Question

part 4 of 4
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 - 1)^2 - 1 )

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 the vertex form of a parabola

The vertex form of a parabola is \( g(x) = a(x - h)^2 + k \), where \((h, k)\) is the vertex and \(a\) determines the direction and width. For \( f(x)=x^2 \), it can be written as \( f(x) = 1(x - 0)^2 + 0 \), so \( a = 1 \), vertex \((0,0)\), axis of symmetry \( x = 0 \). For \( g(x)=(x - 1)^2 - 1 \), comparing to \( a(x - h)^2 + k \), we have \( a = 1 \), \( h = 1 \), \( k = -1 \).

Step2: Determine vertex, axis of symmetry, direction

• Vertex: From \( g(x)=(x - 1)^2 - 1 \), using the vertex form, the vertex is \((h, k)=(1, -1)\).
• Axis of symmetry: The axis of symmetry for a parabola in the form \( y = a(x - h)^2 + k \) is \( x = h \), so for \( g(x) \), the axis of symmetry is \( x = 1 \).
• Direction: Since \( a = 1>0 \), the parabola opens up, which matches the given information.

Step3: Compare the width

The width of a parabola \( y = ax^2 + bx + c \) (or in vertex form \( y = a(x - h)^2 + k \)) is determined by the absolute value of \( a \). If \( |a| = 1 \), it has the same width as \( f(x)=x^2 \) (where \( a = 1 \)). For \( g(x) \), \( a = 1 \), so \( |a| = 1 \), same as \( f(x)=x^2 \). So the graph of \( g(x) \) has the same width as \( f(x)=x^2 \).

Answer:

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