Question

Match the two graphs shown here to the equations $x{y}^2+x=1$and$x{y}^2+x+1=0$Explain your choices.

Short answer

The graph of the function:$x{y}^2+x=1$:

Graph of$x{y}^2+x+1=0$

Step-by-step solution

Step 1. Given information:

The functions are:

$$\begin{gathered} x{y}^2+x=1 \\ 1+x+x{y}^2=0 \end{gathered}$$

Step 2. Find x-intercepts of both functions.

Substitute $y=0$in both functions and solve for $x$.

$$\begin{gathered} x{y}^2+x=1 \\ x(1+y^2)=1 \\ x(1+0)=1 \\ x=1 \end{gathered}$$

So x-intercept of this function is $(1,0)$,

$$\begin{gathered} 1+x+x{y}^2=0 \\ 1+x+x.0=0 \\ 1+x=0 \\ x=-1 \end{gathered}$$

$x-$intercept is $(-1,0)$

Step 3. Match the graphs with their function.

Since $x{y}^2+x=1$, has $x-$intercept is localid="1648443221959" $(1,0)$,so the graph will cross at this point.

The localid="1648443300930" $x-$intercept of the function localid="1648443229137" $1+x+x{y}^2=0$is localid="1648443236418" $(-1,0)$,so the graph will cross localid="1648443244527" $x-$axis at this point.