Video Transcript
Determine the sign of the function 𝑓 of 𝑥 equals two 𝑥 minus one squared.
The sign of a function is a description that tells us whether the function is positive or negative, or possibly even zero. We say that the function is positive when 𝑓 of 𝑥 is greater than zero, and it’s negative for values of 𝑥, where 𝑓 of 𝑥 is less than zero. And of course, we can actually use the graph to help us determine the sign of the function. 𝑓 of 𝑥 is, of course, the output for a given input of 𝑥. So, 𝑓 of 𝑥 is said to be positive when the graph of the function lies above the 𝑥-axis and it’s said to be negative when the graph of 𝑓 of 𝑥 lies below the 𝑥-axis.
So, let’s begin then by sketching the graph of 𝑓 of 𝑥 equals two 𝑥 minus one squared. If we were to distribute the parentheses here, we’d notice that we have a quadratic equation. Specifically, the coefficient of 𝑥 squared, if we were to distribute these parentheses, would be four; it’s positive. And so, we know that not only is it a parabola, it’s a U-shaped parabola since the coefficient of 𝑥 squared is positive.
Next, we can find the values of the 𝑥-intercepts of the graph of our function by letting 𝑓 of 𝑥 equal zero. In doing so, we get zero equals two 𝑥 minus one squared. We take the square root of both sides to get zero equals two 𝑥 minus one. And then if we add one to both sides, our equation is two 𝑥 equals one. Finally, we divide by two. When we do, we find that 𝑥 equals one-half is a solution to the equation 𝑓 of 𝑥 equals zero. This means there is a single 𝑥-intercept, one-half.
So, the graph of the function 𝑦 equals 𝑓 of 𝑥 or 𝑦 equals two 𝑥 minus one squared will look like this. The function is equal to zero at 𝑥 equals one-half. However, for all other values of 𝑥, we notice that the graph of the function lies above the 𝑥-axis. Therefore, 𝑓 of 𝑥 is positive for all other values of 𝑥, for all real values of 𝑥 not including 𝑥 equals one-half.
We can therefore say that the sign of the function is positive when 𝑥 is any real number not including one-half and it’s equal to zero when 𝑥 equals one-half.
