Video: Finding the Value of 𝑥 at Which a Quadratic Function Has a Minimum Value given Its Rule in Vertex Form

At which value of 𝑥 does the function 𝑓(𝑥) = (𝑥 + 3)² + 4 have its minimum?

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Video Transcript

At which value of 𝑥 does the function 𝑓 of 𝑥 equals 𝑥 plus three all squared plus four have its minimum?

So if we take a look at our function, we want the part 𝑥 plus three all squared to be the lowest possible value it can be if we want the value of our function to be as its minimum. Well, we know that 𝑥 plus three all squared is gonna be greater than or equal to zero. And that’s because any number squared is gonna be zero or above. So it’s gonna be zero or a positive number. So therefore, the lowest possible value of 𝑥 plus three all squared has got to be zero.

So before we work out which value of 𝑥 is gonna give us this minimum, we need to decide how do we know it’s gonna be a minimum, not a maximum. We know it’s gonna be a minimum because if we multiply out of our parentheses 𝑥 plus three all squared, we’re gonna have a positive 𝑥 squared term. So therefore, we know the shape of our graph is going to be a parabola, a U-shaped parabola. If it was a negative 𝑥 squared term, then we’d actually have an inverted U-shaped parabola.

So therefore, to find our minimum point, what we want to do is set 𝑥 plus three all squared equal to zero cause we want to find out what’s going to make this equal to zero. So then, if we take the square root of each side of the equation, we’re gonna get 𝑥 plus three is equal to zero. And then, we subtract three from each side of the equation. We’re gonna get 𝑥 is equal to negative three.

And if we put this back in, we would have negative three plus three. Well, that’s just zero. Zero squared is equal to zero. So therefore, we can say that the value of 𝑥 that the function 𝑓 of 𝑥 equals 𝑥 plus three all squared plus four has its minimum is going to be 𝑥 equals negative three.

And it’s worth noting that this minimum point is gonna have the coordinates negative three, four and that’s because our 𝑥-value is negative three. If we have that, then 𝑓 of 𝑥 would be equal to zero plus four. So therefore, the value of 𝑦 was gonna be four. So we’d have the coordinate negative three, four to the minimum point.

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