Video Transcript
Choose the graph that represents the function 𝑓 of 𝑥 equals 0.5 minus two 𝑥 squared.
Let’s have a closer look at this function 𝑓 of 𝑥. First, we observe that it is a quadratic function. This means that its graph will be in the shape of a parabola, either opening upwards or opening downwards. As the leading coefficient, that’s the coefficient of 𝑥 squared, is negative two, which is negative, this tells us that the graph of the function will be an n-shaped parabola, which opens downwards.
Looking closely at the function, we see that it is actually a specific type of quadratic function. It’s of the form 𝑘𝑥 squared plus 𝑐 for some constants 𝑘 and 𝑐. For such a quadratic function, two things are true of its graph. Firstly, its line of symmetry is the 𝑦-axis, or the line with equation 𝑥 equals zero. And secondly, the coordinates of its 𝑦-intercept and its vertex are zero, 𝑐. For this quadratic function, the value of 𝑘 is negative two and the value of 𝑐 is 0.5.
So, we now know that the graph that represents this function will be a parabola that opens downwards. It will have a line of symmetry along the 𝑦-axis. And the coordinates of its 𝑦-intercept, which are also the coordinates of its vertex, are zero, 0.5. Looking carefully then at the five graphs given, we can see that the only one that has these properties is graph (E). So we’ve identified the graph representing the function 𝑓 of 𝑥 equals 0.5 minus two 𝑥 squared. It’s graph (E).
Looking carefully at the other options, we can rule them out for various reasons. Firstly, graphs (C) and (D) are both parabolas that open upwards. So these would both correspond to quadratic functions with positive leading coefficients. Graphs (A) and (B) are both parabolas that open downwards with a line of symmetry along the 𝑦-axis. But they each have their vertex and 𝑦-intercept at the origin. They would, therefore, correspond to functions of the form 𝑓 of 𝑥 equals 𝑘𝑥 squared. The value of 𝑐 in the general form we wrote down would be zero.
So we’ve identified the correct graph by considering the properties of the graph of 𝑓 of 𝑥. But we’ve also ruled all the others out by considering their properties. We found that the graph that represents the function 𝑓 of 𝑥 equals 0.5 minus two 𝑥 squared is graph (E).
