Answer:
17x² - 612x + 2360 = 0
Explanation:
Based on the given information, we can determine that the graph in question is a downward-facing parabola that has a vertex at the point (4, -9). Additionally, the graph extends slightly into the second quadrant and has two endpoints in the fourth quadrant, namely (1, y) and (7, y), where y is a constant that we need to determine.
To derive the equation of the graph in the form ax²+bx+C=0, we can use the standard form of the equation of a parabola, which is given by y = a(x - h)² + k, where (h, k) is the vertex of the parabola and a is a constant that determines the shape and orientation of the parabola.
Using the given information, we can substitute the values of the vertex into the equation and obtain:
y = a(x - 4)² - 9
To determine the value of a, we need to use the given endpoints of the parabola in the fourth quadrant. Since the graph extends slightly into the second quadrant, we can assume that the parabola is symmetrical and that the y-coordinate of the endpoints is the same as the y-coordinate of the point (8, y).
Substituting the value of x=1 and y into the equation, we obtain:
y = a(1 - 4)² - 9
y = 9a - 9
Substituting the value of x=7 and y into the equation, we obtain:
y = a(7 - 4)² - 9
y = 9a - 9
Since both expressions for y are equal to the y-coordinate of the point (8, y), we can set them equal to each other and solve for a:
9a - 9 = 8
a = 17/9
Substituting this value of a into the equation for y, we obtain:
y = (17/9)(x - 4)² - 9
Simplifying this expression, we obtain:
17x²/81 - 68x/9 + 236/9 = 0
Therefore, the equation of the graph in the form ax²+bx+C=0 is:
17x² - 612x + 2360 = 0
In conclusion, to derive the equation of a graph, we can use the standard form of the equation of a parabola and the known points on the graph. For the given graph with a vertex at (4, -9) and two endpoints in the fourth quadrant at (1, y) and (7, y), we can derive the equation as 17x² - 612x + 2360 = 0.