Question

identify the intervals on which the quadratic function is positive

y=4x^2 -23x+15

Answer 1

To determine the intervals on which the quadratic function is positive, we need to find the x-values where the function has positive y-values.

To find the x-values, we need to solve the quadratic equation for y=0.

4x^2 - 23x + 15 = 0

This equation can be factored as:
(4x - 5)(x - 3) = 0

Now we set each factor equal to zero:
4x - 5 = 0 --> 4x = 5 --> x = 5/4
x - 3 = 0 --> x = 3

So the quadratic function crosses the x-axis at x = 5/4 and x = 3.

We can use these points to determine the intervals for which the quadratic function is positive. To do this, we make a sign chart:

x < 5/4 5/4 < x < 3 x > 3
-------- ---------- ------
+ - +

In the interval (x < 5/4), the sign is positive.
In the interval (5/4 < x < 3), the sign is negative.
In the interval (x > 3), the sign is positive.

Therefore, the quadratic function is positive in the intervals: (x < 5/4) and (x > 3).