Answer :
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].
