The vertex of the parabola in the given 𝑥𝑦 plane is zero, 𝑐. Which of the following is true about the parabola with the equation 𝑦 equals negative 𝑎 times 𝑥 plus 𝑏 all squared plus 𝑐? And we’ve been given the graph of a quadratic function 𝑦 equals 𝑎𝑥 squared plus 𝑐. And the four options are. A) The vertex is negative 𝑏, 𝑐 and the graph opens downward. B) The vertex is 𝑏, 𝑐 and the graph opens downward. C) The vertex is negative 𝑏, 𝑐 and the graph opens upward. And D) The vertex is 𝑏, 𝑐 and the graph opens upward.
Let’s just recall what we know about quadratic equations. The general form of the quadratic equation is that 𝑦 is equal to some constant times 𝑥 squared. Let’s call that 𝑎 plus some other constant. Let’s call that 𝑏 times 𝑥 plus some other constant. Let’s call that 𝑐. Now it’s important that the value of 𝑎 is not zero. Because if we had zero 𝑥 squared, then we wouldn’t have a quadratic equation. But either or both of constants 𝑏 and 𝑐 could be equal to zero. You’d still have a quadratic equation.
Now we also know that the shape of the graph of a quadratic is a parabola. And this is either a symmetrical curve like this, which is open upwards, or a symmetrical curve like this, which is open downwards. If the value of 𝑎 in our general equation is greater than zero. If it’s positive, then we have an open upwards curve. You can think of this as being positive and happy. So it looks like a smiley face. But if the value of 𝑎 is negative, then you have an open downward type graph. And you can think of this if you’re negative, you’re sad. And it looks like a sad face. Also, the value of 𝑐 tells us the value of the 𝑦-intercept. That’s the 𝑦-coordinate where it cuts the 𝑦-axis.
Remember, on the 𝑦-axis, the 𝑥-coordinate is zero. So this term is gonna be 𝑎 times zero squared, which will be zero. This term will be 𝑏 times zero, which is zero. So the 𝑦-coordinate will just be equal to whatever this number is, 𝑐. And just another couple of things. The turning point of the curve we call a vertex. And if it’s an open upwards curve, then that vertex will be the minimum 𝑦-value on the graph. And if it’s an open downwards curve, then the vertex will be at the maximum 𝑦-value on the graph. And you also need to remember that you can calculate the 𝑥-coordinate of the vertex by evaluating negative 𝑏 over two 𝑎.
Okay then, let’s look at the graph that they gave us in the question. First, it’s open downward. We’ve got a negative sad curve. That tells us that the value of 𝑎 must be less than zero. 𝑎 is negative. There’s no term for 𝑏 in our equation. So we know that the value of 𝑏 is zero. And the value of 𝑐 is, well, 𝑐. So our 𝑦-intercept is at zero, 𝑐. And the question also told us that the vertex of the parabola is at zero, 𝑐. So this means that our parabola is symmetrical about the 𝑦-axis. Now, the values of 𝑎, 𝑏, and 𝑐 that we’ve been given in our equation 𝑦 equals negative 𝑎 times 𝑥 plus 𝑏 all squared plus 𝑐. Are gonna be the same as the values of 𝑎, 𝑏, and 𝑐 in the graph that we were given. So let’s do a little bit of analysis and see if we can work out what the coefficient of 𝑥 squared is gonna be in our new equation. And also what the 𝑥-coordinate of the vertex of that curve is going to be.
