Let's work through the quadratic function [tex]\( f(x) = 4x^2 + 6x + 1 \)[/tex] step-by-step.
### Step 1: Identifying the coefficients
In the quadratic function [tex]\( f(x) = ax^2 + bx + c \)[/tex], we identify:
- [tex]\( a = 4 \)[/tex]
- [tex]\( b = 6 \)[/tex]
- [tex]\( c = 1 \)[/tex]
### Step 2: Finding the Vertex
The vertex of a quadratic function in standard form [tex]\( ax^2 + bx + c \)[/tex] can be found using the vertex formula. The x-coordinate of the vertex [tex]\( h \)[/tex] is given by:
[tex]\[ h = -\frac{b}{2a} \][/tex]
Plugging in our coefficients [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
[tex]\[ h = -\frac{6}{2 \cdot 4} = -\frac{6}{8} = -0.75 \][/tex]
To find the y-coordinate of the vertex [tex]\( k \)[/tex], substitute [tex]\( h \)[/tex] back into the original function [tex]\( f(x) \)[/tex]:
[tex]\[ k = 4(-0.75)^2 + 6(-0.75) + 1 \][/tex]
[tex]\[ k = 4 \cdot 0.5625 - 4.5 + 1 \][/tex]
[tex]\[ k = 2.25 - 4.5 + 1 = -1.25 \][/tex]
Thus, the vertex is at [tex]\( (-0.75, -1.25) \)[/tex].
### Step 3: Finding the Axis of Symmetry
The axis of symmetry is the vertical line that passes through the vertex. This line is given by the equation:
[tex]\[ x = h \][/tex]
So the axis of symmetry is:
[tex]\[ x = -0.75 \][/tex]
### Step 4: Determining the Range
For a quadratic function [tex]\( y = ax^2 + bx + c \)[/tex], the parabola opens upwards if [tex]\( a > 0 \)[/tex] and downwards if [tex]\( a < 0 \)[/tex]. Since [tex]\( a = 4 \)[/tex] (which is greater than 0), the parabola opens upwards.
The range of the function is from the y-coordinate of the vertex to positive infinity. Thus, the range is:
[tex]\[ [-1.25, \infty) \][/tex]
### Step 5: Writing the Vertex Form
The vertex form of a quadratic equation is [tex]\( f(x) = a(x - h)^2 + k \)[/tex]. With the values we've found:
[tex]\[ a = 4 \][/tex]
[tex]\[ h = -0.75 \][/tex]
[tex]\[ k = -1.25 \][/tex]
Substituting these values, we get:
[tex]\[ f(x) = 4(x - (-0.75))^2 + (-1.25) \][/tex]
[tex]\[ f(x) = 4(x + 0.75)^2 - 1.25 \][/tex]
So, the vertex form of the quadratic function is:
[tex]\[ f(x) = 4(x + 0.75)^2 - 1.25 \][/tex]
### Summary
- Vertex: [tex]\((-0.75, -1.25)\)[/tex]
- Axis of Symmetry: [tex]\( x = -0.75 \)[/tex]
- Range: [tex]\([-1.25, \infty)\)[/tex]
- Vertex Form: [tex]\( f(x) = 4(x + 0.75)^2 - 1.25 \)[/tex]
