# How do you graph and label the vertex and axis of symmetry of y=(x+1)^2+3?

Jan 24, 2018

See below.

#### Explanation:

$y = {\left(x + 1\right)}^{2} + 3$ , is in the form:

$y = a {\left(x - h\right)}^{2} + k$

Where a is the coefficient of ${x}^{2}$, h is the axis of symmetry and k is the maximum/minimum value.

Notice that a $= 1$ is positive, so the parabola will be of this form: $\bigcup$. This means it has a minimum value.

The minimum or maximum value always occurs where the $x$ coordinate is $h$ and the $y$ coordinate is $k$ i.e.:

$\left(h , k\right)$

From our equation this is:

color(blue)((-1 ,3)

Notice that $h = - 1$ and not $1$. This is because we have ${\left(x - h\right)}^{2}$, so when $h$ is negative, ${\left(x - \left(- 1\right)\right)}^{2} = {\left(x + 1\right)}^{2}$

This coordinate can be used for graphing. Other useful points are:

$y$ axis intercept occurs when $x = 0$

$y = {\left(\left(0\right) + 1\right)}^{2} + 3 = 4$

coordinate:

color(blue)((0,4)

$x$ axis intercepts occur when $y = 0$:

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

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

$x = - 1 + \sqrt{- 3}$ ( This is a non real value, so no x axis intercepts ).

Other point for graphing will have to be found by plugging in values for $x$ and calculating the corresponding values of $y$.

Values could be:

x=1->(1+1)^2+3=7->color(blue)((1,7)

x=2->(2+1)^2+3=12->color(blue)((2,12)

x=-3->(-3+1)^2+3=7->color(blue)((-3,7)

x=-4->(-4+1)^2+3=12->color(blue)((-4,12)

The more points you plot the more accurate the curve will be:

Join the points and mark the axis of symmetry ( You could use a dashed line for this ). Mark the minimum value ( You can use a dot for this ).