Question

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graph the function.
$f(x) = \frac{3}{2}(x - 4)^2 + 3$
(graph with y-axis labeled from 6 to 9)

Explanation:

Step1: Identify the vertex form

The function \( f(x)=\frac{3}{2}(x - 4)^{2}+3 \) is in vertex form \( y = a(x - h)^{2}+k \), where the vertex is \( (h,k)=(4,3) \).

Step2: Determine the direction and width

Since \( a=\frac{3}{2}>0 \), the parabola opens upward. The value of \( |a|=\frac{3}{2}>1 \), so the parabola is narrower than the standard \( y = x^{2} \).

Step3: Find additional points

• When \( x = 4 \), \( f(4)=\frac{3}{2}(0)^{2}+3 = 3 \) (vertex).
• When \( x = 5 \), \( f(5)=\frac{3}{2}(1)^{2}+3=\frac{3}{2}+3=\frac{9}{2}=4.5 \).
• When \( x = 3 \), \( f(3)=\frac{3}{2}(-1)^{2}+3=\frac{3}{2}+3 = 4.5 \).
• When \( x = 6 \), \( f(6)=\frac{3}{2}(2)^{2}+3=\frac{3}{2}\times4 + 3=6 + 3=9 \).
• When \( x = 2 \), \( f(2)=\frac{3}{2}(-2)^{2}+3=\frac{3}{2}\times4+3 = 6 + 3=9 \).

Plot the vertex \( (4,3) \), and the points \( (5,4.5) \), \( (3,4.5) \), \( (6,9) \), \( (2,9) \), then draw a parabola opening upward through these points.

Answer:

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

1. Plot the vertex at \( (4, 3) \).
2. Since \( a=\frac{3}{2}>0 \), draw a parabola opening upward, narrower than \( y = x^{2} \).
3. Use additional points (e.g., \( (3, 4.5) \), \( (5, 4.5) \), \( (2, 9) \), \( (6, 9) \)) to sketch the parabola.