To graph the equation [tex]\( y = x^2 + 8x + 15 \)[/tex], follow these steps:
1. Find the Roots: The roots (or x-intercepts) are the points where [tex]\( y = 0 \)[/tex]. The solutions to the equation [tex]\( x^2 + 8x + 15 = 0 \)[/tex] are:
[tex]\[
x = -5 \quad \text{and} \quad x = -3
\][/tex]
Therefore, the roots are [tex]\((-5, 0)\)[/tex] and [tex]\((-3, 0)\)[/tex].
2. Find the Vertex: The vertex of a parabola given by the equation [tex]\( y = ax^2 + bx + c \)[/tex] has its x-coordinate at:
[tex]\[
x = -\frac{b}{2a}
\][/tex]
For the equation [tex]\( y = x^2 + 8x + 15 \)[/tex], here [tex]\( a = 1 \)[/tex] and [tex]\( b = 8 \)[/tex]. Thus, the x-coordinate of the vertex is:
[tex]\[
x = -\frac{8}{2 \cdot 1} = -4
\][/tex]
To find the y-coordinate of the vertex, substitute [tex]\( x = -4 \)[/tex] back into the equation:
[tex]\[
y = (-4)^2 + 8(-4) + 15 = 16 - 32 + 15 = -1
\][/tex]
Hence, the vertex is at [tex]\((-4, -1)\)[/tex].
3. Find the Y-Intercept: The point where the parabola crosses the y-axis is found by setting [tex]\( x = 0 \)[/tex]:
[tex]\[
y = (0)^2 + 8(0) + 15 = 15
\][/tex]
Therefore, the y-intercept is [tex]\((0, 15)\)[/tex].
4. Plot a Another Point: Choose an additional x-value to find another point for accuracy. Let's select [tex]\( x = -4 \)[/tex]:
[tex]\[
y = (-4)^2 + 8(-4) + 15 = 16 - 32 + 15 = -1
\][/tex]
This reaffirms the vertex point [tex]\((-4, -1)\)[/tex]. We can instead choose a new point; for instance, [tex]\( x = -5 \)[/tex]:
[tex]\[
y = (-5)^2 + 8(-5) + 15 = 25 - 40 + 15 = 0
\][/tex]
Now, choosing another point like [tex]\( x = -6 \)[/tex]:
[tex]\[
y = (-6)^2 + 8(-6) + 15 = 36 - 48 + 15 = 3
\][/tex]
So, another point is [tex]\((-6, 3)\)[/tex].
Summary of Points:
1. Root 1: [tex]\((-5, 0)\)[/tex]
2. Root 2: [tex]\((-3, 0)\)[/tex]
3. Vertex: [tex]\((-4, -1)\)[/tex]
4. Y-Intercept: [tex]\((0, 15)\)[/tex]
5. Another Point: [tex]\((-6, 3)\)[/tex]
Using these points, you can plot the following points on the graph set of axes:
- [tex]\((-5, 0)\)[/tex]
- [tex]\((-3, 0)\)[/tex]
- [tex]\((-4, -1)\)[/tex]
- [tex]\((0, 15)\)[/tex]
- [tex]\((-6, 3)\)[/tex]
Finally, connect these points with a smooth curved line to form the parabola.
The vertex of the parabola is at [tex]\((-4, -1)\)[/tex].
