Write the equations of the parabola in Vertex Form using the given information: axis of symmetry : x=-2; points: (-6,44), (1,23); direction of opening : upward

Answer 1

The "generic" form vertex form for a parabola opening either up or down would be y = a(x - h)² + k where "a" is the stretch or compression factor and the vertex is at the point (h, k).

Since we know the parabola opens upward, we know that "a" will be a positive value.

Since we know the axis of symmetry is x = -2, we know that the x-coordinate of the vertex is -2, in other words, h = -2.

So, that means we can write the equation as:

y = a(x - (-2))² + k or y = a(x + 2)² + k

Since we know the points (-6,44) and (1,23) are on the parabola, we can plug those into the equation as x-values and y-values like this:

(1) 44 = a(-6 + 2)² + k

(2) 23 = a(1 + 2)² + k

Equation (1) simplifies to 44 = 16a + k and equation (2) simplifies to 23 = 9a + k. You can solve for "a" by subtracting the two equations like this:

44 = 16a + k

23 = 9a + k

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21 = 7a

a = 3

Then, plug that back into 23 = 9a + k to solve for "k":

23 = 9(3) + k

23 = 27 + k

k = -4

That makes the equation y = 3(x + 2)² - 4