Question

which statements are true about the graph of the function ( f(x) = 6x - 4 + x^2 )? select two options.

• the vertex form of the function is ( f(x) = (x - 2)^2 + 2 ).
• the vertex of the function is ( (-3, -13) ).
• the axis of symmetry for the function is ( x = 3 ).
• the graph increases over the interval ( (-3, infty) ).
• the function does not cross the ( x )-axis.

Explanation:

Step 1: Rewrite the function in vertex form

Given \( f(x) = 6x - 4 + x^2 \), we first reorder the terms: \( f(x)=x^{2}+6x - 4 \). To complete the square, we take the coefficient of \( x \), which is \( 6 \), divide by \( 2 \) to get \( 3 \), and square it to get \( 9 \). So we add and subtract \( 9 \):

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So the vertex form is \( f(x)=(x + 3)^{2}-13 \), not \( (x - 2)^{2}+2 \). So the first statement is false.

Step 2: Find the vertex

For a quadratic function in vertex form \( f(x)=a(x - h)^{2}+k \), the vertex is \( (h,k) \). From \( f(x)=(x + 3)^{2}-13=(x-(-3))^{2}+(-13) \), the vertex is \( (-3,-13) \). So the second statement is true.

Step 3: Find the axis of symmetry

The axis of symmetry for a quadratic function in vertex form \( f(x)=a(x - h)^{2}+k \) is \( x = h \). From \( f(x)=(x + 3)^{2}-13 \), \( h=-3 \), so the axis of symmetry is \( x=-3 \), not \( x = 3 \). So the third statement is false.

Step 4: Analyze the increasing interval

For a quadratic function \( f(x)=ax^{2}+bx + c \) with \( a>0 \) (here \( a = 1>0 \)), the function increases for \( x>h \) (where \( h \) is the x - coordinate of the vertex). The vertex has \( x=-3 \), so the function increases over the interval \( (-3,\infty) \). So the fourth statement is true.

Step 5: Check if the function crosses the x - axis

To check if the function crosses the x - axis, we find the discriminant of the quadratic equation \( x^{2}+6x - 4=0 \). The discriminant \( D=b^{2}-4ac \), where \( a = 1 \), \( b = 6 \), \( c=-4 \). So \( D=6^{2}-4\times1\times(-4)=36 + 16=52>0 \). Since the discriminant is positive, the quadratic equation has two real roots, which means the graph crosses the x - axis. So the fifth statement is false.

Answer:

The vertex of the function is \((-3, -13)\).
The graph increases over the interval \((-3, \infty)\).