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.