Determine the sign of the function 𝑓 of 𝑥 equals negative 𝑥 squared minus two 𝑥 minus seven.
Remember, the sign of a function tells us whether the function is positive, negative, or zero. Since 𝑓 of 𝑥 is the output of the function, we can say that the function is positive if 𝑓 of 𝑥 is greater than zero and it’s negative if 𝑓 of 𝑥 is less than zero. Of course, if 𝑓 of 𝑥 is equal to zero, the function is neither positive nor negative. And in fact, we can determine the sign of a function by looking at its graph. If parts of the graph of the function lie above the 𝑥-axis, then the function is positive over those intervals. Similarly, if there are any portions of the graph that lie below the 𝑥-axis, then these are the intervals on which 𝑓 of 𝑥 is negative.
So, let’s begin by sketching the graph of the function. It’s a quadratic equation with a negative leading coefficient. The coefficient of 𝑥 squared is negative one, so the shape of the curve is an inverted parabola. We can find the location of any 𝑥-intercepts by setting 𝑓 of 𝑥 equal to zero and solving for 𝑥. In other words, we set negative 𝑥 squared minus two 𝑥 minus seven equals zero. To make this easier to work with, lets multiply through by negative one. We get 𝑥 squared plus two 𝑥 plus seven equals zero.
Now, we can’t easily factor this left-hand side since there are no integer numbers that multiply to make seven and add to make two. So we could either use completing the square or the quadratic formula to solve. Before we do, though, let’s just double-check that there are indeed solutions to this equation. And to do so, we’ll use the discriminant. Suppose we have a quadratic equation of the form 𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐 equals zero. The discriminant is the part of the quadratic formula that goes inside the square root. It’s 𝑏 squared minus four 𝑎𝑐.
Now, if the discriminant is negative, if it’s less than zero, then there are no real solutions to the quadratic equation. And what this means in terms of the graph is that there are no 𝑥-intercepts. The graph of the quadratic function lies entirely below or entirely above the 𝑥-axis. Now, for our new quadratic equation 𝑥 squared plus two 𝑥 plus seven equals zero, 𝑎 is one — that’s the coefficient of 𝑥 squared — 𝑏 is the coefficient of 𝑥 — so it’s two — and 𝑐 is the constant term — it’s seven. Then the discriminant is two squared minus four times one time seven, which is equal to negative 24.
Notice that negative 24 is less than zero. It’s negative. This means there are no real solutions to our quadratic equation. And hence, the graph of the function does not pass through the 𝑥-axis. This means the graph of 𝑦 equals 𝑓 of 𝑥 might look a little something like this. We can observe that the graph of 𝑦 equals 𝑓 of 𝑥 lies below the 𝑥-axis for all possible values of 𝑥. This means the function must be negative for all values in the domain of the function. Now, of course, the domain of a quadratic function, the set of possible inputs, is just all real numbers.
So we can say the function is negative for all real numbers. More formally, the function is negative for all values of 𝑥 within the set of real numbers.