# How do you find the axis of symmetry, graph and find the maximum or minimum value of the function  y = x^2 + 4x + 1?

Jan 15, 2016

color(green)("Minimum"->(x_("vertex")" , "y_("vertex")) -> (-2,-3))

$\textcolor{b l u e}{\text{Axis of symmetry} \to x = - 2}$

#### Explanation:

$\textcolor{b l u e}{\text{Axis of symmetry}}$

This a quadratic equation and because $\textcolor{b r o w n}{{x}^{2} \text{ is positive}}$ it is of the $\textcolor{b r o w n}{\text{upward horse shoe shape like a letter U. }}$Thus it has a minimum value which coincides with the axis of symmetry.

As there is no coefficient in front of ${x}^{2}$ the axis of symmetry and ${x}_{\text{vertex}}$ occurs at $\left(- \frac{1}{2}\right) \times \left(+ 4\right)$ where the 4 is from $4 x$.

So the axis of symmetry is at $\textcolor{b l u e}{x = - 2}$

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$\textcolor{b l u e}{\text{Determine the minimum}}$

Substitute the known value for x to determine the associated value of y.

$y = {\left(- 2\right)}^{2} + 4 \left(- 2\right) + 1$

$\textcolor{b l u e}{y = + 4 - 8 + 1 = - 3}$
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color(green)("Minimum"->(x_("vertex")" , "y_("vertex")) -> (-2,-3))