How do you find the axis of symmetry, and the maximum or minimum value of the function y=x^2+10x+21?

May 3, 2017

Using part of the process of completing the square

Axis of symmetry -> x=-5)
Vertex $\to \text{ minimum } \to \left(x , y\right) = \left(- 5 , - 4\right)$

Explanation:

color(blue)("Determine axis of symmetry & " x_("vertex"))

Consider the standardised form of $y = a {x}^{2} + b x + c$

Write as $y = a \left({x}^{2} + \frac{b}{a} x\right) + c$

then we have:

x_("vertex")=" axis of symmetry"=(-1/2)xxa/b

color(blue)(ul(bar(|" "x_("vertex")=" axis of symmetry"=(-1/2)xx10=-5" "|))

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color(blue)("To determine "y_("vertex"))

Substitute $x = - 5$

$y = {x}^{2} + 10 x + 21 \text{ "->" } y = {\left(- 5\right)}^{2} + 10 \left(- 5\right) + 21$

" "color(blue)(ul(bar(|color(white)(2/2)y_("vertex")=-4" "|)))
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$\text{ "color(red)("Vertex } \to \left(x , y\right) = \left(- 5 , - 4\right)$

The coefficient of ${x}^{2} \to + 1$ as positive then the graph is of form $\cup$. $\textcolor{red}{\text{Thus the vertex is a minimum}}$

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