Questions: Identifying The Characteristics of Quadratic Functions For each of the following quadratic functions (Show your work): 1. Calculate the vertex by hand and write it as an ordered pair. 2. Determine the axis of symmetry and write it as a linear equation (x=# or t=#).

Solution Steps

1. To find the vertex of a quadratic function in the form \( f(x) = ax^2 + bx + c \), use the vertex formula \( x = -\frac{b}{2a} \). Substitute this \( x \)-value back into the function to find the \( y \)-coordinate of the vertex.
2. The axis of symmetry for a quadratic function is a vertical line that passes through the vertex. It can be expressed as \( x = -\frac{b}{2a} \).

To find the vertex of the quadratic function \( f(x) = 2x^2 + 3x + 1 \), we use the vertex formula:

\[ x = -\frac{b}{2a} \]

Substituting \( a = 2 \) and \( b = 3 \):

\[ x = -\frac{3}{2 \cdot 2} = -\frac{3}{4} = -0.75 \]

Next, we substitute \( x = -0.75 \) back into the function to find the \( y \)-coordinate:

\[ y = 2(-0.75)^2 + 3(-0.75) + 1 = 2(0.5625) - 2.25 + 1 = 1.125 - 2.25 + 1 = -0.125 \]

Thus, the vertex is:

\[ \text{Vertex} = \left(-0.75, -0.125\right) \]

The axis of symmetry for the quadratic function is given by the same \( x \)-value we calculated for the vertex:

\[ x = -\frac{b}{2a} = -0.75 \]

Thus, the axis of symmetry can be expressed as:

\[ \text{Axis of Symmetry: } x = -0.75 \]

Final Answer