Connecting Graphs with Equations Find a quadratic function f h...

Question

Connecting Graphs with Equations Find a quadratic function f having the graph shown. (Hint: See the Note following Example 3.)

Accepted Answer

Mhm Hey, everyone in this problem, we're asked to determine a quadratic function F of X for the graph given below the graph we're given shows a quadratic function. It opens upwards. We given that the vertex occurs at the 0.3 comma negative nine. And this graph also passes through the origin. We have four answer choices. Option A F of X equals X squared minus six, X, option B F of X equals X squared plus six X, option C F of X equals X squared minus X and option D F FX equals X squared minus six X plus nine. Now, since we're giving the point of the vertex, and we want to write this as a function. Let's use the vertex form of a quadratic so that we can just substitute that vertex indirectly. So we call that the vertex form of A para is Y is equal to a multiplied by X minus H squared plus K where comma K is the vertex. So our vertex from the graph is giving us three common negative nine which tells us that our H value is going to be three and our K value is going to be negative nine, the X and Y coordinate of the vertex. So we can write the function as Y is equal to a multiplied by X minus three squared minus nine. Now, we've got a lot of information in this function just from using that vertex. But we still need to figure out what this constant A is. Now, when I wrote uh talked about the function I mentioned that it passes through the origin, OK. The graph passes through the origin at 00. So we have the 00.0. Let's substitute that into our function. And you use it to solve for bay. So if we substitute the 0.0, we get that zero is equal to a multiplied by zero minus three squared minus nine. We can move the minus nine to the left-hand side by adding it, we get that nine is equal to nine A and dividing both sides by nine gives us that A is equal to one. All right. So we have this a value now. So we used our vertex and our 0.0 and we found that the function is Y is equal to X minus three squared minus nine. We have all of the information we need in this function. But notice that the answer traces are written in standard form. OK? They've been expanded. So we wanna go ahead and expand this function and write it in standard form so that we can compare it to our answer traces. So we have Y is equal to X minus three squared. If we expand that we get X squared minus six X plus nine, and then we have our minus nine at the end. Well, positive nine minus nine gives us zero. And so we're left with our function Y is just equal to X squared minus six X. So using the vertex and one other point, we found that this function F of X is equal to X squared minus six X which corresponds with answer choice. A thanks everyone for watching. I hope this video helped see you in the next one.

Connecting Graphs with Equations Find a quadratic function f having the graph shown. (Hint: See the Note following Example 3.)

Key Concepts

Vertex Form of a Quadratic Function

Determining the Value of 'a' in the Quadratic Function

Graph Interpretation and Coordinate Points

Watch next