Question

The function f is defined by f(x) = (x+3)² - 1.

1.What are the coordinates of the vertex of the graph

2. Does the graph open up or down? Does it mean it has a maximum or minimum

Answer 1

1. Coordinates of the vertex of the graph: (-3, -1)

2. The graph opens up.

2. The graph has a minimum.

Explanation:

Currently, f(x) = (x + 3)^2 - 1 is in the vertex form of a quadratic equation. The general equation of the vertex form is given by:

f(x) = a(x - h)^2 + k, where

• a is a constant determining whether the parabola opens up or down (this also means it determines whether the vertex is a maximum or minimum),
• and (h, k) are the coordinates of the vertex.

1. When dealing with the vertex form, the h coordinate becomes the opposite of what it is inside the parentheses. Thus, h is -3 and k is -1. Therefore, the coordinates of the vertex are (-3, -1).

2. Question #1: We can imagine that there's an imaginary 1 in front of (x + 3)^2 and thus a = 1.

• When a > 0, the parabola opens up.
• When a < 0, the parabola opens down.

Since a = 1 and a > 0, the parabola opens up.

2. Question #2: All parabolas that open up will by definition have a minimum. Thus, the graph of the function f has a minimum.

Answer 2

Explanation:

the general equation of a parabola is

y = f(x) = a(x - h)² + k

with (h, k) being the vertex.

compare it to the given function :

f(x) = (x + 3)² - 1

clearly,

k = -1

h = -3

so, the vertex is (-3, -1).

if a > 0, then the parabola opens up.

if a < 0, then the parabola opens down.

in our case, a = 1, so it opens up.

and that means the vertex is a minimum (as all other points will be "above" it, meaning their y- or f(x)-value will be larger).