The teacher asked the class to change π¦ = π₯^2 β 8π₯ + 11 to vertex form. A classmate does not understand why completing the square is necessary to do this. How would you explain this to your classmate? Change the function to vertex form in your explanation
Completing the square is necessary in order to convert the equation π¦ = π₯^2 β 8π₯ + 11 to vertex form because the vertex form of a quadratic equation is in the form π¦ = π(π₯ β β)^2 + π, where (β, π) represents the coordinates of the vertex.
To start, let's rewrite the given equation π¦ = π₯^2 β 8π₯ + 11.
Now, let's focus on completing the square to transform this equation into vertex form. We want to rewrite the quadratic expression π₯^2 β 8π₯ in the form π(π₯ β β)^2, where β represents the x-coordinate of the vertex.
To do this, let's consider the coefficient of π₯, which is -8 in our equation. We divide this coefficient by 2, giving us -4. Now, we square this value, obtaining (-4)^2 = 16.
To maintain the equivalence of our equation, we need to add and subtract 16 inside the parentheses, like this: π¦ = π₯^2 β 8π₯ + 16 β 16 + 11.
Now, let's rearrange the terms: π¦ = (π₯^2 β 8π₯ + 16) β 16 + 11.
By grouping the first three terms, we have: π¦ = (π₯ β 4)^2 β 5.
Voila! We have successfully completed the square and transformed the equation π¦ = π₯^2 β 8π₯ + 11 into vertex form, which is π¦ = (π₯ β 4)^2 β 5.
In this form, we can easily identify the vertex of the parabola as (4, -5). The vertex form gives us important information about the vertex and the direction of the parabola's opening.
To explain why completing the square is necessary in transforming the given equation to vertex form, let's first understand what vertex form represents.
The vertex form of an equation represents a quadratic function in its vertex coordinates. The vertex coordinates consist of the x-coordinate (h) and the y-coordinate (k), written as (h, k). In vertex form, the equation is given as:
π¦ = π(π₯ β h)^2 + k
To convert the equation π¦ = π₯^2 β 8π₯ + 11 into vertex form, we need to complete the square. Completing the square is important because it allows us to find the vertex of the quadratic function without relying on a graph or any other method.
Now, let's go through the process step-by-step:
Step 1: Group the first two terms together.
π¦ = (π₯^2 β 8π₯) + 11
Step 2: Now, we want to complete the square. To do that, we take half of the coefficient of the x-term and square it. In this case, the coefficient of the x-term is -8, so half of it is -4, and squaring that gives us 16.
Step 3: Add and subtract 16 within the parentheses to maintain the equation's accuracy:
π¦ = (π₯^2 β 8π₯ + 16 - 16) + 11
Step 4: Rearrange the equation:
π¦ = (π₯^2 β 8π₯ + 16) - 16 + 11
Step 5: Rewrite the equation as a perfect square trinomial within the parentheses:
π¦ = (π₯ β 4)^2 - 5
Therefore, the vertex form of the equation π¦ = π₯^2 β 8π₯ + 11 is π¦ = (π₯ β 4)^2 - 5.
By completing the square, we were able to identify the vertex coordinates of the quadratic function. In this case, the vertex is located at (4, -5). Completing the square helps us make transformations and gain insights into the function's behavior without relying on other methods like factoring or using a graph.
Completing the square is necessary to change the given quadratic function, π¦ = π₯^2 β 8π₯ + 11, to vertex form. The vertex form of a quadratic function is π¦ = π(π₯ β π)^2 + π, where (π, π) represents the vertex of the parabola.
To understand why completing the square is required, let's first look at the standard form of the quadratic function we have: π¦ = π₯^2 β 8π₯ + 11.
Step 1: Observe the coefficient of π₯^2, which is 1. Since it's a positive coefficient, we can conclude that the parabola opens upward.
Step 2: Now, let's rewrite the expression by grouping the terms related to π₯^2 and π₯ separately. We have: π¦ = (π₯^2 β 8π₯) + 11.
Step 3: To complete the square, we need to find a constant term that, when added to the expression inside the parentheses, will result in a perfect square trinomial. The constant we are looking for is half the coefficient of π₯, squared.
In this case, the coefficient of π₯ is -8, and half of it is -4. Squaring -4 gives us 16. Therefore, we need to add 16 inside the parentheses: π¦ = (π₯^2 β 8π₯ + 16) + 11 - 16.
Step 4: Simplify the expression inside the parentheses and combine like terms outside the parentheses: π¦ = (π₯^2 β 8π₯ + 16) - 5.
Step 5: Rewrite the expression inside the parentheses as a perfect square trinomial. This can be done by factoring it as (π₯ - 4)^2: π¦ = (π₯ - 4)^2 - 5.
Now we have the quadratic function in vertex form. The vertex is (4, -5), and by converting the function to this form, it becomes easier to identify key characteristics of the parabola, such as the vertex, direction of opening, and the axis of symmetry.