Certainly! For the quadratic function \( f(x) = a x^2 \), we can determine key features such as the vertex and the axis of symmetry. Let's go through the steps:
Step 1: Identify the form of the quadratic function.
The given function is \( f(x) = a x^2 \), which is a standard form of a quadratic function where:
- \( a \) is a constant coefficient.
Step 2: Determine the vertex of the function.
For a quadratic function of the form \( f(x) = a x^2 \), the vertex is a critical point where the function changes direction. This standard form is symmetric about a vertical line through its vertex, and the vertex can be identified as follows:
- For \( f(x) = a x^2 \), the vertex is at the point \((0, 0)\). This is because when \( x = 0 \), the value of \( f(x) \) is also \( 0 \) (i.e., \( f(0) = a \cdot 0^2 = 0 \)).
Step 3: Identify the axis of symmetry.
The axis of symmetry for this quadratic function is a vertical line that runs through the vertex. It essentially splits the graph into two identical halves. For the function \( f(x) = a x^2 \):
- The vertex is at \( (0, 0) \), so the axis of symmetry is the line \( x = 0 \).
Summary:
- Vertex: The vertex of the function \( f(x) = a x^2 \) is \((0, 0)\).
- Axis of Symmetry: The axis of symmetry is the vertical line \( x = 0 \).
So, the answers are:
- The vertex is \( (0, 0) \).
- The axis of symmetry is [tex]\( x = 0 \)[/tex].
