Question

The line with the negative slope is a Reflection of the line with a positive slope. The equation of the line with a positive slope is f(x)=5/4x-2. Find the equation of its Reflection g(x) along with the function rule that describes the Reflection in the coordinate plane.

The Reflection is over the y axis such that g(x)=-4x/5+2. The rule for the Reflection is ry-axis.
The Reflection is over the y axis such that g(x)=-5x/4+2. The rule for the Reflection is ryaxis..
The Reflection is over the x axis such that g(x)=-5x/4+2. The rule for the Reflection is r x axis.
The Reflection is over the x axis such that g(x)=-4x/5+2. The rule for the Reflection is r x axis

Answer 1

Are you sure?

Answer 2

To find the reflection of the line \( f(x) = \frac{5}{4}x - 2 \) and understand the type of reflection involved, we need to consider how reflections work in the coordinate plane.

1. **Reflection over the y-axis**: When a function \( f(x) \) is reflected over the y-axis, the new function \( g(x) \) will be \( g(x) = f(-x) \). This changes the sign of the \( x \) term.

2. **Reflection over the x-axis**: When a function \( f(x) \) is reflected over the x-axis, the new function \( g(x) \) will be \( g(x) = -f(x) \). This changes the sign of the entire function.

### Step 1: Find the reflection over the y-axis
Starting with:
\[ f(x) = \frac{5}{4}x - 2 \]

Reflect over the y-axis:
\[ g(x) = f(-x) = \frac{5}{4}(-x) - 2 = -\frac{5}{4}x - 2 \]

### Step 2: Find the reflection over the x-axis
Using the same original function:
\[ f(x) = \frac{5}{4}x - 2 \]

Reflect over the x-axis:
\[ g(x) = -f(x) = -\left(\frac{5}{4}x - 2\right) = -\frac{5}{4}x + 2 \]

### Step 3: Analyze the given options
From the steps above:

- The reflection over the **y-axis** gives \( g(x) = -\frac{5}{4}x - 2 \).
- The reflection over the **x-axis** gives \( g(x) = -\frac{5}{4}x + 2 \).

Now let's consider the options given:

1. The Reflection is over the **y-axis** such that \( g(x) = -\frac{4}{5}x + 2 \). The rule for the Reflection is \( r_{y-axis} \).
2. The Reflection is over the **y-axis** such that \( g(x) = -\frac{5}{4}x + 2 \). The rule for the Reflection is \( r_{yaxis} \).
3. The Reflection is over the **x-axis** such that \( g(x) = -\frac{5}{4}x + 2 \). The rule for the Reflection is \( r_{x axis} \).
4. The Reflection is over the **x-axis** such that \( g(x) = -\frac{4}{5}x + 2 \). The rule for the Reflection is \( r_{x axis} \).

The correct reflection is:
- Over the **x-axis** with \( g(x) = -\frac{5}{4}x + 2 \) which follows the rule \( r_{x axis} \).

Thus, the correct choice is:
**The Reflection is over the x axis such that \( g(x) = -\frac{5}{4}x + 2 \). The rule for the Reflection is \( r_{x axis} \).**

Answer 3

Let me clarify and ensure accuracy regarding the reflection of the line \( f(x) = \frac{5}{4}x - 2 \).

### Original Function
The equation of the line with a positive slope is:
\[ f(x) = \frac{5}{4}x - 2 \]

### Reflection Types

1. **Reflection Over the y-axis**:
- For any function \( f(x) \), reflecting it over the y-axis results in \( g(x) = f(-x) \).

So we compute:
\[
g(x) = f(-x) = \frac{5}{4}(-x) - 2 = -\frac{5}{4}x - 2
\]

2. **Reflection Over the x-axis**:
- For reflection over the x-axis, we take the negative of the entire function, resulting in:
\[
g(x) = -f(x) = -\left(\frac{5}{4}x - 2\right) = -\frac{5}{4}x + 2
\]

### Analyzing Your Choices
From my calculations:

- Over the y-axis, the reflection results in \( g(x) = -\frac{5}{4}x - 2 \).
- Over the x-axis, the reflection results in \( g(x) = -\frac{5}{4}x + 2 \).

Now let's review your options again:

1. **Over the y-axis such that** \( g(x) = -\frac{4}{5}x + 2 \) (incorrect)
2. **Over the y-axis such that** \( g(x) = -\frac{5}{4}x + 2 \) (incorrect)
3. **Over the x-axis such that** \( g(x) = -\frac{5}{4}x + 2 \) (correct)
4. **Over the x-axis such that** \( g(x) = -\frac{4}{5}x + 2 \) (incorrect)

The accurate answer regarding the reflection of the given line \( f(x) \) reflects over the **x-axis** resulting in:
\[ g(x) = -\frac{5}{4}x + 2 \]

### Conclusion
Thus, I confirm that the correct answer is:
**The Reflection is over the x-axis such that \( g(x) = -\frac{5}{4}x + 2 \). The rule for the Reflection is \( r_{x-axis} \).** Thank you for your patience, and I appreciate your diligence in verifying this!