Q1.
We have learnt that to evaluate the value of an algebraic expression, we substitute a value of the variable in the given expression. Consider Example 3, where the wire is bent to form a rectangle. Here, the area of the rectangle, 10x – x², is a function of x. Can you interpret this as an input-output process? What value does the expression take when x = 6 cm?
Answer
Yes. The machine takes in a length x and returns the area of the rectangle that the 20 cm wire then makes.
input x = 6 → 10 × 6 − 6²
= 60 − 36
= 24 cm²
= 60 − 36
= 24 cm²
Check by the shape: if the length is 6 cm, the width is 10 − 6 = 4 cm, and 6 × 4 = 24 cm². ✓
| Input x (cm) | 2 | 4 | 6 | 8 |
|---|---|---|---|---|
| Output 10x − x² (cm²) | 16 | 24 | 24 | 16 |
| Step | — | +8 | 0 | −8 |
Why this machine is not linear: the steps in the bottom row are not equal — they are +8, 0, −8. Two different inputs, 4 and 6, even give the same output, 24. A linear machine can never do that. The reason is the x² term: subtracting consecutive values leaves an x behind instead of cancelling it, so the step size itself depends on where you are. That is why 10x − x² is called a quadratic function while 2x + 3 is linear.
Try This: feed in x = 5. You get 25 cm², the largest output of all — the square. Beyond it the outputs come back down, which is why the graph of this machine is a curve, not a straight line.