 let f(x)= - 3 (x-1)^2+4 and g(x)  be the function graphed below.

Which statement is a true comparison of the properties of f(x) and g(x)? A. The graph of g(x) has a lower maximum value but a higher y-intercept value than the graph of f(x). B. The graph of g(x) has a higher maximum value and a higher y-intercept value than the graph of f(x). C. The graph of g(x) has a lower maximum value and a lower y-intercept value than the graph of f(x). D. The graph of g(x) has a higher maximum value but a lower y-intercept value than the graph of f(x).


To determine the correct comparison between the properties of \( f(x) \) and \( g(x) \), we need to analyze the properties of \( f(x) = -3(x - 1)^2 + 4 \) and then compare it to the function \( g(x) \) based on its graph (which you would need to provide if it's not described).

### Step 1: Analyze \( f(x) = -3(x - 1)^2 + 4 \)
This is a quadratic function in vertex form, \( f(x) = a(x - h)^2 + k \), where:
- \( a = -3 \)
- \( h = 1 \)
- \( k = 4 \)

#### Properties of \( f(x) \):
- **Vertex**: The vertex of \( f(x) \) is \( (1, 4) \), which is the maximum point since \( a = -3 \) (negative, meaning the parabola opens downwards).
  - **Maximum value**: 4 (since \( k = 4 \), the maximum y-value of \( f(x) \) is 4).
- **Y-intercept**: To find the y-intercept, set \( x = 0 \):
  \[
  f(0) = -3(0 - 1)^2 + 4 = -3(1) + 4 = 1
  \]
  So the y-intercept of \( f(x) \) is \( (0, 1) \).

### Step 2: Analyze \( g(x) \) based on the graph
Without the graph of \( g(x) \), I'll outline how to make the comparison:
- **Maximum value of \( g(x) \)**: From the graph, identify the highest point on \( g(x) \).
- **Y-intercept of \( g(x) \)**: Find where \( g(x) \) crosses the y-axis.

### Step 3: Compare properties of \( f(x) \) and \( g(x) \)
Using the information you gather from the graph of \( g(x) \), compare the following:
- Maximum value of \( g(x) \) vs. \( f(x) \)'s maximum value (4).
- Y-intercept of \( g(x) \) vs. \( f(x) \)'s y-intercept (1).

### Answer selection:
Once you have this information, you can choose the correct statement based on whether \( g(x) \)'s maximum and y-intercept are higher or lower compared to \( f(x) \).

- If \( g(x) \) has a **lower maximum** and **higher y-intercept** than \( f(x) \), select **A**.
- If \( g(x) \) has a **higher maximum** and **higher y-intercept** than \( f(x) \), select **B**.
- If \( g(x) \) has a **lower maximum** and **lower y-intercept** than \( f(x) \), select **C**.
- If \( g(x) \) has a **higher maximum** and **lower y-intercept** than \( f(x) \), select **D**.

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Would you like to provide more details about the graph of \( g(x) \), or should I clarify anything else?

Here are 5 questions to further explore:
1. How do we know the direction of a parabola based on its equation?
2. What impact does the value of \( a \) in \( f(x) = a(x - h)^2 + k \) have on the graph?
3. How do you calculate the vertex of a quadratic function in standard form?
4. What does the y-intercept represent in a function graph?
5. How do you determine if a function has a maximum or minimum?

**Tip**: In vertex form, the vertex \( (h, k) \) immediately gives you the maximum or minimum point, making it easier to analyze quadratic functions.

let f(x) = -3(x - 1)^2 + 4 and g(x) be the function graphed below. Which statement is a true comparison of the properties of f(x) and g(x)? A. The graph of g(x) has a lower maximum value but a higher y-intercept value than the graph of f(x). B. The graph of g(x) has a higher maximum value and a higher y-intercept value than the graph of f(x). C. The graph of g(x) has a lower maximum value and a lower y-intercept value than the graph of f(x). D. The graph of g(x) has a higher maximum value but a lower y-intercept value than the graph of f(x).

Analyze and compare the properties of the quadratic function f(x) = -3(x - 1)^2 + 4 and another graphed function g(x). Explore key concepts like maximum values and y-intercepts, suitable for students in Grades 9-11.

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