write the equation of the parabola that has it's x-intercepts at (-10,0) and (-2,0)
y-intercept at (0,-4)
wrong, done correctly 2 posts above by 2 humans
I apologize for the mistake and any confusion caused. Thank you for bringing it to my attention.
y = (-1/5)(x+10)(x+2) = (-1/5)x^2 - (12/5)x - 4
You are correct. That is the equation of the parabola that has x-intercepts at (-10, 0) and (-2, 0) and a y-intercept at (0, -4). Thank you for pointing out the mistake.
To find the equation of a parabola, you can use the standard form equation: y = a(x - h)^2 + k, where (h, k) represents the vertex of the parabola.
Step 1: Find the vertex of the parabola:
The x-coordinate of the vertex is the average of the x-intercepts: (-10 - 2)/2 = -12/2 = -6.
The y-coordinate of the vertex is the y-intercept, which is -4.
Therefore, the vertex is (-6, -4).
Step 2: Use the vertex form equation:
Substitute the vertex values into the equation:
y = a(x - (-6))^2 - 4.
Simplify the equation:
y = a(x + 6)^2 - 4.
Step 3: Determine the value of 'a':
To find 'a', you need an additional point on the parabola. Since we already have the y-intercept, substitute (0, -4) into the equation:
-4 = a(0 + 6)^2 - 4.
Simplify and solve for 'a':
0 = 36a - 4.
36a = 4.
a = 4/36 = 1/9.
Step 4: Substitute the value of 'a' back into the equation:
y = (1/9)(x + 6)^2 - 4.
Therefore, the equation of the parabola is y = (1/9)(x + 6)^2 - 4.
To write the equation of a parabola, we need to use the standard form of the quadratic equation: y = ax^2 + bx + c, where a, b, and c are constants.
Given that the parabola has x-intercepts at (-10,0) and (-2,0), we can deduce that these are the points where the parabola crosses the x-axis. This means that the y-coordinate at both points is 0.
We can use these two x-intercepts to determine the factors of the equation. Since the x-intercepts are -10 and -2, the factors of the equation will be (x + 10) and (x + 2).
Therefore, we can write the equation of the parabola as follows:
y = a(x + 10)(x + 2)
Now, let's find the value of 'a' using the y-intercept at (0,-4).
The y-intercept is the point where the parabola crosses the y-axis, which means the x-coordinate of this point is 0. Given that the y-coordinate is -4, we can substitute these values into the equation:
-4 = a(0 + 10)(0 + 2)
Simplifying further:
-4 = a(10)(2)
-4 = 20a
To solve for 'a', we divide both sides by 20:
-4/20 = a
-1/5 = a
Now that we have the value of 'a', we can substitute it back into the equation to get the final equation of the parabola:
y = (-1/5)(x + 10)(x + 2)
Hence, the equation of the parabola is y = (-1/5)(x + 10)(x + 2).
To begin, we know the standard form of a parabola is y = a(x - h)^2 + k, where (h, k) is the vertex. Since we have two x-intercepts, we can set up the following equations using the roots:
(x + 10)(x + 2) = 0
x^2 + 12x + 20 = 0
Solving for x using the quadratic formula, we get:
x = (-12 ± √(12^2 - 4(1)(20))) / (2(1))
x = (-12 ± √(64)) / 2
x = -6 ± 4
So the x-intercepts are (-10,0) and (-2,0). To find the vertex, we can use the axis of symmetry, which is the line midway between the x-intercepts:
axis of symmetry = (10 + 2) / 2
axis of symmetry = 6
Therefore, the vertex is (6,-4). Plugging this into our standard form equation, we get:
y = a(x - 6)^2 - 4
To find the value of a, we can use either of the x-intercepts:
0 = a(-10 - 6)^2 - 4
40a = 4
a = 1/10
Therefore, the equation of the parabola is:
y = (1/10)(x - 6)^2 - 4