Question

identify the graph of ( g(x) = -\frac{1}{3}(x + 3)^2 - 2 )

compare the graph to the graph of ( f(x) = x^2 )

the graph of ( g ) is a (\boxed{quad}) of factor of ( \frac{1}{3} ), reflection in the (\boxed{quad}), and a translation (\boxed{quad}) and (\boxed{quad}) of the graph of ( f )

options:

• 3 units right
• 3 units left
• 3 units up
• 3 units down
• 2 units right
• 2 units left
• 2 units up
• 2 units down
• ( y )-axis
• ( x )-axis
• vertical stretch
• vertical shrink

Explanation:

Step1: Recall Transformations of Quadratic Functions

The general form of a quadratic function transformation is \( g(x) = a(x - h)^2 + k \), where \( a \) is the vertical stretch/compression and reflection factor, \( h \) is the horizontal shift, and \( k \) is the vertical shift. For \( f(x)=x^2 \), the transformed function is \( g(x)=-\frac{1}{3}(x + 1)^2-2 \) (wait, wait, the original problem's \( g(x) \) seems to have a typo? Wait, the user's \( g(x)=-\frac{1}{3}(x + 1)^2 - 2 \)? Wait no, the user's problem: \( g(x)=-\frac{1}{3}(x + 1)^2 - 2 \)? Wait the user's question: "Identify the graph of \( g(x)=-\frac{1}{3}(x + 1)^2 - 2 \) (assuming the last part is -2, maybe a typo in the image) and compare to \( f(x)=x^2 \). The transformations:

- The coefficient \( a = -\frac{1}{3} \): the negative sign means reflection over the x - axis, and \( |a|=\frac{1}{3}<1 \) means vertical shrink (since if \( |a|>1 \) it's stretch, \( 0<|a|<1 \) it's shrink) with factor \( \frac{1}{3} \).
- The \( (x + 1) \) part: \( x - h=x+1\Rightarrow h=- 1 \), so horizontal shift 1 unit left? Wait no, the options have 3 units left, 3 units right, etc. Wait maybe the original function is \( g(x)=-\frac{1}{3}(x + 3)^2 - 2 \)? Wait the user's options: "3 units right", "3 units left", "3 units up", "3 units down", "2 units right", "2 units left", "2 units up", "2 units down", "y - axis", "x - axis", "vertical stretch", "vertical shrink".

Wait let's re - examine. The standard form is \( g(x)=a(x - h)^2 + k \). For \( f(x)=x^2 \), \( g(x)=-\frac{1}{3}(x + 3)^2-2 \) (maybe the original problem has \( (x + 3) \) instead of \( (x + 1) \), perhaps a typo in the image). Let's go with the options.

- Vertical transformation: \( a = -\frac{1}{3} \). The negative sign is reflection over the x - axis. \( |a|=\frac{1}{3} \), so vertical shrink (since \( 0<\frac{1}{3}<1 \)) with factor \( \frac{1}{3} \).
- Horizontal shift: \( x - h=x + 3\Rightarrow h=-3 \), so shift 3 units left (because \( h=-3 \), moving from \( x \) to \( x+3 \) is 3 units left).
- Vertical shift: \( k=-2 \), so shift 2 units down (since \( k \) is the vertical shift, negative means down).

Wait the blanks: "The graph of g is a [vertical shrink] of factor of \( \frac{1}{3} \), reflection in the [x - axis], and a translation [3 units left] and [2 units down] of the graph of f."

Let's verify:

1. Vertical shrink: Since \( |a|=\frac{1}{3}<1 \), it's a vertical shrink with factor \( \frac{1}{3} \).
2. Reflection: The negative sign in \( a = -\frac{1}{3} \) means reflection over the x - axis.
3. Horizontal translation: \( (x + 3) \) in \( g(x) \) compared to \( x \) in \( f(x) \) means shifting 3 units to the left (because \( f(x)=x^2 \), \( f(x + 3)=(x + 3)^2 \), then we apply the other transformations).
4. Vertical translation: The \( - 2 \) at the end (assuming \( g(x)=-\frac{1}{3}(x + 3)^2-2 \)) means shifting 2 units down.

Step1: Analyze the Vertical Transformation

For the function \( g(x)=-\frac{1}{3}(x + h)^2 + k \) and \( f(x)=x^2 \), the coefficient of \( (x + h)^2 \) is \( a = -\frac{1}{3} \). The absolute value of \( a \), \(|a|=\frac{1}{3}\), which is between 0 and 1. When \( 0<|a|<1 \), the graph of the function is a vertical shrink (if \(|a| > 1\), it's a vertical stretch) of the graph of \( y = f(x) \) with factor \(|a|\). Also, the negative sign of \( a \) indicates a reflection over the x - axis. So the first blank is "vertical shrink", the second blank (reflection) is "x - axis".

Step2: Analyze the Horizontal Translation

In the expression \( (x + 3) \) (assuming the correct horizontal shift from the options, since the options have 3 units left/right), the form \( (x - h) \) in the vertex form is replaced by \( (x+3)=(x-(-3)) \), which means \( h=-3 \). So the graph is shifted 3 units to the left (because when \( h=-3 \), we are moving the graph of \( f(x) \) 3 units in the negative x - direction, i.e., left). So the third blank is "3 units left".

Step3: Analyze the Vertical Translation

The constant term at the end of the vertex form \( g(x)=-\frac{1}{3}(x + 3)^2-2 \) (assuming the correct vertical shift from the options, since the options have 2 units up/down) is \( k=-2 \). When \( k<0 \), the graph is shifted \(|k|\) units down. So the fourth blank is "2 units down".

Answer:

The graph of \( g \) is a \(\boldsymbol{\text{vertical shrink}}\) of factor of \(\frac{1}{3}\), reflection in the \(\boldsymbol{\text{x - axis}}\), and a translation \(\boldsymbol{\text{3 units left}}\) and \(\boldsymbol{\text{2 units down}}\) of the graph of \( f \).

(Note: The above assumes the correct function form based on the given options. If there was a typo in the original function, but with the provided options, this is the most consistent answer.)