner insight

“Again, candidates did not use their calculators properly, using the trace function rather than the built in max/min function.” “Many candidates correctly found the co-ordinates of the minimum point, although some presumably used the trace facility on the calculator and reached an approximate value, for instance (1.9, 3.2) instead of the exact answer.”

Worked example 13 marks

For the graph of y = 2x² - 5x + 1 for values of x between -2 and 4:(b) Find the x-coordinates of the points where the graph crosses the x-axis.(c) Find the x-coordinate of the local minimum.

  1. 1

    Enter y = 2x² - 5x + 1 and use the zero function.

  2. 2

    x = 0.219 (or 0.2192…) [1]

  3. 3

    x = 2.28 (or 2.280…) [1]

  4. 4

    Use the minimum function: x = 1.25 [1]

4. Intersection of two graphs

To solve an equation with a graph, sketch each side as its own function. The intersection of the graphs gives the solution.

Use the calculator's intersection function. Read the x-coordinate for a solution. Read both coordinates when the question asks for a point.

Show the sketch on your paper with both graphs drawn. Without it, a correct answer may not earn the marks for the method.

Examiner insight

“A large number of candidates gave the correct answer but did not show any evidence that a graphical method had been used.” “Many candidates tried to use an algebraic approach rather than use their sketch to solve the inequality.”

Worked example 13 marks

You have sketched y = x² + 2 and y = x³ + x² - 5x + 2 for values of x between -3 and 3.(c) Use your graph to solve this equation. x³ + x² - 5x + 2 = x² + 2

  1. 1

    Both graphs are on the same axes for -3 ≤ x ≤ 3. The solutions are the x-coordinates of the intersections.

  2. 2

    Use the intersection function three times.

  3. 3

    x = 0 [1]

  4. 4

    x = -2.24 [1]

  5. 5

    x = 2.24 [1]

Recap

  • Sketch: enter the function and set the window to the range of x in the question.
  • Table of values: set the start and the step. Asked to sketch, do not plot points by hand.
  • Zeros: values of x where y = 0. Use the zero function.
  • Local maximum and local minimum: use the built-in functions, never trace.
  • Vertex of a quadratic: its local maximum or local minimum. Give both coordinates.
  • Intersection: sketch both sides, then use the intersection function.
  • Show the sketch. Give answers to 3 significant figures unless told otherwise.

End-of-chapter exercise

Test yourself on the whole chapter. Work through these before moving on.

  1. A graph is a U shape. State the calculator function used to find its lowest point accurately.1 mark
  2. Use a graphic display calculator to find the zeros of y = x² - 5x + 4.2 marks
  3. Make a table of values for y = x² - 3x for x = -1, 0, 1, 2, 3, 4.3 marks
  4. Find the coordinates of the vertex of y = x² - 6x + 5.2 marks
  5. Find the coordinates of the points where y = x² - 2 meets y = x + 4.3 marks
  6. Find the x-coordinates of the points where y = x³ - 4x + 1 crosses the x-axis. Give each answer to 3 significant figures.3 marks
  7. Find the coordinates of the points where y = 3/x meets y = x + 2.4 marks
  8. Use a graphical method to solve 2^(x) = 10 - 2x. Show a sketch of your graph. Give your answer to 3 significant figures.4 marks
  9. The graph of y = x + 4/x is sketched for values of x between -6 and 6. (a) Find the coordinates of the local maximum and of the local minimum. (b) Find the x-coordinates of the points where the graph meets the line y = 6. Give each answer to 3 significant figures.5 marks
  10. A rectangular sheet of card measures 30 cm by 20 cm. A square of side x cm is cut from each corner and the sides are folded up to make an open box. The volume of the box is V cm³, where V = x(30 - 2x)(20 - 2x). (a) Find the greatest volume of the box and the value of x that gives it. (b) Find the two values of x that give a volume of 800 cm³. Give each answer to 3 significant figures.6 marks

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