Question

1. graph $y = 2(x - 5)^2 - 6$

Explanation:

Step1: Identify the vertex form

The equation \( y = 2(x - 5)^2 - 6 \) is in vertex form \( y = a(x - h)^2 + k \), where \((h, k)\) is the vertex. Here, \( h = 5 \) and \( k = -6 \), so the vertex is \((5, -6)\).

Step2: Determine the direction of opening

Since \( a = 2 > 0 \), the parabola opens upward.

Step3: Find the axis of symmetry

The axis of symmetry is the vertical line \( x = h \), so \( x = 5 \).

Step4: Find a few more points

• When \( x = 4 \): \( y = 2(4 - 5)^2 - 6 = 2(1) - 6 = -4 \), so the point is \((4, -4)\).
• When \( x = 6 \): \( y = 2(6 - 5)^2 - 6 = 2(1) - 6 = -4 \), so the point is \((6, -4)\).
• When \( x = 3 \): \( y = 2(3 - 5)^2 - 6 = 2(4) - 6 = 2 \), so the point is \((3, 2)\).
• When \( x = 7 \): \( y = 2(7 - 5)^2 - 6 = 2(4) - 6 = 2 \), so the point is \((7, 2)\).

Step5: Plot the vertex, axis of symmetry, and the additional points, then draw the parabola.

Answer:

To graph \( y = 2(x - 5)^2 - 6 \):

1. Plot the vertex \((5, -6)\).
2. Draw the axis of symmetry \( x = 5 \).
3. Plot the points \((4, -4)\), \((6, -4)\), \((3, 2)\), \((7, 2)\) (and more if needed).
4. Draw a parabola opening upward through these points.