It is known that x and y are related by an equation, $y = ab^x + 4$ where a and b are constants.

1. Plot the given data points on the grid

Plot the points $(1, 0.78)$, $(2, 1.08)$, $(3, 1.38)$, and $(4, 1.68)$ on the axes provided, using the specified scales.

2. Draw the straight line graph

Draw a single straight line that passes through all the plotted points.

1. Express the given equation in linear form

$y = ab^x + 4$
$y - 4 = ab^x$
$\lg(y - 4) = \lg(ab^x)$
$\lg(y - 4) = \lg a + x \lg b$

2. Identify the gradient and $Y$-intercept from the linear form

Comparing $\lg(y - 4) = (\lg b)x + \lg a$ with $Y = mX + c$:
Gradient $m = \lg b$
$Y$-intercept $c = \lg a$

3. Calculate the gradient from the table/graph to find $b$

Using points $(1, 0.778)$ and $(4, 1.68)$:
Gradient $m = \frac{1.68 - 0.778}{4 - 1} = \frac{0.902}{3} \approx 0.3007$
$\lg b = 0.3007$
$b = 10^{0.3007} \approx 2.00$

4. Calculate the $Y$-intercept to find $a$

Using $Y = mX + c$ with point $(1, 0.778)$ and $m = 0.301$:
$0.778 = 0.301(1) + c$
$c = 0.778 - 0.301 = 0.477$
$\lg a = 0.477$
$a = 10^{0.477} \approx 3.00$

1. Calculate the value on the vertical axis corresponding to $y = 33$

When $y = 33$,
$\lg(y - 4) = \lg(33 - 4)$
$\lg(y - 4) = \lg(29) \approx 1.462$

2. Read the corresponding $x$-value from the graph

From the graph of $\lg(y-4)$ against $x$, when the vertical axis value is $1.462$, the corresponding horizontal axis value is read as:
$x \approx 3.27$
(Acceptable range typically $3.25 \leqslant x \leqslant 3.30$ depending on the drawn graph)