Video Transcript
Which of the following functions has been represented by the graph? (A) 𝑓 of 𝑥 equals two 𝑥 squared plus two. (B) 𝑓 of 𝑥 equals two 𝑥 squared minus two. (C) 𝑓 of 𝑥 equals negative two 𝑥 squared minus two. (D) 𝑓 of 𝑥 equals two minus two 𝑥 squared. Or (E) 𝑓 of 𝑥 equals two minus 𝑥 squared.
Let’s begin by identifying some key features of the graph we’ve been given. First, we observe that its shape is a parabola. So the graph represents a quadratic function. The most general form of a quadratic function is 𝑓 of 𝑥 equals 𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐, where 𝑎, 𝑏, and 𝑐 are real constants and 𝑎 is nonzero. We can also observe that this is a U-shaped parabola, or we might describe it as a parabola that opens upwards. And so this tells us that the leading coefficient, that is, 𝑎, the coefficient of 𝑥 squared, is positive.
All parabolas are symmetric. And we can observe that for this parabola, its line of symmetry is the 𝑦-axis or the line with equation 𝑥 equals zero. We can also observe that the vertex of this parabola is on the 𝑦-axis. This tells us that the value of 𝑏 in the general form of a quadratic function that we wrote down is zero. And so the quadratic function we’re looking for is of the form 𝑓 of 𝑥 equals 𝑎𝑥 squared plus 𝑐.
We now need to determine the values of 𝑎 and 𝑐. To find 𝑐 first, we can recall that the value of 𝑐 corresponds to the 𝑦-intercept of the graph. We know this because the 𝑦-intercept occurs when 𝑥 is equal to zero. And 𝑓 of zero is equal to 𝑎 multiplied by zero squared plus 𝑐, which is 𝑐. From the graph, we identify that the 𝑦-intercept is two. And so the value of 𝑐 is two.
We now have 𝑓 of 𝑥 is equal to 𝑎𝑥 squared plus two. To determine the value of 𝑎, which we know to be positive, we can use the coordinates of any other point on the curve. Let’s use the point with coordinates two, 10. We know that when 𝑥 is equal to two, 𝑓 of 𝑥 is equal to 10. So, substituting 10 for 𝑓 of 𝑥, two for 𝑥, and also two for 𝑐, which we’ve just determined, we have the equation 10 equals 𝑎 multiplied by two squared plus two. Two squared is four. And subtracting two from each side, we have eight is equal to four 𝑎. Finally, we can divide both sides of the equation by four, and we find that 𝑎 is equal to two.
So, in the general form 𝑓 of 𝑥 equals 𝑎𝑥 squared plus 𝑐, we found that both 𝑎 and 𝑐 are equal to two. So the equation of the function represented on the graph is 𝑓 of 𝑥 equals two 𝑥 squared plus two, which is option (A).
We can check our answer by using the coordinates of any other point that lies on the curve. Let’s choose the point with coordinates negative three, 20. Substituting 𝑥 equals negative three into the function 𝑓 of 𝑥 equals two 𝑥 squared plus two, we have 𝑓 of negative three equals two multiplied by negative three squared plus two. That’s two multiplied by nine, which is 18, plus two, and this is equal to 20. So this confirms that we’ve chosen the correct function.
We could also look at the other functions and rule them all out for various reasons. For example, we said that because the parabola is a U-shaped parabola which opens upwards, the value of 𝑎, the coefficient of 𝑥 squared, must be positive. This rules out options (C), (D), and (E) because in each of these functions, the coefficient of 𝑥 squared is negative. Option (B) does have the correct coefficient of 𝑥 squared. But in this case, the 𝑦-intercept will be negative rather than positive two. So we can also rule out option (B).
We found that the function represented by the graph is 𝑓 of 𝑥 equals two 𝑥 squared plus two.
