Which of the following is a tranformation that can be used to graph the function f(x)=-7 square root 5(x+9)+12

Answer 1

To graph the function \( f(x) = -7 \sqrt{5}(x + 9) + 12 \), we can break down its transformations step-by-step.

1. Start with the base function: The base function here is \( g(x) = \sqrt{x} \).

2. Horizontal shift: The term \( (x + 9) \) indicates a horizontal shift. Since it is \( (x + 9) \), we shift the graph 9 units to the left.

3. Stretch/Vertical scaling: The multiplication of \( \sqrt{5} \) means that the graph is stretched vertically by a factor of \( \sqrt{5} \) (which affects the output values).

4. Reflection: The negative sign before the \( 7 \) indicates a reflection over the x-axis. This means that the graph will be flipped upside down.

5. Vertical scaling: The \( -7 \) indicates a vertical scaling by a factor of \( 7 \). Since it is negative, it stretches the graph downwards.

6. Vertical shift: The \( +12 \) at the end indicates a vertical shift upwards by \( 12 \) units.

Putting all the transformations together, we can describe the overall transformation process as follows:

• Start with \( g(x) = \sqrt{x} \).
• Shift left by 9 units: \( g(x + 9) = \sqrt{x + 9} \).
• Stretch by \( \sqrt{5} \): \( g(x + 9) \) becomes \( \sqrt{5} \sqrt{x + 9} \).
• Reflect over the x-axis: \( -\sqrt{5} \sqrt{x + 9} \).
• Stretch by 7: \( -7 \sqrt{5} \sqrt{x + 9} \).
• Finally, shift up by 12: \( -7 \sqrt{5} \sqrt{x + 9} + 12 \).

Thus, the transformations applied to \( y = \sqrt{x} \) to arrive at \( f(x) = -7\sqrt{5}(x + 9) + 12 \) are:

1. Horizontal shift left by 9.
2. Vertical stretch and reflection (a vertical stretch by a factor of 7 and reflect over the x-axis).
3. Vertical shift up by 12.

These transformations will result in the graph of the function \( f(x) \).