To graph the parabola [tex]\( y = x^2 + 8x + 14 \)[/tex], we need to determine five key points: the vertex, two points to the left of the vertex, and two points to the right of the vertex. Here are the steps:
1. Find the Vertex:
The vertex of a parabola given by [tex]\( y = ax^2 + bx + c \)[/tex] is found using the formula for the x-coordinate of the vertex [tex]\( x = -\frac{b}{2a} \)[/tex].
For the equation [tex]\( y = x^2 + 8x + 14 \)[/tex]:
[tex]\[
a = 1, \quad b = 8, \quad c = 14
\][/tex]
The x-coordinate of the vertex is:
[tex]\[
x = -\frac{8}{2 \cdot 1} = -4
\][/tex]
To find the y-coordinate, substitute [tex]\( x = -4 \)[/tex] back into the equation:
[tex]\[
y = (-4)^2 + 8(-4) + 14 = 16 - 32 + 14 = -2
\][/tex]
So, the vertex is:
[tex]\[
(-4, -2)
\][/tex]
2. Find Two Points to the Left of the Vertex:
Choose [tex]\( x \)[/tex] values slightly left of the vertex, such as [tex]\( x = -5 \)[/tex] and [tex]\( x = -6 \)[/tex].
For [tex]\( x = -5 \)[/tex]:
[tex]\[
y = (-5)^2 + 8(-5) + 14 = 25 - 40 + 14 = -1
\][/tex]
This gives the point:
[tex]\[
(-5, -1)
\][/tex]
For [tex]\( x = -6 \)[/tex]:
[tex]\[
y = (-6)^2 + 8(-6) + 14 = 36 - 48 + 14 = 2
\][/tex]
This gives the point:
[tex]\[
(-6, 2)
\][/tex]
3. Find Two Points to the Right of the Vertex:
Choose [tex]\( x \)[/tex] values slightly right of the vertex, such as [tex]\( x = -3 \)[/tex] and [tex]\( x = -2 \)[/tex].
For [tex]\( x = -3 \)[/tex]:
[tex]\[
y = (-3)^2 + 8(-3) + 14 = 9 - 24 + 14 = -1
\][/tex]
This gives the point:
[tex]\[
(-3, -1)
\][/tex]
For [tex]\( x = -2 \)[/tex]:
[tex]\[
y = (-2)^2 + 8(-2) + 14 = 4 - 16 + 14 = 2
\][/tex]
This gives the point:
[tex]\[
(-2, 2)
\][/tex]
4. Summary of Points:
The five points to plot on the graph are:
[tex]\[
(-4, -2), \quad (-5, -1), \quad (-6, 2), \quad (-3, -1), \quad (-2, 2)
\][/tex]
5. Graph the Parabola:
Plot these points on the coordinate plane:
- Vertex: [tex]\( (-4, -2) \)[/tex]
- Left of Vertex: [tex]\( (-5, -1) \)[/tex] and [tex]\( (-6, 2) \)[/tex]
- Right of Vertex: [tex]\( (-3, -1) \)[/tex] and [tex]\( (-2, 2) \)[/tex]
After plotting these points, draw a smooth curve through them to complete the graph of the parabola [tex]\( y = x^2 + 8x + 14 \)[/tex].
To complete the graphing process, you can use graphing tools or software to accurately plot these points and visualize the parabola.
