Question

graph the function. \\( g(x) = -\frac{3}{2}\left(x - \frac{5}{2}\
ight)^2 \\)

Explanation:

Step1: Identify the parent function

The given function is \( g(x) = -\frac{3}{2}(x - 2)^2 \). The parent function is \( y = x^2 \), which is a parabola opening upwards with vertex at \((0,0)\).

Step2: Analyze transformations

• Horizontal shift: The \((x - 2)\) inside the square means a horizontal shift 2 units to the right. So the vertex moves from \((0,0)\) to \((2,0)\) initially (before vertical transformations).
• Vertical stretch/compression and reflection: The coefficient \(-\frac{3}{2}\) has two effects. The negative sign reflects the parabola over the x - axis (so it will open downwards), and the absolute value \(\frac{3}{2}\) (greater than 1) vertically stretches the parabola.
• Vertical shift: There is no vertical shift term (the constant term after the square is 0 in terms of vertical shift from the vertex form \(y=a(x - h)^2+k\), here \(k = 0\) initially, but wait, wait, the function is \(g(x)=-\frac{3}{2}(x - 2)^2\), so \(k = 0\). Wait, maybe I misread earlier. Wait the original function is \(g(x)=-\frac{3}{2}(x - 2)^2\)? Or is there a typo? Wait the user's image shows \(g(x)=-\frac{3}{2}(x - 2)^2\)? Wait no, looking at the image, the function is \(g(x)=-\frac{3}{2}(x - 2)^2\)? Wait the text in the image: \(g(x)=-\frac{3}{2}(x - 2)^2\)? Wait maybe the user made a typo, but assuming the function is \(g(x)=-\frac{3}{2}(x - 2)^2\), let's proceed.

Step3: Find key points

• Vertex: From the vertex form \(y=a(x - h)^2+k\), where \((h,k)\) is the vertex. Here \(h = 2\), \(k = 0\), so vertex is \((2,0)\).
• Other points: Let's find two points on either side of the vertex.
• When \(x=2 + 2=4\) (since the horizontal stretch? Wait no, for the parent function \(y = x^2\), when \(x = 1\), \(y = 1\); \(x=2\), \(y = 4\). For our function, let's plug in \(x=0\): \(g(0)=-\frac{3}{2}(0 - 2)^2=-\frac{3}{2}\times4=-6\). So the point \((0,-6)\) is on the graph.
• When \(x = 4\): \(g(4)=-\frac{3}{2}(4 - 2)^2=-\frac{3}{2}\times4=-6\). So the point \((4,-6)\) is on the graph.

Step4: Plot the points and draw the parabola

• Plot the vertex \((2,0)\), the points \((0,-6)\) and \((4,-6)\). Since the coefficient \(a=-\frac{3}{2}<0\), the parabola opens downwards. Draw a smooth curve through these points, symmetric about the vertical line \(x = 2\) (the axis of symmetry, which is \(x=h = 2\)).

Answer:

To graph \(g(x)=-\frac{3}{2}(x - 2)^2\):

1. Vertex: \((2,0)\) (from vertex form \(y = a(x - h)^2+k\), \(h = 2\), \(k = 0\)).
2. Direction: Opens downward (due to \(a=-\frac{3}{2}<0\)).
3. Key Points: \((0,-6)\) (when \(x = 0\), \(g(0)=-\frac{3}{2}(0 - 2)^2=-6\)) and \((4,-6)\) (when \(x = 4\), \(g(4)=-\frac{3}{2}(4 - 2)^2=-6\)).
4. Graph: Plot the vertex and key points, then draw a downward - opening parabola symmetric about \(x = 2\).

(Note: If the function was intended to have a vertical shift, re - evaluate with the correct function. Assuming the given function is \(g(x)=-\frac{3}{2}(x - 2)^2\), this is the graphing process.)