# How do you identify the important parts of y = 1/2(x-3)(x+1) to graph it?

Nov 25, 2015

See stepwise explanation

#### Explanation:

color(green)("Given: "y=1/2(x-3)(x+1).................................(1)

$\textcolor{g r e e n}{\text{Same as: } y = \frac{1}{2} \left({x}^{2} - 2 x - 3\right)}$

color(blue)(x^2 " is positive so" underline(" upwards U-shape")).................(2)

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$\textcolor{b r o w n}{\text{To find "x_("vertex")" Consider the -2 of } - 2 x}$

$\textcolor{b l u e}{{x}_{\text{vertex}} = - \frac{1}{2} \left(- 2\right) = + 1} \ldots \ldots \ldots \ldots \ldots \ldots \ldots . \left(3\right)$
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$\textcolor{b r o w n}{\text{To find "y_("vertex")" Substitute (3) into (1)}}$

$y = \frac{1}{2} \left(1 - 3\right) \left(1 + 1\right) = \frac{1}{2} \left(- 2\right) \left(2\right) = - 2$

$\textcolor{b l u e}{{y}_{\text{vertex}} = - 2} \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \left(4\right)$
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$\textcolor{b r o w n}{\text{To find y-intercept}}$

Substitute $x = 0$ in (1)

$y = \frac{1}{2} \left(0 - 3\right) \left(0 + 1\right) = \frac{1}{2} \left(- 3\right) \left(1\right) = - \frac{3}{2}$

$\textcolor{b l u e}{{y}_{\text{intercept}} = - \frac{3}{2}}$
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$\textcolor{b r o w n}{\text{To find x-intercept}}$

Substitute $y = 0$ in (1)

$0 = \frac{1}{2} \left(x - 3\right) \left(x + 1\right)$

$\left(x - 3\right) = 0 \to x = + 3$
$\left(x + 1\right) = 0 \to x = - 1$

color(blue)(x_("intercept")-> (x ,y) -> (-1 , 0) " and "(+3 , 0))
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