HW Vertex Form 2 .pdf - Algebra II 5.2 Vertex Form Worksheet Name ... - Free Printable

Problem Analysis:

The task involves analyzing a quadratic function and its graph. The given information includes the standard form, vertex form, and intercept form of the quadratic equation, as well as the coordinates of specific points on the graph. The goal is to identify the axis of symmetry of the parabola.

#### Given Information:
1. Standard Form: \( y = x^2 - 4x + 3 \)
2. Vertex Form: \( y = (x - 2)^2 - 1 \)
3. Intercept Form: \( y = (x - 1)(x - 3) \)
4. Points on the Graph:
- Vertex: \( (2, -1) \)
- \( x \)-intercepts: \( (1, 0) \) and \( (3, 0) \)
- \( y \)-intercept: \( (0, 3) \)

Step-by-Step Solution:

#### 1. Understanding the Axis of Symmetry:
The axis of symmetry of a parabola is a vertical line that divides the parabola into two symmetric halves. For a quadratic function in the form \( y = ax^2 + bx + c \), the axis of symmetry is given by the formula:
\[
x = -\frac{b}{2a}
\]

#### 2. Using the Standard Form:
The standard form of the quadratic equation is:
\[
y = x^2 - 4x + 3
\]
Here, \( a = 1 \), \( b = -4 \), and \( c = 3 \). Using the formula for the axis of symmetry:
\[
x = -\frac{b}{2a} = -\frac{-4}{2 \cdot 1} = \frac{4}{2} = 2
\]
Thus, the axis of symmetry is \( x = 2 \).

#### 3. Using the Vertex Form:
The vertex form of the quadratic equation is:
\[
y = (x - 2)^2 - 1
\]
In vertex form, \( y = a(x - h)^2 + k \), the vertex is \( (h, k) \). Here, the vertex is \( (2, -1) \). The axis of symmetry is the vertical line passing through the vertex, which is:
\[
x = 2
\]

#### 4. Using the \( x \)-Intercepts:
The intercept form of the quadratic equation is:
\[
y = (x - 1)(x - 3)
\]
The \( x \)-intercepts are the roots of the equation, which are \( x = 1 \) and \( x = 3 \). The axis of symmetry is the vertical line halfway between the \( x \)-intercepts. The midpoint of \( x = 1 \) and \( x = 3 \) is:
\[
x = \frac{1 + 3}{2} = \frac{4}{2} = 2
\]
Thus, the axis of symmetry is \( x = 2 \).

#### 5. Graphical Confirmation:
From the graph, the parabola is symmetric about the vertical line \( x = 2 \). The vertex \( (2, -1) \) lies on this line, and the \( x \)-intercepts \( (1, 0) \) and \( (3, 0) \) are equidistant from \( x = 2 \).

Final Answer:

The axis of symmetry of the parabola is:
\[
\boxed{x = 2}
\]