Question

which statement is true concerning the vertex and the axis of symmetry of ( g(x) = 5x^2 - 10x )?
the function written in vertex form is ( g(x) = 5(x - 1)^2 - 5 ). the vertex is at ( (1, -5) ) and the axis of symmetry is ( x = 1 ).
the vertex is at ( (1, -5) ) and the axis of symmetry is ( y = 1 ).
the vertex is at ( (0, 0) ) and the axis of symmetry is ( x = 1 ).
the vertex is at ( (0, 0) ) and the axis of symmetry is ( y = 1 ).

Explanation:

Step1: Recall Vertex Form of Quadratic

The vertex form of a quadratic function is \( g(x) = a(x - h)^2 + k \), where \((h, k)\) is the vertex and the axis of symmetry is \( x = h \).

Step2: Identify \( h \) and \( k \) in \( g(x)=5(x - 1)^2 - 5 \)

Here, \( h = 1 \) and \( k = -5 \), so the vertex is \((1, -5)\). The axis of symmetry is \( x = h = 1 \).

Step3: Analyze Other Options

• For vertex \((0, 0)\), the function would be \( g(x)=ax^2 \), but our function has \( -5 \) and \( (x - 1)^2 \), so vertex isn't \((0, 0)\).
• Axis of symmetry for quadratic in vertex form \( a(x - h)^2 + k \) is vertical (\( x = h \)), not horizontal (\( y = 1 \)), so options with \( y = 1 \) are wrong.

Answer:

The vertex is at \((1, -5)\) and the axis of symmetry is \( x = 1 \) (the first correct option among the listed ones, assuming the first option is "The vertex is at \((1, -5)\) and the axis of symmetry is \( x = 1 \)").