Question

ava graphs the function $h(x) = x^2 + 4$. victor graphs the function $g(x) = (x + 4)^2$. which statements are true regarding the two graphs? choose three correct answers. avas graph is a vertical translation of $f(x) = x^2$. victors graph is a vertical translation of $f(x) = x^2$. avas graph moved 4 units from $f(x) = x^2$ in a positive direction.

Explanation:

To solve this, we analyze the transformations of the parent function \( f(x) = x^2 \):

1. Analyze Ava’s function \( h(x) = x^2 + 4 \):

The general form for a vertical translation of \( f(x) = x^2 \) is \( f(x) = x^2 + k \), where \( k > 0 \) shifts up (positive direction) and \( k < 0 \) shifts down. Here, \( h(x) = x^2 + 4 \) (so \( k = 4 \)), which is a vertical translation (shift up 4 units) of \( f(x) = x^2 \). Thus, the statement “Ava’s graph is a vertical translation of \( f(x) = x^2 \)” is true.

2. Analyze Victor’s function \( g(x) = (x + 4)^2 \):

The general form for a horizontal translation of \( f(x) = x^2 \) is \( f(x) = (x - h)^2 \), where \( h > 0 \) shifts right and \( h < 0 \) shifts left. Here, \( g(x) = (x + 4)^2 = (x - (-4))^2 \), which is a horizontal translation (shift left 4 units) of \( f(x) = x^2 \), not a vertical translation. Thus, the statement “Victor’s graph is a vertical translation of \( f(x) = x^2 \)” is false.

3. Analyze “Ava’s graph moved 4 units from \( f(x) = x^2 \) in a positive direction”:

For vertical translations, “positive direction” typically refers to upward (along the \( y \)-axis). Since \( h(x) = x^2 + 4 \) shifts \( f(x) = x^2 \) up 4 units (a positive vertical direction), this statement is true.

(Note: The problem asks to “choose three correct answers,” but only three options are shown here. Assuming there are more options, but based on the visible ones, the two true statements so far are:

• Ava’s graph is a vertical translation of \( f(x) = x^2 \).
• Ava’s graph moved 4 units from \( f(x) = x^2 \) in a positive direction.

If additional options exist (e.g., about horizontal translation for Victor), we would analyze those too, but with the given visible options, these two are true.)

Final Answer (for visible options):

The correct statements are:

• Ava’s graph is a vertical translation of \( f(x) = x^2 \).
• Ava’s graph moved 4 units from \( f(x) = x^2 \) in a positive direction.

(If more options are provided, we would include the third correct one, but with the given text, these are the true ones from the visible choices.)

Answer:

To solve this, we analyze the transformations of the parent function \( f(x) = x^2 \):

1. Analyze Ava’s function \( h(x) = x^2 + 4 \):

The general form for a vertical translation of \( f(x) = x^2 \) is \( f(x) = x^2 + k \), where \( k > 0 \) shifts up (positive direction) and \( k < 0 \) shifts down. Here, \( h(x) = x^2 + 4 \) (so \( k = 4 \)), which is a vertical translation (shift up 4 units) of \( f(x) = x^2 \). Thus, the statement “Ava’s graph is a vertical translation of \( f(x) = x^2 \)” is true.

2. Analyze Victor’s function \( g(x) = (x + 4)^2 \):

The general form for a horizontal translation of \( f(x) = x^2 \) is \( f(x) = (x - h)^2 \), where \( h > 0 \) shifts right and \( h < 0 \) shifts left. Here, \( g(x) = (x + 4)^2 = (x - (-4))^2 \), which is a horizontal translation (shift left 4 units) of \( f(x) = x^2 \), not a vertical translation. Thus, the statement “Victor’s graph is a vertical translation of \( f(x) = x^2 \)” is false.

3. Analyze “Ava’s graph moved 4 units from \( f(x) = x^2 \) in a positive direction”:

For vertical translations, “positive direction” typically refers to upward (along the \( y \)-axis). Since \( h(x) = x^2 + 4 \) shifts \( f(x) = x^2 \) up 4 units (a positive vertical direction), this statement is true.

(Note: The problem asks to “choose three correct answers,” but only three options are shown here. Assuming there are more options, but based on the visible ones, the two true statements so far are:

• Ava’s graph is a vertical translation of \( f(x) = x^2 \).
• Ava’s graph moved 4 units from \( f(x) = x^2 \) in a positive direction.

If additional options exist (e.g., about horizontal translation for Victor), we would analyze those too, but with the given visible options, these two are true.)

Final Answer (for visible options):

The correct statements are:

• Ava’s graph is a vertical translation of \( f(x) = x^2 \).
• Ava’s graph moved 4 units from \( f(x) = x^2 \) in a positive direction.

(If more options are provided, we would include the third correct one, but with the given text, these are the true ones from the visible choices.)