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.
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To determine the coefficients of the quadratic function f, we need to consider the given information. We know that the function is negative only on the open interval (-2, 14). This means that the graph of the function will be above the x-axis for values less than -2 and above the x-axis for values greater than 14.
We are also given that the graph passes through the point (-1, -5). This gives us an additional piece of information that we can use to determine the coefficients of the function.
Let's assume that the quadratic function f takes the form f(x) = ax^2 + bx + c, where a, b, and c are the coefficients we need to find.
To find the coefficients, we can begin by substituting the point (-1, -5) into the function. This gives us the equation -5 = a(-1)^2 + b(-1) + c.
Simplifying this equation, we have -5 = a - b + c.
Next, we need to use the fact that the function is negative only on the open interval (-2, 14). This tells us that the vertex of the quadratic function lies between -2 and 14. The x-coordinate of the vertex is given by x = -b / (2a). Since the parabola is concave upwards (since the graph is negative only), the coefficient a must be positive.
From this information, we know that -2 < -b / (2a) < 14.
By simplifying this inequality, we get -4a < -b < 28a.
Now, we have two equations:
-5 = a - b + c
-4a < -b < 28a
From here, we can solve these equations simultaneously to find the values of a, b, and c.
Now, to sketch the graph of the function f, we can use the information we have.
First, we can find the vertex of the parabola by using the formula x = -b / (2a), which will give us the x-coordinate of the vertex. Then, we can substitute the x-coordinate into the function to find the corresponding y-coordinate.
Next, we can plot the vertex on the graph and use the fact that the function is concave upwards to draw the parabola with the vertex as the minimum point.
Finally, we can use the information about the open interval (-2, 14) to determine whether the graph of the function should be above or below the x-axis in this interval.
By following these steps, we will be able to determine the coefficients of the function f and sketch its graph.