Answer:
To find the equation of the quadratic function g, we can use the general form of a quadratic function:
g(x) = ax^2 + bx + c
We have two points on the graph of g: (-3, 5) and (-2, 3). We can use these points to form a system of equations and solve for the coefficients a, b, and c.
Using the first point (-3, 5):
5 = a(-3)^2 + b(-3) + c
5 = 9a - 3b + c
Using the second point (-2, 3):
3 = a(-2)^2 + b(-2) + c
3 = 4a - 2b + c
Now we have a system of equations:
9a - 3b + c = 5
4a - 2b + c = 3
We can solve this system of equations to find the values of a, b, and c.
Multiplying the second equation by 3, we get:
12a - 6b + 3c = 9
Subtracting the first equation from this new equation, we eliminate c:
12a - 6b + 3c - (9a - 3b + c) = 9 - 5
12a - 9a - 6b + 3b + 3c - c = 4
3a - 3b + 2c = 4
Now we have a new equation:
3a - 3b + 2c = 4
We can rewrite this equation as:
3a - 3b = 4 - 2c
Dividing both sides by 3, we get:
a - b = (4 - 2c)/3
Now we have an expression for a - b in terms of c.
To find the equation of g(x), we need to express a, b, and c in terms of a single variable. We can choose c as the variable.
Let's set c = k, where k is a constant.
Then, a - b = (4 - 2k)/3
We can rewrite this equation as:
a = (4 - 2k)/3 + b
Now we have a in terms of b and k.
Substituting this expression for a into one of the original equations, we can solve for b.
Using the first equation 5 = 9a - 3b + c:
5 = 9((4 - 2k)/3 + b) - 3b + k
5 = (12 - 6k + 9b)/3 - 3b + k
15 = 12 - 6k + 9b - 9b + 3k
15 = 12 - 3k
Simplifying, we find:
3k = 3
k = 1
Now we have the value of k.
Substituting k = 1 into the expression for a, we get:
a = (4 - 2(1))/3 + b
a = 2/3 + b
Now we have a in terms of b.
Finally, we can write the equation of g(x) using the values of a, b, and c:
g(x) = (2/3 + b)x^2 + bx + 1
So, the equation of the quadratic function g is:
g(x) = (2/3 + b)x^2 + bx + 1
Note that the value of b is still unknown, as it was not determined from the given information.