Well, we can start off by remembering that 𝑥 plus 𝑏 all squared means 𝑥 plus 𝑏 times 𝑥 plus 𝑏. And if we multiply each term in the first set of parentheses by each term in the second set of parentheses. And we can see that positive 𝑥 times positive 𝑥 gives us positive 𝑥 squared. Positive 𝑥 times positive 𝑏 gives us positive 𝑥𝑏. Or we can write that as positive 𝑏𝑥. Positive 𝑏 times positive 𝑥 gives us positive 𝑏𝑥. And positive 𝑏 times positive 𝑏 is positive 𝑏 squared. And we’ve still got our negative 𝑎 times all of that. And we’ve got plus 𝑐 on the end. We can add together the like terms, positive 𝑏𝑥 and positive 𝑏𝑥 to give us positive two 𝑏𝑥. Then, we can distribute negative 𝑎 through the parentheses. Negative 𝑎 times positive 𝑥 squared is negative 𝑎𝑥 squared. Negative 𝑎 times positive two 𝑏𝑥 is negative two 𝑎𝑏𝑥. And negative 𝑎 times positive 𝑏 squared is negative 𝑎𝑏 squared. And don’t forget we’ve got positive 𝑐 on the end. So 𝑦 is equal to negative 𝑥 squared minus two 𝑎𝑏𝑥 minus 𝑎𝑏 squared plus 𝑐.
Now, this starts to look a bit confusing because we’ve used 𝑎, 𝑏, and 𝑐 for different things. We’ve got 𝑎, 𝑏, and 𝑐 in our general form of the quadratic equation. And then, we’ve got the 𝑎, 𝑏, and 𝑐 that we’re using for specific values in this question. So I’m just gonna change the 𝑎, 𝑏, and 𝑐 in our general form of the quadratic equation to 𝑑, 𝑒, and 𝑓. And that makes the 𝑥-coordinate of the vertex negative 𝑒, that’s the coefficient of 𝑥, over two times 𝑑. That’s the coefficient of 𝑥 squared. So in a rearranged equation, the coefficient of 𝑥 squared, our equivalent of 𝑑 in the general formula, is negative 𝑎. The coefficient of 𝑥, our equivalent of 𝑒 in the general formula, is negative two 𝑎𝑏. And our constant term, the equivalent of 𝑓 in the general formula, is negative 𝑎𝑏 squared plus 𝑐.
Now we know that the value of 𝑎 is less than zero. So if 𝑎 is negative, then negative of 𝑎 must be positive. Negative 𝑎 must be greater than zero. And we said that when our coefficient of 𝑥 squared is greater than zero, it’s positive. If it’s positive, it’s happy. If it’s happy, it’s smiling. And a smiling parabola is also known as an open upward curve. Now, we need to use this formula to work out the 𝑥-coordinate of our vertex. So that’ll be the negative of the 𝑥-coefficient divided by two times the 𝑥 squared coefficient. And in our case, we said that the 𝑥-coefficient was negative two 𝑎𝑏. And the 𝑥 squared coefficient was negative 𝑎.
Now, we can divide top and bottom by two to give us the negative of negative one 𝑎𝑏 over one times negative 𝑎. So that simplifies to the negative of negative 𝑎𝑏 over negative 𝑎. Well, we don’t need the parentheses on the bottom anymore. So I can divide top and bottom by negative 𝑎. And of course, negative 𝑎 divided by negative 𝑎 is one in each case. So this all simplifies to negative 𝑏. The 𝑥-coordinate of our vertex is negative 𝑏.
Now, let’s just make a little space to calculate the 𝑦-coordinate of the vertex. And to find out the 𝑦-coordinate of the vertex, we simply need to substitute in that value that we found for 𝑥, negative 𝑏, into our original equation. 𝑦 equals negative 𝑎 times 𝑥 plus 𝑏 all squared plus 𝑐. So that 𝑦-value is negative 𝑎 times negative 𝑏 plus 𝑏 all squared plus 𝑐. But of course, negative 𝑏 plus 𝑏 is just equal to zero. So we’ve got negative 𝑎 times zero squared plus 𝑐. Well, zero squared is zero. And zero times negative 𝑎 is just zero. So the 𝑦-coordinate at the vertex is just 𝑐. So we found that the parabola with the equation 𝑦 equals negative 𝑎 times 𝑥 plus 𝑏 all squared plus 𝑐 is open upwards. And it has a vertex at negative 𝑏, 𝑐.
Option A had the correct vertex. But it said the graph was open downwards. Option B had the wrong vertex and said the graph was open downward. Option D said the graph was open upward, which was good. But it did have the wrong vertex. This leaves us with option C. In our case, the vertex is negative 𝑏, 𝑐. And the graph opens upward.