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 \( f(x) = x^2 \):

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

The general form for vertical translation is \( f(x) + k \), where \( k \) is the vertical shift. For \( h(x) = x^2 + 4 \), we have \( k = 4 \). This means \( h(x) \) is a vertical translation of \( f(x) = x^2 \) (specifically, a vertical shift up by 4 units).

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

The general form for horizontal translation is \( f(x + h) \), where \( h \) is the horizontal shift (negative \( h \) means shift right, positive \( h \) means shift left). For \( g(x) = (x + 4)^2 \), we can rewrite it as \( f(x - (-4)) \), so \( h = -4 \), meaning it is a horizontal translation (shift left by 4 units), not a vertical translation.

Step 3: Evaluate the statements

• "Ava’s graph is a vertical translation of \( f(x) = x^2 \).": True (from Step 1, it’s a vertical shift up).
• "Victor’s graph is a vertical translation of \( f(x) = x^2 \).": False (from Step 2, it’s a horizontal shift, not vertical).
• "Ava’s graph moved 4 units from \( f(x) = x^2 \) in a positive direction.": True (vertical shift up is positive \( y \)-direction, 4 units).

(Note: Since the problem asks for three correct answers, we assume there are more options not fully visible. But from the given, the two true ones are:

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

If we consider typical options for this problem, a third true statement might be about Victor’s graph being a horizontal translation, or another about Ava’s vertex, etc. For the given visible options, the two true ones are as above.)

Final Answer (for visible options):

• 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 were provided, we’d include the third, but with the given, these two are correct.)

Answer:

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

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

The general form for vertical translation is \( f(x) + k \), where \( k \) is the vertical shift. For \( h(x) = x^2 + 4 \), we have \( k = 4 \). This means \( h(x) \) is a vertical translation of \( f(x) = x^2 \) (specifically, a vertical shift up by 4 units).

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

The general form for horizontal translation is \( f(x + h) \), where \( h \) is the horizontal shift (negative \( h \) means shift right, positive \( h \) means shift left). For \( g(x) = (x + 4)^2 \), we can rewrite it as \( f(x - (-4)) \), so \( h = -4 \), meaning it is a horizontal translation (shift left by 4 units), not a vertical translation.

Step 3: Evaluate the statements

• "Ava’s graph is a vertical translation of \( f(x) = x^2 \).": True (from Step 1, it’s a vertical shift up).
• "Victor’s graph is a vertical translation of \( f(x) = x^2 \).": False (from Step 2, it’s a horizontal shift, not vertical).
• "Ava’s graph moved 4 units from \( f(x) = x^2 \) in a positive direction.": True (vertical shift up is positive \( y \)-direction, 4 units).

(Note: Since the problem asks for three correct answers, we assume there are more options not fully visible. But from the given, the two true ones are:

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

If we consider typical options for this problem, a third true statement might be about Victor’s graph being a horizontal translation, or another about Ava’s vertex, etc. For the given visible options, the two true ones are as above.)

Final Answer (for visible options):

• 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 were provided, we’d include the third, but with the given, these two are correct.)