https://file.aiquickdraw.com/mathbot/file/17255996502094na490ri.jpg

The uploaded image contains a mathematical function and a corresponding blank chart for analyzing its properties. The function given is:

\[
f(x) = -3\sqrt{-2x} + 1
\]

Let's go over the different aspects of this function step-by-step.

### 1. **Parent Function**:
The parent function is \( f(x) = \sqrt{x} \), which is the square root function.

### 2. **Transformations**:
- **Horizontal Stretch/Compression and Reflection**: The expression inside the square root is \( -2x \). The negative sign inside reflects the graph across the y-axis, and the factor of 2 causes a horizontal compression by a factor of \( \frac{1}{2} \).
- **Vertical Stretch/Compression and Reflection**: The leading coefficient of -3 reflects the graph vertically and stretches it by a factor of 3.
- **Vertical Translation**: The constant term \( +1 \) translates the graph 1 unit upwards.

### 3. **Domain**:
To find the domain, we need the expression inside the square root to be non-negative:
\[
-2x \geq 0 \implies x \leq 0
\]
Thus, the domain is \( (-\infty, 0] \).

### 4. **Range**:
Since the graph is reflected vertically and shifted upwards by 1 unit, the range is \( (-\infty, 1] \).

### 5. **X-intercepts**:
To find the x-intercepts, set \( f(x) = 0 \) and solve for \( x \):
\[
-3\sqrt{-2x} + 1 = 0
\]
\[
\sqrt{-2x} = \frac{1}{3}
\]
\[
-2x = \frac{1}{9} \implies x = -\frac{1}{18}
\]
Thus, the x-intercept is \( \left( -\frac{1}{18}, 0 \right) \).

### 6. **Increasing/Decreasing Intervals**:
- The function is **decreasing** throughout its domain because of the negative leading coefficient.
- There are **no increasing intervals**.

### 7. **End Behavior**:
As \( x \to -\infty \), \( f(x) \to -\infty \).

Would you like me to plot the graph of this function on the grid and complete other parts of the table? If so, I can do that for you! Let me know if you need any further details.

Here are 5 questions related to this:
1. How does a vertical reflection affect the range of a function?
2. Why is the domain restricted to non-positive numbers for this function?
3. What impact does the horizontal compression by 1/2 have on the graph of the square root function?
4. How would the function behave if there were no negative sign in front of the square root?
5. Why does the vertical shift affect the end behavior of the function?

Tip: When dealing with square root functions, always check the inside expression to determine valid x-values (domain).

Analyze the function f(x) = -3√(-2x) + 1, including its parent function, transformations, domain, range, intercepts, increasing/decreasing intervals, and end behavior.

This detailed analysis covers the function f(x) = -3√(-2x) + 1, including transformations such as vertical reflection and horizontal compression. Explore the domain, range, intercepts, and behavior of the function. Suitable for high school students learning advanced algebra and function analysis.

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