Video: Finding the Coordinates of Vertices of Quadratic Functions

Find the coordinates of the vertex of the function 𝑓(π‘₯) = βˆ’7π‘₯Β² + 7π‘₯ + 5.


Video Transcript

Find the coordinates of the vertex of the function 𝑓 of π‘₯ equals negative seven π‘₯ squared plus seven π‘₯ 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 π‘₯ plus π‘Ž all squared plus 𝑏. And we recall that a function of this form must have a vertex at negative π‘Ž, 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 π‘₯ squared plus seven π‘₯ 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 π‘₯ squared minus π‘₯. Initially, we’re just going to focus on completing the square for the expression π‘₯ squared minus π‘₯. We recall that to do that, we halve the coefficient of π‘₯.

Now, the coefficient of π‘₯ 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 π‘₯ squared minus π‘₯ with this expression in our original function. And we find that 𝑓 of π‘₯ is equal to negative seven times π‘₯ minus a half all squared minus a quarter all plus five.

We’re now going to redistribute these parentheses. Negative seven multiplied by π‘₯ minus a half all squared is negative seven π‘₯ 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 π‘₯ 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.

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