We will now look at a new way to write a quadratic function.
Vertex form f(x) = a(x - h)² + k makes it easy to identify the vertex of the parabola. The of value a is the same a value seen in standard form and factored form. The h value is the x-coordinate of vertex (Axis of Symmetry) and the the k value is the y-coordinate of the vertex. Watch the video below as an introduction. |
First lets practice identifying the vertex when it is written in vertex form. Remember the sign change with the h!
Finding x-intercepts using the Square Root Property
We have practiced finding the x-intercepts by writing a quadratic in factored form. Consider the quadratic f(x) = (x - 2)² - 25. Remember that the x-intercepts happen with y = 0, so the equation to solve will be 0 = (x - 2)² - 25. With an equation in this form we can use a method called the square root property. The square root property of equality states that you to take the square root of both sides of an equation and get exactly 2 solutions. When a value is being squared, you can "undo" the square by taking the square root. You do need to remember that there are 2 possible answers, positive and negative, when using the square root property. See the example below for more detail.
Copy the 2 examples into your notes.
Complete the following and check your answers.
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Completing the Square
To re-write a quadratic function from standard form to vertex form, a process is used called completing the square. Recall that a perfect square trinomial is a trinomial that can be factored into 2 equal factors: ex x² - 10x + 25 = (x - 5)(x - 5).
So a function in vertex form: f(x) = (x - 5)² + 2 is the same as f(x) = (x - 5)(x - 5) + 2,
therefore includes a perfect square trinomial. To complete the square you will form a perfect square trinomial by manipulating the function algebraically.
Forming these perfect square trinomials is called completing the square. Watch the video below and complete 5 questions from the practice link.
So a function in vertex form: f(x) = (x - 5)² + 2 is the same as f(x) = (x - 5)(x - 5) + 2,
therefore includes a perfect square trinomial. To complete the square you will form a perfect square trinomial by manipulating the function algebraically.
Forming these perfect square trinomials is called completing the square. Watch the video below and complete 5 questions from the practice link.
To complete the process lets apply the skill from above to a quadratic function. Make sure to watch the entire video below and copy down the following examples.
Now lets use completing the square to find the x-intercepts. Copy the example below and complete practice skill. If you need additional help watch the video provided.