Answer:
Explanation:
The factors of a quadratic function can be determined by analyzing the x-intercepts or roots of the function. These are the values of x where the graph of the function intersects the x-axis.
In this case, the graph of the quadratic function has a maximum value at the point (3,4). This means that the function opens downwards and has two x-intercepts.
To find the x-intercepts, we need to determine the values of x when the graph intersects the x-axis. Since the maximum value occurs at x = 3, the x-intercepts must be symmetrically located around this point.
Let's consider the options given:
a. (x ? 1) and (x ? 5)
b. (-x ? 1) and (x 5)
c. (x ? 1) and (x 5)
d. (-x 1) and (x ? 5)
We can eliminate options b and d because they have incorrect signs.
Now, let's test the remaining options:
a. (x ? 1) and (x ? 5)
If we substitute x = 1 into the first factor, we get (1 ? 1) = 0. This means that (x ? 1) is a factor of the quadratic function.
Similarly, if we substitute x = 5 into the second factor, we get (5 ? 5) = 0. This means that (x ? 5) is also a factor of the quadratic function.
Therefore, option a. (x ? 1) and (x ? 5) is the correct answer.