write an equation for the function graphed below. the y intercept is at (0,0.4)

Explanation:

Step1: Identify rational function form

The graph has vertical asymptotes at $x=-1$ and $x=3$, so the denominator is $(x+1)(x-3)$. It has a double root at $x=1$ (touches x-axis), so the numerator has $(x-1)^2$. The general form is: $$f(x)=\frac{A(x-1)^2}{(x+1)(x-3)}$$

Step2: Solve for A using y-intercept

Use $(0, 0.4)$: substitute $x=0$, $f(0)=0.4$ $$0.4=\frac{A(0-1)^2}{(0+1)(0-3)}$$ $$0.4=\frac{A(1)}{(1)(-3)}$$ $$A=0.4\times(-3)=-1.2=-\frac{6}{5}$$

Step3: Substitute A into the function

$$f(x)=\frac{-\frac{6}{5}(x-1)^2}{(x+1)(x-3)}$$ Simplify the denominator and rewrite: $$f(x)=\frac{-6(x-1)^2}{5(x+1)(x-3)}$$ Or expand numerator/denominator if needed: $$f(x)=\frac{-6(x^2-2x+1)}{5(x^2-2x-3)}=\frac{-6x^2+12x-6}{5x^2-10x-15}$$

Answer:

$\boldsymbol{f(x)=\frac{-6(x-1)^2}{5(x+1)(x-3)}}$ (or expanded form $\boldsymbol{f(x)=\frac{-6x^2+12x-6}{5x^2-10x-15}}$)