Question

Let g be a translation 1 unit down and 5 units right, followed by a reflection in the x-axis of the graph of f(x) = -1/2 4*√x +3/2 + Write a

rule for g described by the transformations of the graph of f.

Answer 1

`\[ g(x) = (√(4x - 20) - 1)/(2) \]` is the final rule for the given composition of transformations applied to the function
`\(f(x) = -(1)/(2) √(4x) + (3)/(2)\).`

To determine the rule for the composition of transformations described by the translation 1 unit down and 5 units right, followed by a reflection in the x-axis, applied to the graph of
`\(f(x) = -(1)/(2) √(4x) + (3)/(2)\),` we need to apply each transformation sequentially.

1. **Translation Down and Right:**

The translation 1 unit down and 5 units right can be expressed as
`\(g_1(x) = f(x-5) - 1\).` This shifts the graph of f one unit downward and five units to the right.

2. **Reflection in the X-axis:**

The reflection in the x-axis is given by
`\(g_2(x) = -g_1(x)\)`, which flips the graph vertically.

Combining these transformations, we get the rule for the composition of transformations:

`\[ g(x) = -\left(-(1)/(2) √(4(x-5)) + (3)/(2) - 1\right) \]`

Let's simplify the given expression step by step:

**Distribute the negative sign:**

`\[ g(x) = (1)/(2) √(4(x-5)) - (3)/(2) + 1 \]`

**Simplify the square root:**

`\[ g(x) = (1)/(2) √(4x - 20) - (3)/(2) + 1 \]`

**Multiply the fraction:**

`\[ g(x) = (√(4x - 20))/(2) - (3)/(2) + 1 \]`

**Combine the fractions:**

`\[ g(x) = (√(4x - 20) - 3 + 2)/(2) \]`

**Combine like terms in the numerator:**

`\[ g(x) = (√(4x - 20) - 1)/(2) \]`

So, the simplified expression for the composition of transformations is:

`\[ g(x) = (√(4x - 20) - 1)/(2) \]`

This is the final rule for the given composition of transformations applied to the function
`\(f(x) = -(1)/(2) √(4x) + (3)/(2)\).`