The quadratic function f is negative only on the open interval (−2, 1 4 ) and its graph passes through the point (−1, −5). Determine the coefficients of f.
since the vertex is between the roots, we have
f(x) = a(x+2)(x-1 4)
Not sure what "1 4 " means, but now just plug in f(-1) = -5 to solve for a.
To determine the coefficients of the quadratic function, we need to first write the equation of the function in the standard form: f(x) = ax² + bx + c, where a, b, and c are the coefficients.
We know that the quadratic function is negative only on the open interval (-2, 14). This means that the graph of the function opens downward (since the leading coefficient a is positive) and has x-intercepts at x = -2 and x = 14.
To find the equation of the quadratic function, we can start by finding the roots (x-intercepts). The roots can be calculated by solving the equation f(x) = 0.
Since the quadratic function passes through the point (-1, -5), we can use this information to set up an equation as well.
Let's start by finding the roots:
Since the roots of the quadratic are at x = -2 and x = 14, we can set up the following equations:
(x + 2)(x - 14) = 0
Expanding this equation gives us:
x² - 12x - 28 = 0
Now, let's use the point (-1, -5) to set up another equation:
Substituting x = -1 and f(x) = -5 into the equation of the quadratic function, we get:
a(-1)² + b(-1) + c = -5
a - b + c = -5
Now we have a system of equations:
x² - 12x - 28 = 0
a - b + c = -5
To solve this system of equations, we can use various methods such as substitution or elimination. Let's use substitution:
From the first equation, we can use the quadratic formula to solve for x:
x = (-(-12) ± √((-12)² - 4(1)(-28))) / (2(1))
Simplifying, we get:
x = (12 ± √(144 + 112)) / 2
x = (12 ± √256) / 2
x = (12 ± 16) / 2
This gives us two potential solutions for x:
x1 = (12 + 16) / 2 = 14
x2 = (12 - 16) / 2 = -2
Since we already know that x = -2 and x = 14 are the roots of the quadratic equation, we can use these values to determine the coefficients.
Let's substitute x = -2 into the equation of the quadratic function:
(-2)² - 12(-2) - 28 = 0
4 + 24 - 28 = 0
0 = 0
This means that the equation is satisfied when x = -2.
Now, let's substitute x = 14 into the equation of the quadratic function:
14² - 12(14) - 28 = 0
196 - 168 - 28 = 0
0 = 0
This means that the equation is also satisfied when x = 14.
From this, we can conclude that the equation of the quadratic function is:
f(x) = a(x + 2)(x - 14)
Now, let's substitute the point (-1, -5) into the equation:
a(-1 + 2)(-1 - 14) = -5
a(1)(-15) = -5
-a(15) = -5
15a = -5
a = -5/15
a = -1/3
Therefore, the coefficient a is -1/3.