Find the coordinates of the vertex of the function 𝑓 of 𝑥 equals negative seven 𝑥 squared plus seven 𝑥 plus five.
Whenever we’re looking to find the location of the vertex of a quadratic function, we should be thinking about representing our equation in completed square form. That’s usually of the form 𝑥 plus 𝑎 all squared plus 𝑏. And we recall that a function of this form must have a vertex at negative 𝑎, 𝑏. In fact, we can multiply everything in this set of parentheses by some constant value 𝑐 and the vertex itself doesn’t change. So how do we write the equation negative seven 𝑥 squared plus seven 𝑥 plus five in completed square form?
Well, we’ll begin by factoring negative seven from the first two terms. So the first two terms become negative seven times 𝑥 squared minus 𝑥. Initially, we’re just going to focus on completing the square for the expression 𝑥 squared minus 𝑥. We recall that to do that, we halve the coefficient of 𝑥.
Now, the coefficient of 𝑥 here is technically negative one. So halving that, we get negative one-half. We then subtract this value squared. Well, negative a half squared is a quarter. So we’re going to subtract a quarter from this bracket. We replace 𝑥 squared minus 𝑥 with this expression in our original function. And we find that 𝑓 of 𝑥 is equal to negative seven times 𝑥 minus a half all squared minus a quarter all plus five.
We’re now going to redistribute these parentheses. Negative seven multiplied by 𝑥 minus a half all squared is negative seven 𝑥 minus a half all squared. We then multiply negative seven by negative a quarter. We get seven-quarters. And, of course, we have this plus five to deal with. This is almost in completed square form. Our final step is going to be to find the sum of seven-quarters and five.
To do that, we write five as five over one. And we create a common denominator by multiplying both the numerator and denominator of this fraction by four. And that gives us seven-quarters plus twenty-quarters, which is of course 27 over four. So our completed square form is negative seven times 𝑥 minus a half all squared plus 27 over four. We can convert 27 over four into a mixed number by dividing 27 by four to get six and then writing the remainder as a fraction. So 27 over four is the same as six and three-quarters.
To find the coordinates of the vertex, we begin by taking the negative value of negative one-half. So that’s a half. And then, the 𝑦-coordinate is simply six and three-quarters. And so, the coordinate we’re looking for is a half, six and three-quarters.