# What are the vertex, axis of symmetry, maximum or minimum value, domain, and range of the function, and x and y intercepts for f(x)=(x-5)^2 - 9?

Apr 29, 2015

This is an equation of a parabola, so we can find all the requests easily.

$y = {\left(x - 5\right)}^{2} - 9$

(y-y_v=a(x-x_v)^2)

• The vertex is $V \left(5 , - 9\right)$.
• The axis of symmetry is a vertical line passing from the vertex, so: $x = 5$.
• The minimum is in the vertex (it is concave up!) and the maximum doesn't exist (it goes to $+ \infty$).
• The domain is $\mathbb{R}$, because it is a polynomial function.
• The range is $\left[5 , + \infty\right)$.
• The x intercepts are points whose ordinate are $0$, so:
$0 = {\left(x - 5\right)}^{2} - 9 \Rightarrow {\left(x - 5\right)}^{2} = 9 \Rightarrow x - 5 = \pm 3 \Rightarrow$
$x = 5 \pm 3 \Rightarrow A \left(8 , 0\right) \mathmr{and} B \left(2 , 0\right)$.
• The y intercept is a point whose ascissa is $0$, so:
$y = {\left(0 - 5\right)}^{2} - 9 \Rightarrow 25 - 9 = 16 \Rightarrow C \left(0 , 16\right)$.