Question Video: Finding the Solution Set of a Quadratic Equation Graphically Mathematics • Third Year of Preparatory School

Video Transcript

The diagram shows the graph of 𝑦 is equal to 𝑓 of 𝑥. What is the solution set of the equation 𝑓 of 𝑥 is equal to zero?

In this question, we’re given the graph of a function 𝑦 is equal to 𝑓 of 𝑥. And we’re asked to use this graph to determine the solution set of the equation 𝑓 of 𝑥 is equal to zero. Before we start answering this question, we can note that the graph 𝑦 is equal to 𝑓 of 𝑥 has a parabolic shape which opens upwards. This means it’s likely that 𝑓 of 𝑥 is a quadratic function with positive leading coefficient. However, this is not strictly necessary. And we don’t need this information to answer the question. Instead, we can start by recalling the solution set of an equation is the set of all solutions to the equation. In this case, it will be the set of all values of 𝑥 which satisfy the equation 𝑓 evaluated at 𝑥 is equal to zero.

In other words, we’re looking for all of the values of 𝑥 where the output of the function is zero. And we can find these 𝑥-values by using the given diagram. We recall when we graph a function, the 𝑥-coordinate of any point on the curve tells us the input value of the function and the corresponding 𝑦-coordinate tells us the output value of the function. For example, we can see the point with coordinates two, five lies on the graph of 𝑦 is equal to 𝑓 of 𝑥. The 𝑥-coordinate of this point is the input value of the function, and the 𝑦-coordinate is the corresponding output. 𝑓 evaluated at two must be equal to five.

We want to find the input values of the function where the output of the function is zero. So we want to find the points on the curve whose 𝑦-coordinate is equal to zero. And the 𝑦-coordinate will be equal to zero whenever the function crosses the 𝑥-axis. In other words, the solutions to this equation are the 𝑥-intercepts of the curve.

And we can see that there are two 𝑥-intercepts of this parabola: negative three and one. In particular, 𝑓 evaluated at negative three is equal to zero and 𝑓 evaluated at one is also equal to zero because the 𝑦-coordinate of these points on the curve is zero. And remember, we need to write this as the set of all 𝑥-values which solve the equation. So, that’s the set containing negative three and one. And it’s worth noting that this method holds in general. The solution set of the equation 𝑓 of 𝑥 is equal to zero is always the same as the set of 𝑥-intercepts of the graph of the function 𝑦 is equal to 𝑓 of 𝑥.

Therefore, we were able to show by using the diagram of the graph of 𝑦 is equal to 𝑓 of 𝑥, the solution set of the equation 𝑓 of 𝑥 is equal to zero is the set containing negative three and one.