A viewing window is the rectangle of the graph the calculator displays, set by Xmin, Xmax, Ymin and Ymax. A useful window shows every feature the question asks about: the roots, the turning point, the intercepts or the intersection. It appears whenever you plot a graph during graphic display calculator interpretation.
How do you choose the window before plotting?
The standard window, usually x and y from −10 to 10, suits small numbers only. Exam contexts often have larger values, so work the window out from the equation in four steps.
- Find the y-intercept. Put x = 0 into the function. It tells you roughly how high or low the graph starts.
- Estimate the roots or the domain. If the question gives a range such as 0 ≤ x ≤ 30, use that for the x-axis.
- Find the turning point if there is one. For y = ax² + bx + c, the turning point has x = −b/(2a). Substitute to get its y-value.
- Add a margin. Set the window slightly wider than the features so nothing sits on the edge.
Worked example
Choose a window for y = x² − 40x + 300 that shows the roots and the turning point.
Step 1, y-intercept: x = 0 gives y = 300.
Step 2, roots: x² − 40x + 300 = (x − 10)(x − 30), so the roots are x = 10 and x = 30.
Step 3, turning point: x = 40/2 = 20. Then y = 400 − 800 + 300 = −100. The turning point is (20, −100), a minimum.
Step 4, window: the x-values that matter run from 10 to 30, and the graph starts at x = 0. Take Xmin = 0 and Xmax = 40. The y-values run from −100 up to 300, so take Ymin = −120 and Ymax = 320.
Check: at x = 40, y = 1600 − 1600 + 300 = 300, which fits inside Ymax = 320. In the standard window you would see almost nothing, because the graph is at y = 300 when x = 0 and dips below the screen quickly.
The mistake to watch for
A common move is to shrink Ymax to, say, 10 because “graphs are small” and then read the curve crossing the x-axis.
Mistaken window: Xmin = −10, Xmax = 10, Ymin = −10, Ymax = 10 for y = x² − 40x + 300.
The screen shows only a steep line falling near the left edge. The student concludes “no roots” or reads a wrong crossing.
The window does not match the graph’s size. The correction is to get the y-intercept and the turning point first, then set the window around them. The function was never the problem.
Check yourself
Give a window for each graph without plotting, then check your answers.
1. y = x² − 20x + 75. Show both roots and the turning point.
Show answer
x² − 20x + 75 = (x − 5)(x − 15), so the roots are 5 and 15. The turning point is at x = 10, y = 100 − 200 + 75 = −25. The y-intercept is 75. A good window is Xmin = 0, Xmax = 20, Ymin = −30, Ymax = 80. At x = 20, y = 75, which fits.
2. A tank holds 50 litres and loses 2 litres per minute: y = 50 − 2x. Choose a window that shows the tank until it is empty.
Show answer
y = 0 when x = 25, so the tank empties at 25 minutes. The y-intercept is 50. A sensible window is Xmin = −2, Xmax = 30, Ymin = −5, Ymax = 55. The margin beyond 25 and below 0 shows that the graph really crosses the axis.
3. y = −x² + 8x + 84. Find the roots and the maximum point so you can choose a window.
Show answer
Solve x² − 8x − 84 = 0: the discriminant is 64 + 336 = 400, so x = (8 ± 20)/2, giving 14 and −6. The turning point is at x = 4, y = −16 + 32 + 84 = 100. A window such as Xmin = −10, Xmax = 20, Ymin = −20, Ymax = 120 shows both roots and the maximum.
Where this leads next
With the window set, the next risk is a misleading picture: see how a poor window hides a root. The quadratic structure explorer shows the turning point and roots for coefficients you choose, so you can practise predicting a window before you look.
Some students find the graphs easy and the settings confusing, which is a different problem from not knowing the maths. A teacher in online one-to-one Mathematics tuition can work through your own calculator model with you.