Westonci.ca is the ultimate Q&A platform, offering detailed and reliable answers from a knowledgeable community. Discover solutions to your questions from experienced professionals across multiple fields on our comprehensive Q&A platform. Join our Q&A platform to connect with experts dedicated to providing accurate answers to your questions in various fields.
Sagot :
Sure! Let's graph the equation [tex]\( y = -x^2 + 12x - 35 \)[/tex].
1. Vertex of the Parabola:
Parabolas have a highest or lowest point called the vertex. For a quadratic function in the form [tex]\( y = ax^2 + bx + c \)[/tex], the x-coordinate of the vertex is given by [tex]\( x = -\frac{b}{2a} \)[/tex].
For our equation [tex]\( y = -x^2 + 12x - 35 \)[/tex]:
- [tex]\( a = -1 \)[/tex]
- [tex]\( b = 12 \)[/tex]
- So, the x-coordinate [tex]\( x = -\frac{12}{2(-1)} = 6 \)[/tex].
To find the y-coordinate of the vertex, plug [tex]\( x = 6 \)[/tex] back into the equation:
[tex]\[ y = -(6)^2 + 12(6) - 35 = -36 + 72 - 35 = 1 \][/tex]
Thus, the vertex is at [tex]\( (6, 1) \)[/tex].
2. Roots (x-intercepts):
To find the x-intercepts, solve for [tex]\( x \)[/tex] when [tex]\( y = 0 \)[/tex]:
[tex]\[ -x^2 + 12x - 35 = 0 \][/tex]
The solutions to this quadratic equation are:
[tex]\[ x = 5 \quad \text{and} \quad x = 7 \][/tex]
So, the roots are [tex]\( (5, 0) \)[/tex] and [tex]\( (7, 0) \)[/tex].
3. Two Additional Points:
We can choose any other two x-values to find additional points. Here, we'll take [tex]\( x = 4 \)[/tex] and [tex]\( x = 8 \)[/tex].
For [tex]\( x = 4 \)[/tex]:
[tex]\[ y = -(4)^2 + 12(4) - 35 = -16 + 48 - 35 = -3 \][/tex]
So, the point is [tex]\( (4, -3) \)[/tex].
For [tex]\( x = 8 \)[/tex]:
[tex]\[ y = -(8)^2 + 12(8) - 35 = -64 + 96 - 35 = -3 \][/tex]
So, the point is [tex]\( (8, -3) \)[/tex].
4. Summary of Points:
- Vertex: [tex]\( (6, 1) \)[/tex]
- Roots: [tex]\( (5, 0) \)[/tex] and [tex]\( (7, 0) \)[/tex]
- Additional Points: [tex]\( (4, -3) \)[/tex] and [tex]\( (8, -3) \)[/tex]
You can plot these points on a coordinate axis:
- [tex]\( (4, -3) \)[/tex]
- [tex]\( (5, 0) \)[/tex]
- [tex]\( (6, 1) \)[/tex]
- [tex]\( (7, 0) \)[/tex]
- [tex]\( (8, -3) \)[/tex]
Once you plot these points, draw a smooth curve passing through them to complete the graph of the equation [tex]\( y = -x^2 + 12x - 35 \)[/tex]. This curve will be a downward-opening parabola.
1. Vertex of the Parabola:
Parabolas have a highest or lowest point called the vertex. For a quadratic function in the form [tex]\( y = ax^2 + bx + c \)[/tex], the x-coordinate of the vertex is given by [tex]\( x = -\frac{b}{2a} \)[/tex].
For our equation [tex]\( y = -x^2 + 12x - 35 \)[/tex]:
- [tex]\( a = -1 \)[/tex]
- [tex]\( b = 12 \)[/tex]
- So, the x-coordinate [tex]\( x = -\frac{12}{2(-1)} = 6 \)[/tex].
To find the y-coordinate of the vertex, plug [tex]\( x = 6 \)[/tex] back into the equation:
[tex]\[ y = -(6)^2 + 12(6) - 35 = -36 + 72 - 35 = 1 \][/tex]
Thus, the vertex is at [tex]\( (6, 1) \)[/tex].
2. Roots (x-intercepts):
To find the x-intercepts, solve for [tex]\( x \)[/tex] when [tex]\( y = 0 \)[/tex]:
[tex]\[ -x^2 + 12x - 35 = 0 \][/tex]
The solutions to this quadratic equation are:
[tex]\[ x = 5 \quad \text{and} \quad x = 7 \][/tex]
So, the roots are [tex]\( (5, 0) \)[/tex] and [tex]\( (7, 0) \)[/tex].
3. Two Additional Points:
We can choose any other two x-values to find additional points. Here, we'll take [tex]\( x = 4 \)[/tex] and [tex]\( x = 8 \)[/tex].
For [tex]\( x = 4 \)[/tex]:
[tex]\[ y = -(4)^2 + 12(4) - 35 = -16 + 48 - 35 = -3 \][/tex]
So, the point is [tex]\( (4, -3) \)[/tex].
For [tex]\( x = 8 \)[/tex]:
[tex]\[ y = -(8)^2 + 12(8) - 35 = -64 + 96 - 35 = -3 \][/tex]
So, the point is [tex]\( (8, -3) \)[/tex].
4. Summary of Points:
- Vertex: [tex]\( (6, 1) \)[/tex]
- Roots: [tex]\( (5, 0) \)[/tex] and [tex]\( (7, 0) \)[/tex]
- Additional Points: [tex]\( (4, -3) \)[/tex] and [tex]\( (8, -3) \)[/tex]
You can plot these points on a coordinate axis:
- [tex]\( (4, -3) \)[/tex]
- [tex]\( (5, 0) \)[/tex]
- [tex]\( (6, 1) \)[/tex]
- [tex]\( (7, 0) \)[/tex]
- [tex]\( (8, -3) \)[/tex]
Once you plot these points, draw a smooth curve passing through them to complete the graph of the equation [tex]\( y = -x^2 + 12x - 35 \)[/tex]. This curve will be a downward-opening parabola.
Thank you for trusting us with your questions. We're here to help you find accurate answers quickly and efficiently. We appreciate your time. Please come back anytime for the latest information and answers to your questions. Get the answers you need at Westonci.ca. Stay informed with our latest expert advice.