Question: Using the drawn graph of $y = (1.5)^x + 1$, find the $x$-coordinate of the point where the gradient of the curve is $1$.

Step 1. The gradient of a curve at a point means the gradient of its tangent there — the straight line touching the curve at just that point.

Step 2. A gradient of $1$ rises 1 unit for every 1 across. Slide a ruler along the curve until it touches at exactly one point while climbing at that $45°$-style rate (matching the axes' scales), and draw the tangent.

Step 3. Measure the tangent's gradient from two well-separated points on the ruler line using $\dfrac{\text{rise}}{\text{run}}$, and adjust the touching point until the measured gradient is $1$.

Step 4. Read off the $x$-coordinate of the point of contact:

$$x \approx 2.2 \quad (\text{accept } 2.1 \text{ to } 2.3)$$

⚠ Watch out: A tangent touches the curve at one point — a line that cuts through gives a chord's gradient, not the curve's. Take rise and run from far-apart points on the tangent (small triangles magnify reading error), use the axes' scales (1 cm rarely equals 1 unit), and show the tangent on the graph — the drawn line carries the method mark.

Question: The graph of $y = (1.5)^x + 1$ has been drawn from a table of values for $-2 \le x \le 5$ (at $x = -1$, $p = 1.67$). By drawing a suitable straight line, solve $(1.5)^x = 2 - x$.

Step 1. The drawn curve is $y = (1.5)^x + 1$, but the equation has bare $(1.5)^x$. Rearrange the equation until one side is exactly the drawn curve: add 1 to both sides:

$$(1.5)^x + 1 = 3 - x$$

Step 2. So the line to draw is

$$y = 3 - x$$

(gradient $-1$, $y$-intercept $3$ — plot $(0,3)$ and $(3,0)$ and rule through).

Step 3. Read the $x$-coordinate of the intersection of line and curve:

$$x \approx 0.7 \quad (\text{accept } 0.6 \text{ to } 0.8)$$

Step 4. Why it works: at the intersection, curve height = line height, i.e. $(1.5)^x + 1 = 3 - x$, which is the target equation rearranged. One intersection → one solution.

⚠ Watch out: The most common error is drawing $y = 2 - x$ — forgetting to adjust for the $+1$ already inside the drawn curve. Always transform the equation to match the curve you have, then read $x$ (not $y$) at the crossing; graphical answers carry a tolerance, so quote sensible precision.