Answer
Vertex = (-2, -1)
Axis of symmetry: x = -2
The graph of the function is presented below
Step-by-step explanation
The vertex of a quadratic equation is the point where the graph of the quadratic equation changes from sloping negatively to sloping positively and vice-versa.
The axis of symmetry represents the straight line that divides the graph of the quadratic equation into two mirror parts that are similar to and are mirror images of each other. This axis of symmetry usually passes through the vertex.
To find the vertex, it is usually at the turning point where the first derivative of the quadratic equation is equal to 0.
(df/dx) = 0
f(x) = (x + 2)² - 1
f(x) = x² + 4x + 4 - 1
f(x) = x² + 4x + 3
At the vertex, (df/dx) = 0
(df/dx) = 2x + 4
2x + 4 = 0
2x = -4
Divide both sides by 2
(2x/2) = (-4/2)
x = -2
We can then obtain the corresponding y-coordinate of the vertex
f(x) = (x + 2)² - 1
f(-2) = (-2 + 2)² - 1
f(-2) = 0² - 1
f(-2) = -1
So, the vertex is given as
Vertex = (-2, -1)
Although, one can obtain the vertex from the form in which that equation is given, the general form is that
f(x) = (x - x₁)² + y₁
Comparing that with
f(x) = (x + 2)² - 1
we see that,
x₁ = -2, y₁ = -1
So, Vertex: (-2, -1)
Then, the axis of symmetry will be at the point of the vertex.
Axis of symmetry: x = -2
And for the graph, we just need to obtain a couple of points on the line to sketch that.
when x = 0
f(x) = (x + 2)² - 1
f(0) = (0 + 2)² - 1
f(0) = 4 - 1 = 3
(0, 3)
when y = 0
x = -3 and x = -1
So,
(-3, 0) and (-1, 0)
(-2, -1), (0, 3), (-3, 0) and (-1, 0)
So, with these points, we can sketch the graph.
The graph of this function is presented under answer above.
Hope this Helps!!!