Question

1. find the vertex, axis of symmetry, and direction of opening of the parabola. use this information to sketch the graph.

(a) ( y = (x - 2)^2 + 1 )
(b) ( y = -\frac{1}{2}(x + 4)^2 + 0 )
(c) ( y = 2(x + 1)^2 - 8 )

Explanation:

Let's solve part (a) first: \( y = (x - 2)^2 + 1 \)

Step1: Recall vertex form of parabola

The vertex form of a parabola is \( y = a(x - h)^2 + k \), where \((h, k)\) is the vertex, the axis of symmetry is \( x = h \), and if \( a > 0 \) it opens upward, if \( a < 0 \) it opens downward.
For \( y = (x - 2)^2 + 1 \), we compare with \( y = a(x - h)^2 + k \). Here, \( a = 1 \), \( h = 2 \), \( k = 1 \).

Step2: Find vertex

From the vertex form, the vertex \((h, k)\) is \((2, 1)\).

Step3: Find axis of symmetry

The axis of symmetry is \( x = h \), so \( x = 2 \).

Step4: Find direction of opening

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

Now part (b): \( y = -\frac{1}{2}(x + 4)^2 + 0 \) (assuming the \( +0 \) is just \( y = -\frac{1}{2}(x + 4)^2 \))

Step1: Recall vertex form

Using \( y = a(x - h)^2 + k \), here we can rewrite \( x + 4 \) as \( x - (-4) \), so \( h = -4 \), \( k = 0 \), \( a = -\frac{1}{2} \).

Step2: Find vertex

Vertex \((h, k)\) is \((-4, 0)\).

Step3: Find axis of symmetry

Axis of symmetry is \( x = h = -4 \).

Step4: Find direction of opening

Since \( a = -\frac{1}{2} < 0 \), the parabola opens downward.

Part (c): \( y = 2(x + 1)^2 - 8 \)

Step1: Recall vertex form

Using \( y = a(x - h)^2 + k \), rewrite \( x + 1 \) as \( x - (-1) \), so \( h = -1 \), \( k = -8 \), \( a = 2 \).

Step2: Find vertex

Vertex \((h, k)\) is \((-1, -8)\).

Step3: Find axis of symmetry

Axis of symmetry is \( x = h = -1 \).

Step4: Find direction of opening

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

Answer:

s:

Part (a)

• Vertex: \((2, 1)\)
• Axis of Symmetry: \( x = 2 \)
• Direction of Opening: Upward

Part (b)

• Vertex: \((-4, 0)\)
• Axis of Symmetry: \( x = -4 \)
• Direction of Opening: Downward

Part (c)

• Vertex: \((-1, -8)\)
• Axis of Symmetry: \( x = -1 \)
• Direction of Opening: Upward