Question

In the following exercise, graph the equation$y=6x^2+2x-1$

Short answer

The required standard equation and graph is ${\left(x+\frac{1}{6}\right)}^2=\frac{1}{6}\left(y+\frac{7}{6}\right)$

Step-by-step solution

Step 1, Given the information    

The given equation of the curve$y=6x^2+2x-1$

Step 2. Convert into a standard form  

$$\begin{gathered} y=6x^2+2x-1 \\ y=6\left(x^2+2.\frac{x}{6}-\frac{1}{6}\right) \\ y=6\left(x^2+2.\frac{x}{6}+\frac{1}{36}-\frac{1}{36}-\frac{1}{6}\right) \\ y=6{\left(x+\frac{1}{6}\right)}^2-1-\frac{1}{6} \\ y=6{\left(x+\frac{1}{6}\right)}^2-\frac{7}{6} \\ 6{\left(x+\frac{1}{6}\right)}^2=y+\frac{7}{6} \\ {\left(x+\frac{1}{6}\right)}^2=\frac{1}{6}\left(y+\frac{7}{6}\right) \end{gathered}$$

which is a parabola

Step 3. Draw a graph 

The standard equation of a parabola is ${\left(x+\frac{1}{6}\right)}^2=\frac{1}{6}\left(y+\frac{7}{6}\right)$

Whose vertex is $\left(\frac{-1}{6},-\frac{7}{6}\right)$