Using the graph, determine the coordinates of the x-intercepts of the parabola.
(-4, 8), (-9, 3)
To find the x-intercepts of the parabola, we need to find the values of x where the parabola intersects the x-axis. Since the y-coordinate of points on the x-axis is 0, we can set the equation of the parabola equal to 0 and solve for x.
The general equation of a parabola in the standard form is y = ax^2 + bx + c. Given two points on the parabola (-4, 8) and (-9, 3), we can form two equations:
8 = a(-4)^2 + b(-4) + c
3 = a(-9)^2 + b(-9) + c
Simplifying these equations, we have:
8 = 16a - 4b + c
3 = 81a - 9b + c
We can solve these two equations simultaneously to find the values of a, b, and c. Subtracting the second equation from the first equation, we get:
8 - 3 = 16a - 81a - 4b + 9b + c - c
5 = -65a + 5b
Dividing both sides of the equation by 5, we have:
a - b = -1
Now, we can substitute the value of a from this equation into one of the original equations. Let's use the first equation:
8 = 16a - 4b + c
8 = 16(-b - 1) - 4b + c
8 = -16b - 16 - 4b + c
From the second equation, we know that c = 3 - 81a + 9b. Substituting this into the above equation, we have:
8 = -16b - 16 - 4b + 3 - 81a + 9b
8 = -11b - 81a - 13
Now, we can rearrange this equation to solve for a and b:
-11b - 81a = 8 + 13
-11b - 81a = 21
We can express a in terms of b:
-11b - 81(-1 + b) = 21
-11b + 81 - 81b = 21
-92b = -60
b = -60 / -92
b = 15 / 23
Substituting the value of b into the equation -11b - 81a = 21:
-11(15 / 23) - 81a = 21
-165 / 23 - 81a = 21
-165 - 81a = 21 * 23
-165 - 81a = 483
-81a = 483 + 165
-81a = 648
a = 648 / -81
a = -8
Now that we have the values of a, b, and c, we can determine the equation of the parabola:
y = ax^2 + bx + c
y = -8x^2 + (15 / 23)x + 21
To find the x-intercepts, we need to set y = 0:
0 = -8x^2 + (15 / 23)x + 21
We can use the quadratic formula to solve for x:
x = (-b ± sqrt(b^2 - 4ac)) / 2a
Substituting the values of a, b, and c into the quadratic formula, we have:
x = (-(15 / 23) ± sqrt((15 / 23)^2 - 4(-8)(21))) / (2(-8))
x = (-15 ± sqrt(225 / 529 + 672 / 23)) / -16
x = (-15 ± sqrt((225 * 23 + 672) / 529)) / -16
x = (-15 ± sqrt(5205 / 529)) / -16
x = (-15 ± sqrt(5205) / 23) / -16
So, the coordinates of the x-intercepts of the parabola are approximately:
x ≈ (-15 + sqrt(5205) / 23) / -16
x ≈ (-15 - sqrt(5205) / 23) / -16