Which of the following is the equation of the function drawn on the graph?
Here, we can see that we have a U-shaped graph. A quadratic function is a function where the greatest power of a variable is two, and its graph is U-shaped. Opening upward, if its leading coefficient is positive and opening downward, if its leading coefficient is negative. Therefore, we need to have a quadratic equation. This means we can automatically eliminate option A. Because its greatest power of a variable is one, this means it’s a linear function; we need a quadratic function.
Next, we can see that our graph is opening upward. That means we need to have a positive leading coefficient. A leading coefficient is the number in front of the first term — the leading term. So all of our options have either a positive one in front of our 𝑥 squared or a negative one in front of our 𝑥 squared. Options B and C have a positive one in front of 𝑥 squared and options D and E have a negative one in front of 𝑥 squared. Therefore, we can eliminate options D and E because they have a negative one as their leading coefficient. We need a positive leading coefficient.
This means we need to decide between option B: 𝑓 of 𝑥 equals 𝑥 squared plus eight or option C: 𝑓 of 𝑥 equals 𝑥 squared minus eight. Looking at our two options, it would be best to go ahead and take a look at the points on our graph. Here are a few easy points to work with for our graph. Let’s go ahead and pick one. Here, we have zero, negative eight. This means if we would plug in zero for 𝑥, we should get negative eight for a 𝑦. 𝑓 of 𝑥 in 𝑦 actually means the same thing, so keep that in mind for when we plug in 𝑥.
So let’s go ahead and plug in zero for 𝑥 for both equations and see which one would give us an answer of negative eight. Here we can see we have zero squared plus eight and zero squared is zero and zero plus eight is equal to eight. Over here, we have zero squared minus eight and zero squared is zero and zero minus eight is negative eight, which is what we wanted. So option C, 𝑓 of 𝑥 equals 𝑥 squared minus eight, would be the correct equation for the function drawn on a graph.
Now it’s always good to double check your work. So let’s go ahead and just try another point from our graph and make sure that it works. Let’s plug in the point three, one. So let’s plug in three for 𝑥 and make sure we get one for 𝑦. Here we can see when we plug in three, three squared is nine and nine minus eight is indeed one. Therefore, C, 𝑓 of 𝑥 equals 𝑥 squared minus eight, is the equation of the function drawn on the graph.