The vertex of a parabola is(-2,-4). One x-intercept is 7. What is the other x-intercept?
We can find this without knowing the equation, by using the property that the x of the vertex is midway between the x interxepts
so let the other one be a
then (a + 7)/2 = -2
a+7 = -4
a = -11
the other one is -11
Well, it seems that the parabola is playing a little game of hide and seek with its x-intercepts. Since we already know one x-intercept is 7, let's see if we can trick the parabola into revealing the location of the other x-intercept!
The vertex is like the parabola's hangout spot, and it's located at (-2, -4). To find the other x-intercept, we're going to need to do a little detective work.
Since the vertex is equidistant from both x-intercepts, we can calculate the distance between the vertex and one x-intercept, and then subtract that distance from the x-coordinate of the other x-intercept. In this case, the distance between the vertex (-2, -4) and the given x-intercept (7, 0) is 9 units.
So, to find the other x-intercept, we subtract 9 from 7, giving us -2. The other x-intercept is hiding at (-2, 0). And just like that, we caught the sneaky parabola!
To find the other x-intercept of the parabola, we first need to determine the equation of the parabola.
Since we know the vertex is (-2, -4), the equation of the parabola can be written in the vertex form as:
y = a(x - h)^2 + k
Where (h, k) represents the vertex, and a is a constant.
Substituting the given values, we have:
y = a(x - (-2))^2 + (-4)
y = a(x + 2)^2 - 4
Now, we know that one x-intercept is 7. This means that when y = 0, x = 7. We can use this information to find the constant a.
Substituting y = 0 and x = 7 into the equation, we get:
0 = a(7 + 2)^2 - 4
0 = a(9)^2 - 4
0 = 81a - 4
To find the value of a, we solve the equation:
81a = 4
a = 4/81
Now that we know the value of a, we can substitute it back into the equation:
y = (4/81)(x + 2)^2 - 4
To find the other x-intercept, we set y = 0 and solve for x:
0 = (4/81)(x + 2)^2 - 4
Multiplying through by 81 to clear the fraction, we get:
0 = 4(x + 2)^2 - 324
Rearranging the equation, we have:
4(x + 2)^2 = 324
Dividing both sides by 4, we have:
(x + 2)^2 = 81
Taking the square root of both sides, we get:
x + 2 = ±9
Solving for x, we have:
x = -2 + 9 = 7
x = -2 - 9 = -11
Therefore, the other x-intercept of the parabola is x = -11.
To find the other x-intercept of the parabola, we can use the vertex form of a quadratic equation, which is given by:
y = a(x - h)^2 + k
Where (h, k) represents the vertex of the parabola.
Given that the vertex of the parabola is (-2,-4), we can substitute these values into the equation to find the value of a:
-4 = a(-2 - (-2))^2 + (-4)
-4 = a(0)^2 - 4
-4 = a(0) - 4
-4 = -4a
a = 1
So, the equation of the parabola is:
y = (x - (-2))^2 - 4
Now, to find the x-intercepts of the parabola, we set y equal to zero and solve for x:
0 = (x - (-2))^2 - 4
0 = (x + 2)^2 - 4
To solve this quadratic equation, we can isolate the squared term:
(x + 2)^2 = 4
Next, apply the square root to both sides of the equation:
√((x + 2)^2) = √4
x + 2 = ±2
Now, solve for x in both cases:
Case 1: x + 2 = 2
x = 2 - 2
x = 0
Case 2: x + 2 = -2
x = -2 - 2
x = -4
Therefore, the x-intercepts of the parabola are x = 0 and x = -4.