This set covers the five skills in graphic display calculator interpretation: choosing a window, finding hidden roots, comparing graph and exact answers, rounding, and writing clear method. Questions run from easier to harder. All numbers are original.
Write full working on paper before you open the answer, and use a calculator only where a question says so. Keep your full calculator value until the last step, then round once.
Practice questions
1. Choose a window that shows both roots and the turning point of y = x² − 8x + 12.
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Factorise: (x − 2)(x − 6), so the roots are 2 and 6. The turning point is at x = 4, y = 16 − 32 + 12 = −4. The y-intercept is 12. A suitable window is Xmin = −2, Xmax = 10, Ymin = −6, Ymax = 35. At x = 10, y = 100 − 80 + 12 = 32, so it fits.
2. Water drains from a container: y = 100 − 4x litres after x minutes. Choose a window that shows the container until it is empty.
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y = 0 when x = 25. The container starts at 100 litres. A suitable window is Xmin = −2, Xmax = 30, Ymin = −10, Ymax = 110.
3. The standard window shows y = x² − 31x + 30 crossing the x-axis at x = 1 only. Find the other root and choose a window that shows it.
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The roots add to 31, so the other root is 30. Check the product: 1 × 30 = 30. The turning point is at x = 15.5, y = 240.25 − 480.5 + 30 = −210.25. A suitable window is Xmin = −5, Xmax = 40, Ymin = −250, Ymax = 100.
4. y = x² − 2x + 0.96 looks as if it touches the x-axis near x = 1. Decide whether it has one root or two, and find them.
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Discriminant = 4 − 4 × 0.96 = 4 − 3.84 = 0.16, which is positive, so there are two roots. x = (2 ± 0.4)/2, so x = 0.8 and x = 1.2. Check: 0.8 + 1.2 = 2 and 0.8 × 1.2 = 0.96.
5. A student plots y = x² + 4x + 5 and cannot find a root however far they zoom out. Explain why.
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Discriminant = 16 − 20 = −4, which is negative, so there are no real roots. The turning point is at x = −2, y = 4 − 8 + 5 = 1, so the minimum value is 1 and the curve never reaches the x-axis.
6. Solve x² = 3x + 10 exactly. State the points where y = x² meets y = 3x + 10.
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x² − 3x − 10 = 0, so (x − 5)(x + 2) = 0 and x = 5 or x = −2. When x = 5, y = 25. When x = −2, y = 4. The intersections are (5, 25) and (−2, 4). Check: 3(5) + 10 = 25 and 3(−2) + 10 = 4.
7. Solve y = x² and y = 2x + 3 graphically, then confirm by algebra.
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The intersection tool gives x = −1 and x = 3. Algebra: x² − 2x − 3 = 0, so (x − 3)(x + 1) = 0. The points are (−1, 1) and (3, 9). Check: 2(−1) + 3 = 1 and 2(3) + 3 = 9.
8. Solve x² − 6x + 2 = 0, giving the exact answers and then answers to 2 decimal places.
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x = (6 ± √(36 − 8))/2 = (6 ± √28)/2 = 3 ± √7. With √7 ≈ 2.645751: 5.645751 and 0.354249. The exact answers are x = 3 + √7 and x = 3 − √7. To 2 decimal places: 5.65 and 0.35.
9. (a) Give 0.0062841 to 2 significant figures. (b) A rectangle measures 7.3 cm by 4.6 cm. Find its area to 3 significant figures.
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(a) The first significant figure is 6, the second is 2, and the next digit is 8, so round up: 0.0063.
(b) 7.3 × 4.6 = 33.58. To 3 significant figures: 33.6 cm².
10. A ball’s height is h = 20t − 5t² metres after t seconds. Find when h = 18, giving answers to 3 significant figures, and write a short method.
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Equation: 20t − 5t² = 18, so 5t² − 20t + 18 = 0.
Formula: t = (20 ± √(400 − 360))/10 = (20 ± √40)/10 = 2 ± 0.6325.
Values: t = 1.3675 or 2.6325. To 3 significant figures: t = 1.37 s and t = 2.63 s.
Check: at t = 1.3675, 20t = 27.35 and 5t² = 5 × 1.87 = 9.35, so h = 18.0. Graph set-up: y = 20x − 5x² and y = 18, window 0 ≤ x ≤ 4 and 0 ≤ y ≤ 25.
11. A student answers “x = 4.4 (from graph)” for the equation x² − 3x − 4.4 = 0 and gets no credit. Write the missing working and give the roots to 2 decimal places.
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Discriminant = 9 + 17.6 = 26.6, so x = (3 ± √26.6)/2. √26.6 ≈ 5.15752. So x ≈ 4.07876 or x ≈ −1.07876. To 2 decimal places: x = 4.08 or x = −1.08. The student also gave only one root and did not show an equation, window, check or rounding. Check: 4.08² = 16.6464, then 16.6464 − 12.24 − 4.4 = 0.0064, close to 0 as expected after rounding.
If you got these wrong
- The curve looked empty or flat (questions 1 to 3): go back to selecting a useful viewing window.
- You stopped at one root or called two close roots one (questions 3 to 5): revisit hidden roots caused by a poor window.
- Your answer was a decimal when the question wanted algebra (questions 6 to 8): see comparing graphical and exact solutions.
- The digits were right but the rounding was not (questions 8 to 10): practise reading numerical results with appropriate precision.
- The answer was correct but the working was thin (questions 10 and 11): rewrite using documenting the calculator-supported method.
The quadratic structure explorer and the graph-model explorer let you test your predictions, and the mistake log helps you track which error types repeat.
If you keep making the same kind of slip, a teacher in online one-to-one Mathematics tuition can go through your attempt and work on that pattern directly.