Question: Sketch the graph of $y = 2x^3 - 1$, labelling the axis intercepts.

Step 1. Recognise the family first: $y = 2x^3$ is a cubic through the origin — climbing steeply from bottom-left to top-right, flattening momentarily at the origin.

Step 2. The $-1$ shifts the whole curve down by 1 unit — same shape, new position.

Step 3. Find the $y$-intercept (sub $x = 0$):

Step 4. Find the $x$-intercept (sub $y = 0$):

Sketch: the standard cubic shape, cutting the $y$-axis at $-1$ and the $x$-axis just left of $1$, flattening around its crossing of $x \approx 0.8$… with the bend now centred at $(0, -1)$.

[diagram in original sheet: shifted cubic with both intercepts marked]

⚠ Watch out: A sketch is judged on shape + labelled intercepts, not plotted accuracy — but the intercepts must be calculated, not guessed. Adding a constant shifts vertically; only a minus sign in front of the whole function ($y = -2x^3$) flips the shape.

Question: Four unlabelled sketches show $y = x^2$, $y = x^3$, $y = \dfrac{1}{x}$ and $y = 2^x$. Match each equation to its sketch.

[diagram in original sheet: gallery of the standard y = xⁿ and y = aˣ shapes]

Step 1. $y = x^2$ — the U-shaped parabola: symmetric about the $y$-axis, touching the origin, never below the $x$-axis.

Step 2. $y = x^3$ — the S-shaped cubic: negative for $x < 0$, positive for $x > 0$, flattening as it passes through the origin.

Step 3. $y = \dfrac{1}{x}$ — the hyperbola in two pieces: one branch in the first quadrant, one in the third, never touching either axis (the axes are its asymptotes; $x = 0$ is not allowed).

Step 4. $y = 2^x$ — the exponential: always above the $x$-axis, passing through $(0, 1)$, hugging the $x$-axis on the left and exploding upwards on the right.

Quick discriminators: touches the origin → power graph; passes through $(0,1)$ → exponential; two separate branches → $\frac{1}{x}$ or $\frac{1}{x^2}$ (the latter has both branches above the axis).

⚠ Watch out: $y = 2^x$ and $y = x^2$ both grow quickly for large $x$ — the giveaway is at $x = 0$: the exponential is at height $1$, the parabola at $0$. And a minus sign in front of any of these reflects the whole shape in the $x$-axis.

Question: Water is poured at a constant rate into (a) a cylinder, (b) a cone standing on its vertex (point down). Sketch the graph of water depth against time for each container.

Step 1. The reasoning tool: at a constant pouring rate, the depth rises quickly where the container is narrow and slowly where it is wide. The graph's steepness at any moment mirrors the container's width at the current water level.

Step 2. (a) A cylinder has the same width at every height, so the depth climbs at one constant speed:

a straight line from the origin, sloping up.

Step 3. (b) The vertex-down cone is narrowest at the bottom, widening all the way up: the depth shoots up fast at first, then rises ever more slowly:

a curve that starts steep and flattens (still always increasing, never level).

Step 4. Both graphs start at the origin and end at the brim's depth at the same final time if the volumes match. The cone's curve stays above the straight line in between — early depth outruns the cylinder's.

⚠ Watch out: "Rises more slowly" never means the graph turns downward or flat — depth always increases while pouring continues; only the steepness changes. Flip the cone (point up) and the story reverses: slow first, then steep. Sketches are marked on shape, start point and labels, not plotted precision.