Question

the graph of the function f(x) = (x + 2)(x + 6) is shown below. which statement about the function is true? the function is positive for all real values of x where x > -4. the function is negative for all real values of x where -6 < x < -2. the function is positive for all real values of x where x < -6 or x > -3. the function is negative for all real values of x where x < -2.

Explanation:

Step1: Analyze the roots of the function

The function \( f(x)=(x + 2)(x + 6) \) has roots at \( x=-6 \) and \( x=-2 \) (since when \( x+2 = 0 \) or \( x + 6=0 \), \( f(x)=0 \)). The graph is a parabola opening upwards (because the coefficient of \( x^2 \) when expanded is positive, as \( (x + 2)(x + 6)=x^2+8x + 12 \), and the coefficient of \( x^2 \) is \( 1>0 \)).

Step2: Analyze the sign of the function in different intervals

• For \( x < -6 \): Let's take \( x=-7 \). Then \( f(-7)=(-7 + 2)(-7 + 6)=(-5)(-1)=5>0 \), so the function is positive here.
• For \( -6 < x < -2 \): Let's take \( x=-4 \). Then \( f(-4)=(-4 + 2)(-4 + 6)=(-2)(2)=-4<0 \), so the function is negative here.
• For \( x > -2 \): Let's take \( x = 0 \). Then \( f(0)=(0 + 2)(0 + 6)=12>0 \), so the function is positive here.

Now let's check each option:

• Option 1: "The function is positive for all real values of \( x \) where \( x > - 4 \)". For \( -4 \) (which is in \( -6 < x < -2 \)) we saw \( f(-4)=-4<0 \), so this is false.
• Option 2: "The function is negative for all real values of \( x \) where \( -6 < x < -2 \)". From our test with \( x=-4 \) (in \( -6 < x < -2 \)) we got \( f(-4)=-4<0 \), and since the parabola opens upwards, between the two roots the function is negative. So this is true.
• Option 3: "The function is positive for all real values of \( x \) where \( x < -6 \) or \( x > -3 \)". For \( x=-4 \) (which is in \( x > - 3 \) when \( -3 > -2 \)? No, \( -4 < -2 \), wait \( x > -3 \): take \( x=-2.5 \) (in \( -6 < x < -2 \) and \( x > -3 \)). \( f(-2.5)=(-2.5 + 2)(-2.5 + 6)=(-0.5)(3.5)=-1.75<0 \), so this is false.
• Option 4: "The function is negative for all real values of \( x \) where \( x < -2 \)". For \( x=-7 \) (which is \( x < -2 \)), \( f(-7)=5>0 \), so this is false.

Answer:

The function is negative for all real values of \( x \) where \( -6 < x < -2 \) (the second option: "The function is negative for all real values of \( x \) where \( -6 < x < -2 \)")