To identify the intervals of the function y = 4x^2 - 31x + 21, we can analyze the quadratic equation and determine where it is positive or negative.
First, find the x-intercepts by setting y = 0 and solving for x:
0 = 4x^2 - 31x + 21
This equation can be factored as:
0 = (4x - 3)(x - 7)
Setting each factor equal to zero:
4x - 3 = 0 -> x = 3/4
x - 7 = 0 -> x = 7
So, the x-intercepts are (3/4, 0) and (7, 0).
Next, we can determine the sign of the quadratic equation for different intervals of x. We can do this by creating a sign chart using the x-intercepts as the points of reference.
Interval 1: (-∞, 3/4)
We can choose a value for x between -∞ and 3/4, such as x = 0. Plugging this value into the equation, we get:
y = 4(0)^2 - 31(0) + 21 = 21
Since the calculated value of y is positive, the quadratic equation is positive in this interval.
Interval 2: (3/4, 7)
Choosing a value for x within this interval, such as x = 5, we can calculate y:
y = 4(5)^2 - 31(5) + 21 = -44
Since the calculated value of y is negative, the quadratic equation is negative in this interval.
Interval 3: (7, +∞)
Picking a value for x, such as x = 8, we calculate y:
y = 4(8)^2 - 31(8) + 21 = 83
The calculated value of y is positive, indicating that the quadratic equation is positive in this interval.
Therefore, the intervals of the function y = 4x^2 - 31x + 21 are:
(-∞, 3/4) and (7, +∞)
identify the intervals of y=4x^2-31x+21
1 answer
1 answer