To find the vertex of the given parabola and subsequently graph it, follow these steps:
### Step 1: Simplify the Equation
We start with the given equation:
[tex]\[
\frac{1}{2}(y + 1) = (x - 4)^2
\][/tex]
### Step 2: Eliminate the Fraction
Multiply both sides of the equation by 2 to clear the fraction:
[tex]\[
y + 1 = 2(x - 4)^2
\][/tex]
### Step 3: Isolate [tex]\( y \)[/tex]
Subtract 1 from both sides to isolate [tex]\( y \)[/tex]:
[tex]\[
y = 2(x - 4)^2 - 1
\][/tex]
### Step 4: Identify the Vertex Form
The equation is now in the standard vertex form of a parabola, which is:
[tex]\[
y = a(x - h)^2 + k
\][/tex]
where [tex]\( (h, k) \)[/tex] is the vertex of the parabola.
### Step 5: Determine the Vertex
By comparing the rearranged equation [tex]\( y = 2(x - 4)^2 - 1 \)[/tex] with the standard vertex form [tex]\( y = a(x - h)^2 + k \)[/tex]:
- [tex]\( a = 2 \)[/tex]
- [tex]\( h = 4 \)[/tex]
- [tex]\( k = -1 \)[/tex]
### Vertex
Thus, the vertex of the parabola is:
[tex]\[
(4, -1)
\][/tex]
### Graphing the Parabola:
1. Plot the Vertex:
Plot the point [tex]\( (4, -1) \)[/tex] on the coordinate plane. This is the vertex of the parabola.
2. Determine the Direction:
Since [tex]\( a = 2 \)[/tex] (which is positive), the parabola opens upwards.
3. Symmetry and Additional Points:
Use the vertex symmetry to plot additional points. For example, you can choose [tex]\( x \)[/tex]-values around the vertex [tex]\( x = 4 \)[/tex] and calculate corresponding [tex]\( y \)[/tex]-values.
4. Draw the Parabola:
Sketch the curve by connecting the points smoothly, ensuring the vertex is the minimum point (since the parabola opens upwards).
### Check the Vertex:
Verify that the vertex [tex]\( (4, -1) \)[/tex] fits the parabola equation:
Substitute [tex]\( x = 4 \)[/tex]:
[tex]\[
y = 2(4 - 4)^2 - 1 = 2(0)^2 - 1 = -1
\][/tex]
The point [tex]\( (4, -1) \)[/tex] satisfies the equation.
### Final Answer:
The vertex of the parabola is [tex]\( (4, -1) \)[/tex].
