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