# How do you find the important points to graph y = (x - 3)² + 1?

Nov 2, 2015

See explanation

#### Explanation:

Standard form of the equation is
$y = a {x}^{2} + b x + c$
Note that $a$ could be of value 1
If $a$ is positive then the graph is a horse shoe shape with the curve at the bottom. If negative then the other way round.

Square the bracket and then add the 1 giving:
$y = {x}^{2} - 6 x + 4$

Using standard form equation

$x \text{ of "(x,y)_("minimum")" is } \left(- \frac{1}{2}\right) \times \frac{b}{a}$

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So for your case we have:

color(green)(x" of "(x,y)_("minimum") =(-1/2) times (-6) = color(green)(+3))

color(red)("substitute "x=3" in your equation to find "y_("minimum"))

$y = {\left(3\right)}^{2} - 6 \left(3\right) + 4$
$y = 13 - 18$
$y = - 5$
$\textcolor{g r e e n}{y \text{ of " (x,y)_("minimum}} = \left(- 5\right)$

Putting it all together

color(green)( (x,y)_("minimum")=(3,-5)
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$\textcolor{g r e e n}{\text{To find x-intercepts substitute y=0 and solve}}$

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$\textcolor{g r e e n}{\text{To find y-intercepts substitute x=0 and solve}}$

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