Determine the equation of a parabola with x-intercepts +- 4 and passing through (3,6)
Find the two factors using the roots (the intercepts) and then test the point (3,6) in your answer to see if you need to change the sign of everything.
The parabola is
y = a(x-4)(x+4)
plug in (3,6) to find the value of a.
replace a in my equation.
Expand if you feel like it.
Why did the parabola go see a therapist? Because it needed help finding its equation!
To determine the equation of a parabola, we need to determine its vertex and the coefficient in front of the squared term.
Given that the x-intercepts are ±4, we can conclude that the parabola has the equation in the form:
y = a(x - x₁)(x - x₂),
where x₁ and x₂ are the x-intercepts.
Since the x-intercepts are ±4, x₁ = -4 and x₂ = 4, which gives us:
y = a(x + 4)(x - 4).
Next, we need to find the value of 'a.' We can do this by using the coordinates (3, 6) that the parabola passes through. Substituting these values into the equation, we have:
6 = a(3 + 4)(3 - 4).
Simplifying further:
6 = a(7)(-1).
Now, let's solve for 'a':
6 = -7a.
Dividing both sides by -7:
a = -6/7.
So, the equation of the parabola is:
y = -6/7(x + 4)(x - 4).
To determine the equation of a parabola, we need to know its vertex and at least one other point on the graph. In this case, we have the x-intercepts at -4 and 4, which means the vertex is at the midpoint of these x-intercepts, which is (0, 0).
Now we can substitute the coordinates (3, 6) into the equation of the parabola to find its vertex form.
The vertex form of a parabola is given by the equation: y = a(x - h)^2 + k, where (h, k) represents the coordinates of the vertex.
Substituting the vertex coordinates (0, 0) and the point (3, 6) into the equation, we have:
6 = a(3 - 0)^2 + 0
Simplifying this equation, we get:
6 = 9a
Now, solve for 'a' by dividing both sides by 9:
a = 6/9
a = 2/3
Therefore, the equation of the parabola with x-intercepts of +4 and -4 and passing through (3, 6) is:
y = (2/3)(x - 0)^2 + 0
Simplifying the equation further, we get:
y = (2/3)x^2
So, the equation of the parabola is y = (2/3)x^2.