Question

va 25 - 26: algebra ii q3
graphing radical functions
describe how the graph of the parent function $y = \sqrt{x}$ is transformed when graphing $y = -3\sqrt{x - 6}$.
the graph is translated 6 units right
the graph is reflected over the x - axis.
the graph has a vertical stretch by a factor of 3.
the function $y = -3\sqrt{x - 6}$ is represented by graph b
a.
b.
c.
d.

Explanation:

To analyze the transformation of the parent function \( y = \sqrt{x} \) to \( y = -3\sqrt{x - 6} \), we follow these steps:

Step 1: Horizontal Translation

The general form for horizontal translation is \( y = \sqrt{x - h} \), where \( h \) is the number of units shifted horizontally. For \( y = \sqrt{x - 6} \), \( h = 6 \). Since \( h > 0 \), the graph is translated 6 units to the right.

Step 2: Reflection

The negative sign in front of the radical (\( -3\sqrt{x - 6} \)) indicates a reflection. In the form \( y = a\sqrt{x - h} \), a negative \( a \) reflects the graph over the \( x \)-axis. Here, \( a = -3 \), so the graph is reflected over the \( x \)-axis.

Step 3: Vertical Stretch

The coefficient \( |a| \) determines the vertical stretch or compression. For \( y = -3\sqrt{x - 6} \), \( |a| = 3 \). Since \( |a| > 1 \), the graph is vertically stretched by a factor of 3.

Step 4: Identifying the Graph

• The parent function \( y = \sqrt{x} \) has a domain \( x \geq 0 \) and starts at \( (0, 0) \).
• After translation, the vertex moves to \( (6, 0) \).
• Reflection over the \( x \)-axis flips the graph upside - down.
• Vertical stretch by a factor of 3 makes the graph steeper.
• Among the given graphs (A, B, C, D), graph B matches these transformations: it is shifted right, reflected over the \( x \)-axis, stretched vertically, and has its vertex at \( (6, 0) \) with the correct shape.

Final Answer

• The graph is translated 6 units \(\boldsymbol{\text{right}}\).
• The graph is \(\boldsymbol{\text{reflected over the } x\text{-axis}}\).
• The graph has a vertical \(\boldsymbol{\text{stretch by a factor of 3}}\).
• The function \( y=-3\sqrt{x - 6} \) is represented by graph \(\boldsymbol{B}\).

Answer:

To analyze the transformation of the parent function \( y = \sqrt{x} \) to \( y = -3\sqrt{x - 6} \), we follow these steps:

Step 1: Horizontal Translation

The general form for horizontal translation is \( y = \sqrt{x - h} \), where \( h \) is the number of units shifted horizontally. For \( y = \sqrt{x - 6} \), \( h = 6 \). Since \( h > 0 \), the graph is translated 6 units to the right.

Step 2: Reflection

The negative sign in front of the radical (\( -3\sqrt{x - 6} \)) indicates a reflection. In the form \( y = a\sqrt{x - h} \), a negative \( a \) reflects the graph over the \( x \)-axis. Here, \( a = -3 \), so the graph is reflected over the \( x \)-axis.

Step 3: Vertical Stretch

The coefficient \( |a| \) determines the vertical stretch or compression. For \( y = -3\sqrt{x - 6} \), \( |a| = 3 \). Since \( |a| > 1 \), the graph is vertically stretched by a factor of 3.

Step 4: Identifying the Graph

• The parent function \( y = \sqrt{x} \) has a domain \( x \geq 0 \) and starts at \( (0, 0) \).
• After translation, the vertex moves to \( (6, 0) \).
• Reflection over the \( x \)-axis flips the graph upside - down.
• Vertical stretch by a factor of 3 makes the graph steeper.
• Among the given graphs (A, B, C, D), graph B matches these transformations: it is shifted right, reflected over the \( x \)-axis, stretched vertically, and has its vertex at \( (6, 0) \) with the correct shape.

Final Answer

• The graph is translated 6 units \(\boldsymbol{\text{right}}\).
• The graph is \(\boldsymbol{\text{reflected over the } x\text{-axis}}\).
• The graph has a vertical \(\boldsymbol{\text{stretch by a factor of 3}}\).
• The function \( y=-3\sqrt{x - 6} \) is represented by graph \(\boldsymbol{B}\).