Graph each piecewise-defined function.

Question

f (x) = {\begin{matrix} - \frac{1}{2} x^{2} + 2 & if x \leq 2 \\ \frac{1}{2} x & if x > 2 \end{matrix}

Accepted Answer

Hey, everyone here we are asked to graph the following function F of X is equal to X squared minus two. If X is less than or equal to one and two, X, if X is greater than one here we have four answer choice options. ABC and D, each of which display a graph representing our given function. To begin solving this problem. We first want to consider just the first part of our given function where F of X is equal to X squared minus two for when X is less than or equal to one. So with this function, we can actually rewrite it as F of X is equal to a times the quantity of X minus zero squared minus two. And now we see that this rewritten form of our function follows the vertex form of a quadratic function which if we recall is F of X is equal to a times the quantity of X minus H squared plus K. And now we also can recall that the vertex can be found using this formula since the vertex is the point H K. So now we can compare our formula with our rewritten function to find the vertex of our own function. So doing this, we see our vertex is at the zero and -2. So now with our vertex identified, we can move on to finding more points that lie on our function so that we can then graph the function. So we can consider the point for when X is equal to one because we want to consider X values that fit within our interval or where X is less than, or equal to one. So again, starting for when X is equal to one, we just need to substitute in one for X from our given function. So we have F of one is equal to the quantity of one squared minus two. Simplifying we get the value of negative one. So we see that another ordered pair that we have for our function is the 10.1 and negative. Now we want to repeat this process for a few more values of X. So we chose a value for when X is equal to one. So the next values have to be less than one. So we can choose the value X is equal to negative one. So finding F of negative one, we have the quantity of negative one squared minus two. Simplifying we get the value of negative one. So we see another ordered pair is at negative one and negative one repeating this process one more time, we can choose the value of next. So finding M of negative two, we have the quantity of negative two squared minus two. Simplifying we get positive two. So we see 1/4 ordered pair is at negative two and two. So now we have found four different ordered pairs that are within our function. So the next step is to just graph these four points. And so graphing these four points, we see them lie individually on our graph. And so we have begun graphing the first part of our function. So the next step is to just draw a smooth curve line that connects all of our points. And in doing this, we have graphed our first part of our function where again, F of X is equal to X squared minus two when X is less than, or equal to one. And now it is important to note here that we have an end point at one and negative one because of our interval defining our first part of the function. So the next step is to consider the second part of our original function, which was that F of X is equal to two X when X is greater than one. So this statement says that when X is greater than one, we have this line of two X. So now we just want to graph this line on the same graph of as our other part of the function. So doing this, we see that we in fact have a line but we see that the end point of this line is an open circle. And this is because F of X is equal to two X is undefined for when X is equal to one because our interval or the inequality states that X must be greater than one. So with this finalized graph, we see that the first part of our function gave us part of a parabola. And the second part of our function gave us a, an increasing line. So with this finalized graph, we see here from our answer choice options that we are left with answer choice. C Thanks for watching. I hope you found this video helpful and I'll see you again next time.

Graph each piecewise-defined function.
$f (x) = {\begin{matrix} - \frac{1}{2} x^{2} + 2 & if x \leq 2 \\ \frac{1}{2} x & if x > 2 \end{matrix}$

Key Concepts

Piecewise-Defined Functions

Graphing Quadratic and Linear Functions

Domain Restrictions and Continuity

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