Question

f(x) = \frac{3}{2}(x - 4)^2 + 3

Explanation:

Step1: Identify the vertex form of a parabola

The vertex form of a parabola is \( f(x) = a(x - h)^2 + k \), where \((h, k)\) is the vertex. For the given function \( f(x) = \frac{3}{2}(x - 4)^2 + 3 \), we compare it to the vertex form.

Step2: Determine the vertex coordinates

By comparing \( f(x) = \frac{3}{2}(x - 4)^2 + 3 \) with \( f(x) = a(x - h)^2 + k \), we see that \( h = 4 \) and \( k = 3 \). So the vertex of the parabola should be at the point \((4, 3)\).

Step3: Analyze the graph

Looking at the provided graph, we check the coordinates of the vertex. The green dot (vertex) should be at \((4, 3)\) based on the function. Let's verify the grid: the x - coordinate 4 and y - coordinate 3 on the grid. The graph of the parabola \( f(x)=\frac{3}{2}(x - 4)^2+3 \) is a parabola that opens upwards (since \( a=\frac{3}{2}>0 \)) with vertex at \((4, 3)\). The graph in the image, when we check the vertex (the green dot), if we assume the grid has integer coordinates, the vertex should be at (4,3). Also, the direction of the parabola (opening upwards) matches the positive coefficient \( \frac{3}{2} \) of the squared term.

Answer:

The vertex of the parabola \( f(x)=\frac{3}{2}(x - 4)^2 + 3 \) is at \((4, 3)\) and the graph is a parabola opening upwards with this vertex, which is consistent with the given function. If the question was about identifying the vertex or analyzing the graph, the vertex is \((4, 3)\).