# What are the important points needed to graph y= 2 (x + 1) (x - 4)?

Apr 20, 2016

See explanantion

#### Explanation:

$\textcolor{b l u e}{\text{Determine "x_("intercepts}}$

The graph crosses the x-axis at $y = 0$ thus:

x_("intercept")" at " y=0

Thus we have $\textcolor{b r o w n}{y = 2 \left(x + 1\right) \left(x - 4\right)} \textcolor{g r e e n}{\to 0 = 2 \left(x + 1\right) \left(x - 4\right)}$

Thus color(blue)(x_("intercept") -> (x,y)-> (-1,0)" and "(+4,0))
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color(blue)("Determine "x_("vertex"))

If you multiply out the right hand side you get:

$\text{ } y = 2 \left({x}^{2} - 3 x - 4\right) \to$

From this we have two options to determine x_("vertex")

$\textcolor{b r o w n}{\text{Option 1:}}$ This is the allowed format to apply:
color(blue)(" "x_("vertex")=(-1/2)xx(-3) = +3/2)

$\textcolor{b r o w n}{\text{Option 1:}}$ Take the mean of x_("intercepts")" "(x" values only)"

color(blue)(" "x_("vertex")= ((-1)+(+4))/2 = +3/2)
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color(blue)("Determine "y_("vertex"))

Substitute for $x$ in original equation using ${x}_{\text{vertex")" to find "y_("vertex}}$

$\textcolor{b l u e}{\implies {y}_{\text{vertex}} = 2 \left(\frac{3}{2} + 1\right) \left(\frac{3}{2} - 4\right) = - 12 \frac{1}{2} = - \frac{25}{2}}$
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color(blue)("Determine "y_("intercept"))

The graph crosses the y-axis at x=0. Substituting x=0 giving:
$\textcolor{b l u e}{{y}_{\text{intercept}} = 2 \left(0 + 1\right) \left(0 - 4\right) = - 8}$

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$\textcolor{b l u e}{\text{Determine general shape of the graph}}$

If you totally multiply out the right hand side and look at the highest order you have:

$y = 2 {x}^{2} - \ldots . .$

The coefficient of ${x}^{2}$ is positive (+2)

$\textcolor{g r e e n}{\text{So the general shape of the graph is: } \cup}$

color(blue)("Thus we have a "underline("minimum")->(x,y)->(3/2,-24/2 ))#

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