Question

identify the vertex, axis of symmetry, and direction of the graph of each function. compare the width of the graph to the width of the graph of ( f(x)=x^2 ).( g(x)=(x - 5)^2 - 4 )
the graph opens up.
compare the width of the graph to the width of the graph of ( f(x)=x^2 ).
a. the graph has the same width as ( f(x)=x^2 ).
b. the graph is narrower than ( f(x)=x^2 ).
c. the graph is wider than ( f(x)=x^2 ).

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, \(x = h\) is the axis of symmetry, and the sign of \(a\) determines the direction. The absolute value of \(a\) determines the width: if \(|a| = 1\), same width as \(f(x)=x^2\) (\(a = 1\) for \(f(x)\)); if \(|a|>1\), narrower; if \(|a|<1\), wider.

Step2: Analyze \(g(x)=(x - 5)^2 - 4\)

For \(g(x)=(x - 5)^2 - 4\), we can rewrite it as \(g(x)=1\times(x - 5)^2 - 4\). Here, \(a = 1\), \(h = 5\), \(k=-4\). Since \(|a| = |1| = 1\), which is the same as the \(a\) value (\(a = 1\)) in \(f(x)=x^2\), the width of the graph of \(g(x)\) is the same as the width of the graph of \(f(x)=x^2\).

Answer:

A. The graph has the same width as \(f(x)=x^2\)