Question

21/41 graph $y = (x - 3)^2 - 4$

Explanation:

Step1: Identify the vertex form

The equation \( y=(x - 3)^{2}-4 \) is in vertex form \( y = a(x - h)^{2}+k \), where \( (h,k) \) is the vertex. Here, \( h = 3 \) and \( k=-4 \), so the vertex is \( (3,-4) \).

Step2: Determine the direction of opening

Since \( a = 1>0 \), the parabola opens upward.

Step3: Find the axis of symmetry

The axis of symmetry is the vertical line \( x = h \), so \( x = 3 \).

Step4: Find additional points

• When \( x = 3 \), \( y=-4 \) (vertex).
• When \( x = 2 \), \( y=(2 - 3)^{2}-4=1 - 4=-3 \).
• When \( x = 4 \), \( y=(4 - 3)^{2}-4=1 - 4=-3 \).
• When \( x = 1 \), \( y=(1 - 3)^{2}-4=4 - 4=0 \).
• When \( x = 5 \), \( y=(5 - 3)^{2}-4=4 - 4=0 \).

Plot the vertex \( (3,-4) \), the points \( (2,-3) \), \( (4,-3) \), \( (1,0) \), \( (5,0) \), and draw a smooth upward - opening parabola with the axis of symmetry \( x = 3 \).

Answer:

The graph is a parabola with vertex at \((3, - 4)\), opening upward, axis of symmetry \(x = 3\), and passing through points like \((2,-3)\), \((4,-3)\), \((1,0)\), \((5,0)\) (plotted and connected smoothly).