Parabola properties Consider the general quad...

Question

Parabola properties Consider the general quadratic function ƒ(x) = ax² + bx + c , with a ≠ 0.

a. Find the coordinates of the vertex of the graph of the parabola y= ƒ(x) in terms of a, b, and c.

Accepted Answer

Welcome back everyone. In this problem, we want to write the formula for the vertex of the parabola y equals px squared plus qx plus r in terms of p, q, and r where p is not equal to 0. For our answer choices, a is negative q divided by 2p and r minus q squared divided by 2p. B is the coordinate negative q divided by 2p and r minus q squared divided by p. C is the coordinate negative q divided by p and r minus q squared divided by 4p. And d is the coordinate negative q divided by 2p and r minus q squared divided by 4p. Now what do we want to know here? Well, we want the formula for the vertex of the parabola. And visually, what that means is that for our parabola, we have some point that at the vertex that if we call it h k, for our parabola in the general form, axsquaredplusbxplusc, then the x coordinate of our vertex, in this case, h would be equal to negative b divided by 2 a. That's the general formula for the x coordinate of the vertex for the parabola. So using that idea, that means for all equation y equals px squared plus qx plus r, then the x coordinate the x coordinate of the vertex would be equal to negative b. In this case, that would have been q. So, that's negative q divided by 2 a, where in this case, a would be p. So now we have the x coordinate of our vertex and all we need to do is figure out the y coordinate. Now, to find the y coordinate, we can just plug our x value into our equation. So using x equals negative q divided by 2p to solve for the y coordinate of the vertex. Then our equation will now become y equals p multiplied by x, which is negative q divided by 2p, all squared, plus q multiplied by x, which again is negative q divided by 2 p. And then we're adding r to that. So let's go ahead to see if we can solve for that y coordinate. Now, negative q divided by 2p squared means we multiply negative q by negative q, that's q squared and 2p multiplied by 2p which is 4p squared. Q multiplied by negativeq is negativeq squared and that is being divided by 2p and we're still adding r. Now, since p is a common factor in both terms for our first term, if we cancel out p, then we can rewrite it as q squared divided by 4p and we write minus q squared divided by 2p+r. Now, if we look at those first two terms, notice that both of them have 2 P R 4 P, rather, as a common multiple for the denominator. So by combining both of those 4p into let me write it out in the next line. Okay. So 4p into 4p goes one time. So that would be q squared. 2p into 4p goes two times. So I'm going to be subtracting q squared multiplied by 2, which is minus 2q2qsquared. And then I'm adding r to all of that. Q squared minus 2q squared, that's negativeq squared, all divided by 4p. And then I'm adding r, which I can rewrite as r minus q squared divided by 4p. So, r minus q squared divided by 4p would have been the y coordinate of the vertex. And, recall that the x coordinate is negative q divided by 2p. So that means our vertex the coordinates of the vertex rather would have been negative q divided by 2p and r minus q squared divided by 2p. And when we look on our answer choices, that q squared divided by 4p. My apologies. And when we look on our answer choices, that would have been answer choice d. Thanks a lot for watching everyone. I hope this video helped.

Parabola properties Consider the general quadratic function ƒ(x) = ax² + bx + c , with a ≠ 0.

a. Find the coordinates of the vertex of the graph of the parabola y= ƒ(x) in terms of a, b, and c.

Key Concepts

Quadratic Functions

Vertex of a Parabola

Axis of Symmetry

Watch next