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The function \( f\left( x \right) = x^{2} \) is the parent quadratic function. It is written in the standard form of a quadratic equation: \( y(x) = ax^{2} + bx + c \). For \( f\left( x \right) = x^{2} \), the values of b and c are both zero.
To graph a quadratic function in standard form, you generally have to choose values for x, substitute those values into the function, and solve for the output. Then you need to plot the ordered pairs on the coordinate plane and connect them using a smooth curve. Although not difficult, this can be time consuming. It is easier to graph a quadratic equation when it is written in vertex form.
Reminder
The vertex form of a quadratic equation is \( f\left( x \right) = a{(x - h)}^{2} + k, \) where the values of h and k represent the x and y values of the parabola's vertex.
You can convert any quadratic equation from standard form into vertex form by completing the square.
In the video below, the instructor will review the process of completing the square. He will show you two examples. In Example 1, the value of \( a = 1; \) and in Example 2, the value of \( a \neq 1 \). Pay close attention to how to work through each type of problem.
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Hello! In this video, I would like to share with you ways to transform the graphs of quadratic functions. First off, it’s important to note that parent functions are the simplest form of a family of functions. In other words, a parent function can be modified, or transformed, in shape to meet the needs of a particular set of conditions. For quadratic functions in standard form, the parent form that preserves the general shape of the graph is, f of x equals x-squared.
We use the complete the square method to establish the Vertex form of a quadratic that is written in standard form. The Vertex form is the best way to identify transformations of the parent graph into its new form. The following steps outline the general process for rewriting a quadratic function into vertex form, but we may not have to utilize each step. Step one is to isolate “c” by grouping the first two terms of the quadratic in standard form. Then we identify a greatest common factor in this group in order to reduce the a-value to one. In step three, we utilize a mini-formula to complete the square trinomial. We will add half of b, squared inside the grouping symbol. Since we are systematically selecting a value to add inside the parentheses, in step four, we must modify the outside of the parentheses to keep the equation balanced. Be careful at this step, since the number you modify on the outside might not be the same as the number inside. We’ll see this happen in example two below. Finally, in step five, we rewrite the quadratic in vertex form, by factoring the square trinomial we just created.
Here are two examples. In number one, we could isolate c easily (pause), and after that we would skip step two since the a-value is already one. I’ll rewrite the function now with a little more space in the grouping symbols to make room for step three (pause). Our b-value is negative four, and half of that is negative two, which squared is four. This will be the number we use to complete the square trinomial (pause). Immediately, I will subtract four so that our net result is zero and the equation remains balanced (pause). Finally, factor the square trinomial (pause), combine like terms on the outside, and observe the vertex form of the quadratic function!
In example two, after grouping the first two terms, we should factor negative two from the first two terms (pause). After completing the square (pause), we should add eighteen to the outside of the grouping symbol. We do this because of the nine written inside, is affected by the negative two distributing from the outside. Negative two times nine is negative eighteen and to offset that, we add eighteen on the outside. And after factoring we are left with a function written in vertex form.
Question
In Example 2, the video instructor added 9 inside the parentheses but added 18 outside the parentheses when completing the square. Explain how these actions kept the equation balanced.
Since the value of a in the standard form of the equation was not 1, you had to divide the a and the b terms by the value of a. You then write the number that you divided by outside the set of parentheses, in this case, -2.
Because of the distributive property, the number added inside the parentheses is affected by the number that is outside the parentheses. Since the video instructor added 9 inside the parentheses, he had to multiply (-2)(9) = -18 and then add 18 to the c term, -19, to keep the equation in balance.
