Anthony G. answered • 10/07/19
Algebra 1 Expert
Hi Ted! Let's start with the first question:
a) We are given the parabola's focus at (3, 12) and its vertex at (-2, 12). Remember that the "vertex" is the tip of the parabola, and that the focus of a parabola is inside the mouth of the parabola. Drawing a quick sketch, we can see that this must be a horizontal parabola opening to the right.
The formula for a horizontal parabola opening to the right is: x + h = 1/(4p) (y - k)2.
(h,k) is the coordinate of the vertex
p is the distance from the vertex to the focus (the focal length).
We know the vertex is at (-2, 12), so h = -2 and k = 12.
We know the focus is at (3, 12), so the distance from the vertex to the focus is 3 - (-2) = 5, so p = 5.
The equation of the parabola must be x - 2 = 1/20 (y - 12)2
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b) In this problem, we are given the focus at (0, -5) and the directrix x = 14. Since the directrix's equation is defined in terms of x, we know this is another horizontal parabola (if it was an equation in terms of y, it would be a vertical parabola).
If only we knew the vertex... then we could solve it the same way we solved part a. But wait! The vertex is always halfway between the focus and directrix. Since the focus is at x = 0 and the directrix is at x = 14, the vertex must be at x = 7. Since we know this will be a horizontal parabola, the vertex's y-value will be the same as the focus. The vertex is at (7, -5).
We can now solve exactly the same as we did in part a).
Vertex coordinate (7, -5) means h = 7, k = -5.
Distance between vertex and focus 7 - 0 = 7, so p = 7.
The equation is x - 7 = 1/28 (y + 5)2.
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c) We're almost done! For this final question, the problem changes again! Now we have the x-intercepts (1,0) and (3,0) as well as the y-intercept (0,-3).
How can we use the intercepts to find the parabola's equation? Well, here's an idea. Usually, we are given the equation and asked to find the x-intercepts. To do this, we would first factor the parabola's equation into something like (x - a)(x - b) = 0. Then we could find the solutions x = a and x = b. These are the x-intercepts.
Great... so... how does that help us? Well, we can use the same idea but just apply it backwards:
We know that the x-intercepts are x = 1 and x = 3. So, the factors must be (x - 1)(x - 3) = 0. But, then, we can find the parabola's original equation! (x - 1)(x - 3) = x2 - 3x - x +3 = x2 - 4x + 3.
The solution is: y = x2 - 4x + 3.
Wow! That's a lot of work! Hopefully that makes sense! Good luck, and let me know if there's anything that you'd like more explanation on.
