Question

In the following exercises, graph each equation by using its properties:

$y=2x^2+10x+7$

Short answer

The graph can be given as

Step-by-step solution

Given information

The equation of the parabola is$y=2x^2+10x+7$

Check whether the parabola is upward or downward 

As the value of coefficient of$x^2$is positive the parabola is upward

Find the axis of symmetry

To find the axis of symmetry put $x=-\frac{b}{2a}$

On substituting the values we get

$$\begin{gathered} x=-\frac{10}{4} \\ x=-\frac{5}{2} \end{gathered}$$

 Find the vertex of the parabola 

To find the vertex put $x=-\frac{5}{2}$in the parabola equation

We get

$$\begin{gathered} y=2x^2+10x+7 \\ y=2{(-\frac{5}{2})}^2+10(-\frac{5}{2})+7 \\ y=\frac{25}{2}-\frac{50}{2}+7 \\ y=-\frac{11}{2} \end{gathered}$$

The vertex is at$(-\frac{5}{2},-\frac{11}{2})$

To find the y-intercept put x=0

Put $x=0$

We get

$$\begin{gathered} y=0+0+7 \\ y=7 \end{gathered}$$

The coordinate of parabola is $(0,7)$

As this coordinate is $\frac{5}{2}units$to right of axis of symmetry the other coordinates is $\frac{5}{2}units$left of the parabola.

Hence the coordinate is$(-5,7)$

Step 6; We write the x-intercept 

To find the x-intercept we put $y=0$

We get,

$2x^2+10x+7=0$

On solving we get

$x=-4.158andx=-0.842$

The coordinates are$(-4.158,0),(-0.842,0)$

plot the graph 

The graph can be given as

Conclusion

The graph can be given as