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 and sketch its graph.
To find the coefficients of the quadratic function, we need to consider the given information and apply the properties of quadratic functions.
First, let's determine the general form of the quadratic function f(x) using the coefficients a, b, and c:
f(x) = ax^2 + bx + c
We know that the graph of f(x) is negative only on the open interval (-2, 1/4). This means that the graph is below the x-axis between -2 and 1/4, and above the x-axis outside of this interval.
From this information, we can deduce that the quadratic function has a downward-facing parabola, since it is negative. This gives us a hint that the coefficient 'a' must be negative.
Next, we are given that the graph passes through the point (-1, -5). We can substitute these coordinates into the equation to get:
-5 = a(-1)^2 + b(-1) + c
-5 = a - b + c
Now, let's combine this equation with the information about the open interval. Since the graph is negative only on the open interval (-2, 1/4), we can plug the left and right endpoints into the equation to get two additional conditions:
When x = -2:
0 = a(-2)^2 + b(-2) + c
0 = 4a - 2b + c
When x = 1/4:
0 = a(1/4)^2 + b(1/4) + c
0 = 1/16a + 1/4b + c
Now, we have a system of three equations with three unknowns (a, b, and c):
-5 = a - b + c
0 = 4a - 2b + c
0 = 1/16a + 1/4b + c
To solve this system of equations, we can use various methods such as substitution or elimination. Solving this system will give us the values of a, b, and c, which are the coefficients of the quadratic function f(x).
Once we have the coefficients, we can sketch the graph of f(x) by using the vertex form of the quadratic equation:
f(x) = a(x - h)^2 + k
where (h, k) represents the vertex of the parabola.
well, halfway between x axis intercepts is
x = 12/2 = 6 so that is where the vertex must be
y - b = (x-6)^2
-5 - b = (-7)^2 = 49
b = -54
y = (x-6)^2 - 54
y = x^2 - 12 x + 36 - 54
y = x^2 - 12 x - 18
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check:
x^2 - 12 x = y+18
x^2 - 12 x + 36 = y+ 54
(x-6)^2 = y + 54
(-1-6)^2 = -5 + 54
7^2 = 49 sure enough