Certainly! Let's break down the solution to understand the given quadratic equation [tex]\( y = 4x^2 + 16x + 1 \)[/tex].
### Step 1: Identify the Form of the Equation
The general form of a quadratic equation is:
[tex]\[ ax^2 + bx + c \][/tex]
In this equation:
- [tex]\( a = 4 \)[/tex]
- [tex]\( b = 16 \)[/tex]
- [tex]\( c = 1 \)[/tex]
### Step 2: Analyze the Components
1. Coefficient of [tex]\( x^2 \)[/tex] (a): This is [tex]\( 4 \)[/tex]. Since it is positive, the parabola (graph of the quadratic equation) opens upwards.
2. Coefficient of [tex]\( x \)[/tex] (b): This is [tex]\( 16 \)[/tex]. This term affects the location of the vertex and the axis of symmetry.
3. Constant term (c): This is [tex]\( 1 \)[/tex]. This term affects the y-intercept, which is the point where the parabola crosses the y-axis.
### Step 3: Find the Vertex of the Parabola
The vertex form of a quadratic equation [tex]\( y = ax^2 + bx + c \)[/tex] can be found by using the vertex formula:
[tex]\[ x = -\frac{b}{2a} \][/tex]
Plugging in our values:
[tex]\[ x = -\frac{16}{2 \cdot 4} = -\frac{16}{8} = -2 \][/tex]
To find the y-coordinate of the vertex, substitute [tex]\( x = -2 \)[/tex] back into the equation:
[tex]\[ y = 4(-2)^2 + 16(-2) + 1 \][/tex]
Calculate step-by-step:
[tex]\[ y = 4 \cdot 4 + 16 \cdot (-2) + 1 \][/tex]
[tex]\[ y = 16 - 32 + 1 \][/tex]
[tex]\[ y = -16 + 1 \][/tex]
[tex]\[ y = -15 \][/tex]
So the vertex of the parabola is at [tex]\( (-2, -15) \)[/tex].
### Step 4: Axis of Symmetry
The axis of symmetry is the vertical line that passes through the vertex of the parabola. Its equation is:
[tex]\[ x = -2 \][/tex]
### Step 5: Y-Intercept
The y-intercept is found by setting [tex]\( x = 0 \)[/tex] in the equation and solving for [tex]\( y \)[/tex]:
[tex]\[ y = 4 \cdot 0^2 + 16 \cdot 0 + 1 \][/tex]
[tex]\[ y = 1 \][/tex]
So the y-intercept is at [tex]\( (0, 1) \)[/tex].
### Step 6: Roots of the Equation (if needed)
The roots or solutions to the quadratic equation [tex]\( 4x^2 + 16x + 1 = 0 \)[/tex] can be found using the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Let's calculate the discriminant first:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
[tex]\[ \Delta = 16^2 - 4 \cdot 4 \cdot 1 \][/tex]
[tex]\[ \Delta = 256 - 16 \][/tex]
[tex]\[ \Delta = 240 \][/tex]
Since the discriminant is positive, we will have two distinct real roots:
[tex]\[ x = \frac{-16 \pm \sqrt{240}}{8} \][/tex]
[tex]\[ x = \frac{-16 \pm 4\sqrt{15}}{8} \][/tex]
[tex]\[ x = \frac{-16 \pm 4\sqrt{15}}{8} \][/tex]
[tex]\[ x = -2 \pm \frac{\sqrt{15}}{2} \][/tex]
### Summary
The quadratic equation [tex]\( y = 4x^2 + 16x + 1 \)[/tex] yields the following:
- Vertex: [tex]\((-2, -15)\)[/tex]
- Axis of Symmetry: [tex]\( x = -2 \)[/tex]
- Y-intercept: [tex]\( (0, 1)\)[/tex]
- Roots: [tex]\( x = -2 \pm \frac{\sqrt{15}}{2} \)[/tex]
This detailed analysis provides a comprehensive understanding of the quadratic function.
