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 of a parabola, \( y = a(x - h)^2 + k \), where \((h,k)\) is the vertex. Here, \( h = 3 \), \( k=-4 \), so the vertex is \((3,-4)\).

Step2: Determine the direction of opening

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

Step3: Find the axis of symmetry

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

Step4: Find the y - intercept

Set \( x = 0 \): \( y=(0 - 3)^2-4=9 - 4 = 5 \), so the y - intercept is \((0,5)\).

Step5: Find the x - intercepts

Set \( y = 0 \): \( 0=(x - 3)^2-4 \)
\((x - 3)^2=4\)
\(x - 3=\pm2\)
Case 1: \( x - 3 = 2\Rightarrow x=5\)
Case 2: \( x - 3=-2\Rightarrow x = 1\)
So the x - intercepts are \((1,0)\) and \((5,0)\).

Step6: Plot the points and draw the parabola

Plot the vertex \((3,-4)\), the y - intercept \((0,5)\), the x - intercepts \((1,0)\) and \((5,0)\), and use the axis of symmetry \( x = 3 \) to draw the parabola opening upwards.

Answer:

To graph \( y=(x - 3)^2-4 \):

1. Vertex: \((3,-4)\) (from vertex form \( y=a(x - h)^2 + k \)).
2. Direction: Opens upward (\( a = 1>0 \)).
3. Axis of Symmetry: \( x = 3 \).
4. Intercepts:

• Y - intercept: \((0,5)\) (when \( x = 0 \)).
• X - intercepts: \((1,0)\) and \((5,0)\) (when \( y = 0 \)).

Plot these points and draw a parabola opening upward with the vertex at \((3,-4)\), symmetric about \( x = 3 \).