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Video: Figuring Out the Rule of a Quadratic Function given Its Graph

Kathryn Kingham

Which of the following graphs represent 𝑓(𝑥) = 𝑥² + 1? [A] Graph (a) [B] Graph (b) [C] Graph (c) [D] Graph (d)


Video Transcript

Which of the following graphs represents 𝑓 of 𝑥 equals 𝑥 squared plus one?

Almost immediately, I notice that the leading coefficient here is positive. What do I mean by that? This 𝑥 squared is a positive number. You can imagine it as one times 𝑥 squared, so its coefficient is one, positive one.

And why does that matter when we’re graphing quadratic equations? The leading coefficient with our 𝑥 squared will affect which direction our parabola opens. Positive values means our parabola opens upward; negative values means our parabola opens downward.

Since our 𝑥 squared value is positive, our parabola will open upward. This means that both (b) and (d) are not possible options for graphs of this function. But now we need an additional point to help us figure out if (a) is our graph or (c) is our graph.

To do that, I want to choose a point that we can see on our graph. Let’s plug in zero for 𝑥. If we plug in zero for 𝑥, we get zero 𝑥 squared plus one. This function at 𝑥 equals zero equals one. The point zero, one is on this parabola, so let’s graph it. There’s point zero, one on graph (a), and here’s point zero, one on graph (c). Zero, one is a point on both (a) and (c), so we’ll need to choose an additional value to check.

Let’s choose another number. Let’s check the value of the function when 𝑥 equals one. That would be equal to one squared plus one.

The value of this function when 𝑥 equals one is two. We found an additional point of one, two. Let’s graph that one. Here’s the point one, two on graph (a), and here’s the point one, two on graph (c). The point one, two doesn’t fall on the line in graph (c); we can eliminate it. Graph (a) represents the function 𝑓 of 𝑥 equals 𝑥 squared plus one.