13.1 Cubic, reciprocal and exponential graphs — Practise with help

$ (x-3)(x+3)$; $8$; $(0,2)$; $x=-3$ or $x=3$.

Set $y=0$ and factor out $x$.

$x(x^2-4)=0$, then factor the difference of squares.

$x(x-2)(x+2)=0$, so $x=-2,0,2$. The intercepts are $(-2,0),(0,0),(2,0)$.

Substitute each input. Consider what happens when the input or output is zero.

$1/(-2)=-1/2$ and $1/2=1/2$.

The outputs are $-1/2$ and $1/2$. The asymptotes are $x=0$ and $y=0$.

Substitute the four inputs. Recall that a negative index gives a reciprocal.

Evaluate $2^{-1},2^0,2^1,2^2$.

The outputs are $1/2,1,2,4$. Each unit step right doubles the output.

The $y$-intercept occurs when $x=0$.

Write the point as $(0,5)$, then substitute zero into $2^x$.

The intercept is $(0,5)$. The graph $y=2^x$ instead passes through $(0,1)$ because $2^0=1$.