write an equation for the parabola with the vertex that passes throught the given point?

Question

1.vertex: (4, -36)
point: (0, -20)

2. Vertex: (3, -1)
point: (2,0)

3. Vertex: (-2, 2)
POint: (-1, 3)

i will greatly appreciate it

Answer 1

GENERAL SOLUTION:

Start with vertex form:
y = a(x - h)² + k

The vertex is (h, k), so plug those in.

Finally, plug in your point (x, y) and solve for the remaining variable 'a'.

PROBLEM 1:
y = a(x - h)² + k

Vertex (h, k) = (4, -36):
y = a(x - 4)² - 36

Now plug in point (x, y) = (0, -20) and solve for a:
-20 = a(0 - 4)² - 36
-20 = 16a - 36
16 = 16a
a = 1

So the final equation is:
y = (x - 4)² - 36

PROBLEM 2:
y = a(x - h)² + k

Vertex (h, k) = (3, -1):
y = a(x - 3)² - 1

Now plug in point (x, y) = (2, 0) and solve for a:
0 = a(2 - 3)² - 1
0 = a(-1)² - 1
0 = a - 1
a = 1

So the final equation is:
y = (x - 3)² - 1

PROBLEM 3:
y = a(x - h)² + k

Vertex (h, k) = (-2, 2):
y = a(x - (-2))² + 2
y = a(x + 2)² + 2

Now plug in point (x, y) = (-1, 3) and solve for a:
3 = a(-1 + 2)² + 2
3 = a(1)² + 2
3 = a + 2
a = 1

So the final equation is:
y = (x + 2)² + 2

Note: they were really nice to always make a = 1, but that won't always be the case...

Answer 2

Use vertex form: y = a(x−k)² + h, where vertex = (h,k) Vertex: (−2,−3) y = a(x+2)² − 3 Passing through point (−4, −5) Plug values of x and y into equation, then solve for a: −5 = a(−4+2)² − 3 −5 = 4a − 3 −2 = 4a a = −1/2 y = −1/2 (x + 2)² − 3 y = −1/2 (x² + 4x + 4) − 3 y = −1/2 x² − 2x − 5 ------------------------------ Use same method for second problem: Vertex (2, 0) point (1, 4) y = a(x−2)² + 0 4 = a(1−2)² + 0 a = 4 y = 4(x−2)² y = 4x² − 16x + 16

Answer 3

A general parabola in standard form is given by the expression

y = a(x - h)^2 + k, where (h,k) is the coordinates of the vertex. All we have to do is plug in h and k, and we get,

y = a(x - 4) - 36

But we require the value a. To solve for a, we plug in our given point (0,-20) for x and y.

-20 = a(0 - 4) - 36
-20 = a(-4) - 36
-20 = -4a - 36
16 = -4a, therefore a = -4

So our parabola for (1) is y = -4(x - 4) - 36

You do the two others the EXACT same way.