Explore Westonci.ca, the premier Q&A site that helps you find precise answers to your questions, no matter the topic. Our platform connects you with professionals ready to provide precise answers to all your questions in various areas of expertise. Join our Q&A platform to connect with experts dedicated to providing accurate answers to your questions in various fields.
Sagot :
Certainly! Let's find the intercepts for the given equation [tex]\(y = x^2 - 9\)[/tex] and then plot the points to graph the equation.
### Finding the Intercepts:
1. Y-Intercept:
The y-intercept is found by setting [tex]\(x = 0\)[/tex] in the equation and solving for [tex]\(y\)[/tex].
[tex]\[ y = (0)^2 - 9 = -9 \][/tex]
So, the y-intercept is [tex]\((0, -9)\)[/tex].
2. X-Intercepts:
The x-intercepts are found by setting [tex]\(y = 0\)[/tex] in the equation and solving for [tex]\(x\)[/tex].
[tex]\[ 0 = x^2 - 9 \][/tex]
This implies:
[tex]\[ x^2 = 9 \][/tex]
Taking the square root of both sides, we get:
[tex]\[ x = \pm 3 \][/tex]
So, the x-intercepts are at [tex]\((-3, 0)\)[/tex] and [tex]\((3, 0)\)[/tex].
### Plotting Points to Graph the Equation:
Now, we can plot points to understand the shape of the graph. We'll choose a few values for [tex]\(x\)[/tex] and find their corresponding [tex]\(y\)[/tex] values.
- When [tex]\(x = -4\)[/tex],
[tex]\[ y = (-4)^2 - 9 = 16 - 9 = 7 \][/tex]
Point: [tex]\((-4, 7)\)[/tex]
- When [tex]\(x = -2\)[/tex],
[tex]\[ y = (-2)^2 - 9 = 4 - 9 = -5 \][/tex]
Point: [tex]\((-2, -5)\)[/tex]
- When [tex]\(x = -1\)[/tex],
[tex]\[ y = (-1)^2 - 9 = 1 - 9 = -8 \][/tex]
Point: [tex]\((-1, -8)\)[/tex]
- When [tex]\(x = 1\)[/tex],
[tex]\[ y = (1)^2 - 9 = 1 - 9 = -8 \][/tex]
Point: [tex]\((1, -8)\)[/tex]
- When [tex]\(x = 2\)[/tex],
[tex]\[ y = (2)^2 - 9 = 4 - 9 = -5 \][/tex]
Point: [tex]\((2, -5)\)[/tex]
- When [tex]\(x = 4\)[/tex],
[tex]\[ y = (4)^2 - 9 = 16 - 9 = 7 \][/tex]
Point: [tex]\((4, 7)\)[/tex]
### Graphing:
Using the points we calculated:
[tex]\[ (-4, 7), (-2, -5), (-1, -8), (0, -9), (1, -8), (2, -5), (4, 7) \][/tex]
we can plot these on a coordinate plane. Connect these points smoothly in a parabolic shape to get the graph of the equation [tex]\(y = x^2 - 9\)[/tex].
The graph of this equation will be a parabola opening upwards with its vertex at [tex]\((0, -9)\)[/tex], intersecting the x-axis at [tex]\((-3, 0)\)[/tex] and [tex]\((3, 0)\)[/tex], and the y-axis at [tex]\((0, -9)\)[/tex].
This gives a complete understanding of the intercepts and the plot of the equation [tex]\(y = x^2 - 9\)[/tex].
### Finding the Intercepts:
1. Y-Intercept:
The y-intercept is found by setting [tex]\(x = 0\)[/tex] in the equation and solving for [tex]\(y\)[/tex].
[tex]\[ y = (0)^2 - 9 = -9 \][/tex]
So, the y-intercept is [tex]\((0, -9)\)[/tex].
2. X-Intercepts:
The x-intercepts are found by setting [tex]\(y = 0\)[/tex] in the equation and solving for [tex]\(x\)[/tex].
[tex]\[ 0 = x^2 - 9 \][/tex]
This implies:
[tex]\[ x^2 = 9 \][/tex]
Taking the square root of both sides, we get:
[tex]\[ x = \pm 3 \][/tex]
So, the x-intercepts are at [tex]\((-3, 0)\)[/tex] and [tex]\((3, 0)\)[/tex].
### Plotting Points to Graph the Equation:
Now, we can plot points to understand the shape of the graph. We'll choose a few values for [tex]\(x\)[/tex] and find their corresponding [tex]\(y\)[/tex] values.
- When [tex]\(x = -4\)[/tex],
[tex]\[ y = (-4)^2 - 9 = 16 - 9 = 7 \][/tex]
Point: [tex]\((-4, 7)\)[/tex]
- When [tex]\(x = -2\)[/tex],
[tex]\[ y = (-2)^2 - 9 = 4 - 9 = -5 \][/tex]
Point: [tex]\((-2, -5)\)[/tex]
- When [tex]\(x = -1\)[/tex],
[tex]\[ y = (-1)^2 - 9 = 1 - 9 = -8 \][/tex]
Point: [tex]\((-1, -8)\)[/tex]
- When [tex]\(x = 1\)[/tex],
[tex]\[ y = (1)^2 - 9 = 1 - 9 = -8 \][/tex]
Point: [tex]\((1, -8)\)[/tex]
- When [tex]\(x = 2\)[/tex],
[tex]\[ y = (2)^2 - 9 = 4 - 9 = -5 \][/tex]
Point: [tex]\((2, -5)\)[/tex]
- When [tex]\(x = 4\)[/tex],
[tex]\[ y = (4)^2 - 9 = 16 - 9 = 7 \][/tex]
Point: [tex]\((4, 7)\)[/tex]
### Graphing:
Using the points we calculated:
[tex]\[ (-4, 7), (-2, -5), (-1, -8), (0, -9), (1, -8), (2, -5), (4, 7) \][/tex]
we can plot these on a coordinate plane. Connect these points smoothly in a parabolic shape to get the graph of the equation [tex]\(y = x^2 - 9\)[/tex].
The graph of this equation will be a parabola opening upwards with its vertex at [tex]\((0, -9)\)[/tex], intersecting the x-axis at [tex]\((-3, 0)\)[/tex] and [tex]\((3, 0)\)[/tex], and the y-axis at [tex]\((0, -9)\)[/tex].
This gives a complete understanding of the intercepts and the plot of the equation [tex]\(y = x^2 - 9\)[/tex].
We appreciate your visit. Hopefully, the answers you found were beneficial. Don't hesitate to come back for more information. We hope you found this helpful. Feel free to come back anytime for more accurate answers and updated information. Get the answers you need at Westonci.ca. Stay informed by returning for our latest expert advice.
