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

Question

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

Graph of a quadratic function with points (0, 9) and (2, 13) marked.

Accepted Answer

Mhm Hey, everyone in this problem, we're asked to determine a quadratic function F FX for the graph given below. Now we're showing the graph of a quadratic function it opens downwards and we're given two points on the graph. One is zero comma nine and one is two comma 13. We have four answer choices A through D OK. All four of them are functions F FX that start with negative X squared plus four X and they each have a different constant at the end. So option A has plus nine. Option B has plus four, option C has plus 17 and option D has minus 70. Now, the 0.2 13 were given is right at the vertex of this quadratic of this parabola. So when we're writing the function, let's start with our vertex form because we'll be able to substitute that point in directly. So we're called that the vertex form of our para Y is equal to a multiplied by X minus H squared plus K where H comma K is the vertex. So according to our graph, we're given the vertex of two comma 13. OK. So our vertex is two comma 13 which tells us that H is equal to two and K is equal to 13. So substituting in that, those sorry into our function, we get that Y is equal to a multiplied by X minus two squared plus 13. And so that gives us a lot of information about this function. The only thing we have left that we don't know is this value of A oh, we're given another 0.0 comma nine on the graph of our function. So let's substitute that into what we have so far to solve for A. So we have an X value of zero and A Y value of nine, we get nine is equal to a multiplied by zero minus two squared was 13. Simplifying this is nine is equal to four. A was 13. We can subtract 13 from both sides. We get that four A is equal to negative four. And finally, we can divide both sides by four to get that A is equal to negative one. So we used our vertex to write this equation in vertex form. We've used this second point to find our A value. And so far we have that Y is equal to negative X minus two, all squared plus 30. Now our answer choices are in standard form. So this expanded form. So we want to expand this simplify it so that we can compare to our answer choices. So we get Y is equal to negative X squared minus four X plus four and then plus 13. And so writing this out with that negative applied to each of those terms, we get that Y is equal to negative X squared plus four, X minus four plus 13, which is equal to negative X squared plus four X plus nine. And so given just the vertex and one other point on this graph, we were able to find that the equation of the function is F of X equals negative X squared plus four X plus nine, 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

Quadratic Functions

Vertex of a Parabola

Finding Quadratic Equations from Points

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