To graph the function [tex]\( f(x) = x^2 + 4x + 6 \)[/tex], we need to determine its key features, such as the vertex and another point on the graph. Here is a detailed step-by-step solution:
### Step 1: Identify the Vertex
The function [tex]\( f(x) = x^2 + 4x + 6 \)[/tex] is a quadratic function and its graph is a parabola. The standard form of a quadratic function is [tex]\( ax^2 + bx + c \)[/tex]. For this function, [tex]\( a = 1 \)[/tex], [tex]\( b = 4 \)[/tex], and [tex]\( c = 6 \)[/tex].
The vertex of a parabola given by [tex]\( ax^2 + bx + c \)[/tex] can be found using the formula for the x-coordinate of the vertex:
[tex]\[ x = -\frac{b}{2a} \][/tex]
### Step 2: Calculate the X-Coordinate of the Vertex
Substitute [tex]\( a = 1 \)[/tex] and [tex]\( b = 4 \)[/tex] into the formula:
[tex]\[ x = -\frac{4}{2(1)} = -2 \][/tex]
So, the x-coordinate of the vertex is [tex]\( x = -2 \)[/tex].
### Step 3: Calculate the Y-Coordinate of the Vertex
To find the y-coordinate of the vertex, substitute [tex]\( x = -2 \)[/tex] back into the function [tex]\( f(x) \)[/tex]:
[tex]\[ f(-2) = (-2)^2 + 4(-2) + 6 \][/tex]
[tex]\[ f(-2) = 4 - 8 + 6 \][/tex]
[tex]\[ f(-2) = 2 \][/tex]
Hence, the vertex of the parabola is [tex]\((-2, 2)\)[/tex].
### Step 4: Identify a Second Point
To plot the graph accurately, we need another point. Choose an easy value for [tex]\( x \)[/tex] to calculate [tex]\( f(x) \)[/tex]. Let's choose [tex]\( x = 0 \)[/tex].
[tex]\[ f(0) = 0^2 + 4(0) + 6 \][/tex]
[tex]\[ f(0) = 6 \][/tex]
The point obtained is [tex]\((0, 6)\)[/tex].
### Step 5: Graph the Function
- Start by plotting the vertex [tex]\((-2, 2)\)[/tex] on the graph.
- Next, plot the second point [tex]\((0, 6)\)[/tex].
By connecting these points and noting the parabolic shape that opens upwards (since the coefficient of [tex]\( x^2 \)[/tex] is positive), you will have a good sketch of the graph of the function [tex]\( f(x) = x^2 + 4x + 6 \)[/tex].
### Summary of Points
- Vertex: [tex]\((-2, 2)\)[/tex]
- Second Point: [tex]\((0, 6)\)[/tex]
Using these points, you can accurately graph the function [tex]\( f(x) = x^2 + 4x + 6 \)[/tex].
