Sketch the graph of $y=\frac{2}{x^2}$ on the axes below. The point (1, 1) is shown in the diagram. Answer ... Sketch the graph of $y=5^x$ on the axes below. Indicate clearly where the graph intersects with the axes on your sketch. Answer

1. Identify the shape, symmetry and asymptotes of the graph

For $y=\dfrac{2}{x^2}$, $x\neq 0$ and $y>0$ for all allowed $x$.

Also,
$\frac{2}{(-x)^2}=\frac{2}{x^2},$
so the graph is symmetric about the $y$-axis.

As $x\to 0$, $y\to \infty$, so the $y$-axis is an asymptote.

As $|x|\to \infty$, $y\to 0$, so the $x$-axis is an asymptote.

2. Use key points to position the sketch accurately

Some useful points are
$x=1 \Rightarrow y=\frac{2}{1^2}=2,$
so the curve passes through $(1,2)$.

By symmetry, it also passes through $(-1,2)$.

Also,
$y=1 \Rightarrow \frac{2}{x^2}=1 \Rightarrow x^2=2 \Rightarrow x=\pm\sqrt{2},$
so the curve crosses the horizontal level $y=1$ at $x=\pm\sqrt{2}$.

Draw smooth branches in Quadrants I and II only, approaching but not touching both axes.

1. Identify the intercepts and key shape of the exponential graph

For the $y$-intercept, put $x=0$:

$y=5^0=1$

So the graph intersects the $y$-axis at $(0,1)$.

Also, $5^x>0$ for all real $x$, so the graph does not intersect the $x$-axis. It is an increasing exponential curve with horizontal asymptote $y=0$.