Cubic, reciprocal, exponential and trig graphs

$y = x^2$ is the U shape: D.

$y = \frac{1}{x}$ has two separate branches and never touches the axes: A.

$y = x^3$ rises from bottom left to top right: C.

$y = -x^3$ is its reflection, falling from top left: B.

(a) $y = x^3$ passes through the origin, is negative for negative $x$ and rises steeply: graph F.

(b) $y = \frac{1}{x}$ never touches the axes, with branches in the top-right and bottom-left: graph D.

(i) $y = x^2 - 4$ is a U-shaped parabola crossing the $y$-axis at $-4$: H

(ii) $y = -x^3$ is a cubic that falls from top left to bottom right through the origin: F

(iii) $y = -\frac{5}{x}$ is a reciprocal graph in the second and fourth quadrants: J

$y = x^2 - 4x$ is a U-shaped curve through the origin: B

$y = x + 3$ slopes up and crosses the $y$-axis above O: D

$y = x^3 - 2$ is the S-shaped cubic: E

$y = \frac{1}{x}$ has two separate branches: C

$y = 5 - 2x$ slopes down: A

(a) $x = -2$: $6(-2) - (-2)^3 = -12 + 8 = -4$

$x = -1$: $-6 + 1 = -5$; $x = 0$: $0$; $x = 1$: $6 - 1 = 5$

(b) Plot $(-3, 9), (-2, -4), (-1, -5), (0, 0), (1, 5), (2, 4), (3, -9)$ and join with a smooth curve.

For positive $x$, $y$ is positive and gets smaller as $x$ grows, so draw a curve in the top-right quadrant.

For negative $x$, $y$ is negative, so draw the matching curve in the bottom-left quadrant.

The curves get close to the axes but never touch them, and there is no point at $x = 0$.

(a) $x = -2$: $5 - (-8) = 13$; $x = 0$: $5$; $x = 1$: $5 - 1 = 4$; $x = 2$: $5 - 8 = -3$

(b) Plot the five points and join them with a smooth curve, not straight lines.

When $x = 0$, $y = 3^0 = 1$.

Sana's curve starts at $(0, 0)$, so it is wrong there: it should start at $(0, 1)$.

The cosine graph starts at $(0, 1)$ and falls to its lowest value of $-1$ at $x = 180°$.

So $P = (180, -1)$

- $f(x) \propto \frac{1}{x}$: two branches that never touch the axes, B.
- Trigonometric: a wave through $O$, A.
- Exponential: flat on the left, cuts the $y$-axis above 0 and shoots up, D.
- $f(x) \propto \sqrt{x}$: starts at $O$ and rises more and more slowly, C.

(a) Draw $y = 0.3$ across the graph. It meets the curve twice, at about $x = 18$ and $x = 180 - 18 = 162$.

(b) $\sin\theta = 0$ when $\theta = 0$ or $180$. So $x + 20 = 0$ gives $x = -20$ (or $x + 20 = 180$ gives $x = 160$).

$2^x$ is always positive, so the curve never touches the x-axis.

When $x = 0$, $y = 2^0 = 1$, so it crosses the y-axis at $(0, 1)$.

It gets close to the x-axis on the left and rises ever more steeply on the right.

$\tan x°$ is 0 at 0°, 180° and 360°.

It rises towards $+\infty$ as $x$ approaches 90° and 270°, and comes up from $-\infty$ just after them.

Draw vertical dashed asymptotes at 90 and 270, and three increasing branches through $(0,0)$, $(180,0)$, $(360,0)$.