the points (-9,0) and (19,0) lie on parabola.

a.) determine an equation for its axis of symmetry: X=5
b.) the y-coordinate of the vertex is -28. determine an equation for the parabola in factored form.
c.)write your equation in part b) in standard form.

a.) To determine the equation for the axis of symmetry, we can find the x-coordinate for the midpoint of the two given points. Since the points (-9, 0) and (19, 0) both lie on the axis of symmetry, their x-coordinates are the same. Therefore, the equation for the axis of symmetry is x = (−9 + 19) / 2 = 10 / 2 = 5.

b.) To find the vertex of the parabola, we use the x-coordinate of the axis of symmetry and the given y-coordinate. So, the vertex is at the point (5, -28).

The factored form of the equation for the parabola is given by:
y = a(x - h)^2 + k, where (h, k) represents the vertex.

Substituting the values, we have:
y = a(x - 5)^2 - 28.

c.) To write the equation in standard form, we expand and simplify the equation:
y = a(x^2 - 10x + 25) - 28
y = ax^2 - 10ax + 25a - 28

So, the equation in standard form is:
y = ax^2 - 10ax + (25a - 28)

To determine the answers to these questions, we need to understand some key concepts related to parabolas.

a.) The axis of symmetry of a parabola is a vertical line that passes through the vertex and divides the parabola into two symmetrical halves. To find the equation for the axis of symmetry, we can use the x-coordinate of any two points that lie on the parabola.

Given the points (-9, 0) and (19, 0) on the parabola, we can observe that the x-coordinate is always 0 while the y-coordinate changes. This indicates that the vertex lies on the line x = 5 (since the x-coordinate of the vertex is halfway between -9 and 19). Therefore, the equation for the axis of symmetry is X = 5.

b.) The vertex form of a parabola is given by y = a(x-h)^2 + k, where (h, k) represents the vertex of the parabola. In this case, we are given that the y-coordinate of the vertex is -28, which means our equation becomes y = a(x-5)^2 - 28.

To find the value of a, we need another point on the parabola. Since the given points lie on the x-axis (y = 0), we can substitute one of the points into the equation:

0 = a(-9-5)^2 - 28
0 = a(-14)^2 - 28
0 = 196a - 28

Solving this equation, we can find the value of a:

196a = 28
a = 28 / 196
a = 1/7

Therefore, the equation for the parabola in factored form is y = (1/7)(x-5)^2 - 28.

c.) To rewrite the equation in standard form, we expand the equation:

y = (1/7)(x-5)^2 - 28
y = (1/7)(x^2 - 10x + 25) - 28
y = (1/7)x^2 - (10/7)x + 25/7 - 28
y = (1/7)x^2 - (10/7)x - 147/7

Finally, simplifying the equation, we have:

y = (1/7)x^2 - (10/7)x - 21