Answer
a. $2(x+3)^{2} - 8$
b. $Vertex: (-3,-8)$
$x-intercepts: -1, -5$
$y-intercept: 10$
c. Sketch like in the graph
d. Domain is all real numbers and range is from [-8, infinity)
Work Step by Step
a.
$2x^{2} + 12x + 10 = 2(x^{2} + 6x + 9) - 9\times2 + 10 = 2(x+3)^{2} - 8$
b.
$Vertex: (-3,-8)$
$x-intercepts: f(x)=0$
$2x^{2} + 12x + 10 = 0$
=> $x= \frac{-12 +\sqrt (12^{2}-4\times2\times10)}{2\times2}$ and $x= \frac{-12 -\sqrt (12^{2}-4\times2\times10)}{2\times2}$
=> $x= \frac{-12 +\sqrt (144-80)}{4}$ and $x= \frac{-12 -\sqrt (144-80)}{4}$
=> $x= -1$ and $x= -5$
$y-intercept: f(0) = 2\times0^{2} + 12\times0 + 10 = 0$
c. Sketch like in the graph
d.
Domain is all real numbers
Range is from [-8, infinity)
