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Vertex form of Quadratic Functions orsini-gotha.com Topical Outline | Algebra 1 Outline | MathBits" Teacher Resources Terms of Use liên hệ Person: Donna Roberts
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The vertex size of a quadratic function is given by f (x) = a(x - h)2 + k, where (h, k) is the vertex of the parabola.
FYI: Different textbooks have different interpretations of the reference "standard form" of a quadratic function. Some say f (x) = ax2 + bx + c is "standard form", while others say that f (x) = a(x - h)2 + k is "standard form". Lớn avoid confusion, this site will not refer lớn either as "standard form", but will reference f (x) = a(x - h)2 + k as "vertex form" and will reference f(x) = ax2 + bx + c by its full statement.

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When written in "vertex form": • (h, k) is the vertex of the parabola, and x = h is the axis of symmetry. • the h represents a horizontal shift (how far left, or right, the graph has shifted from x = 0).

• the k represents a vertical shift (how far up, or down, the graph has shifted from y = 0).

• notice that the h value is subtracted in this form, & that the k value is added. If the equation is y = 2(x - 1)2 + 5, the value of h is 1, và k is 5. If the equation is y = 3(x + 4)2 - 6, the value of h is -4, và k is -6.

to Convert from f (x) = ax2 + bx + c khung to Vertex Form: Method 1: Completing the Square lớn convert a quadratic from y = ax2 + bx + c form to lớn vertex form, y = a(x - h)2+ k, you use the process of completing the square. Let"s see an example.

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Convert y = 2x2 - 4x + 5 into vertex form, và state the vertex.


Equation in y = ax2 + bx + c form.
y = 2x2 - 4x + 5
Since we will be "completing the square" we will isolate the x2 và x terms ... So move the + 5 lớn the other side of the equal sign.
y - 5 = 2x2 - 4x
We need a leading coefficient of 1 for completing the square ... So factor out the current leading coefficient of 2.
y - 5 = 2(x2 - 2x)
Get ready lớn create a perfect square trinomial. BUT be careful!! In previous completing the square problems with a leading coefficient not 1, our equations were phối equal khổng lồ 0. Now, we have to khuyễn mãi giảm giá with an additional variable, "y" ... So we cannot "get rid of " the factored 2. When we showroom a box lớn both sides, the box will be multiplied by 2 on both sides of the equal sign.

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Find the perfect square trinomial. Take half of the coefficient of the x-term inside the parentheses, square it, & place it in the box.
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Simplify & convert the right side to a squared expression.
y - 3 = 2(x - 1)2
Isolate the y-term ... So move the -3 khổng lồ the other side of the equal sign.
y = 2(x - 1)2 + 3
In some cases, you may need to transform the equation into the "exact" vertex khung of y = a(x - h)2 + k, showing a "subtraction" sign in the parentheses before the h term, & the "addition" of the k term. (This was not needed in this problem.)
y = 2(x - 1)2 + 3 Vertex khung of the equation. Vertex = (h, k) = (1, 3) (The vertex of this graph will be moved one unit to lớn the right and three units up from (0,0), the vertex of its parent y = x2.)
Here"s a sneaky, quick tidbit:
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When working with the vertex form of a quadratic function, và .The "a" & "b" referenced here refer to lớn f (x) = ax2 + bx + c.

Method 2: Using the "sneaky tidbit", seen above, khổng lồ convert lớn vertex form:


y = ax2 + bx + c form of the equation.
y = 2x2 - 4x + 5
Find the vertex, (h, k). và . <f (h) means to plug your answer for h into the original equation for x.>

a = 2 và b = -4

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Vertex: (1,3)


Write the vertex form. y = a(x - h)2 + k
y = 2(x - 1)2 + 3

khổng lồ Convert from Vertex form to y = ax2 + bx + c Form:


Simply multiply out & combine lượt thích terms:

y = 2(x - 1)2 + 3 y = 2(x2 - 2x + 1) + 3 y = 2x2 - 4x + 2 + 3 y = 2x2 - 4x + 5

Graphing a Quadratic Function in Vertex Form:


1. Start with the function in vertex form:y = a(x - h)2 + k
y = 3(x - 2)2 - 4
2. Pull out the values for h and k. If necessary, rewrite the function so you can clearly see the hk values. (h, k) is the vertex of the parabola. Plot the vertex.
y = 3(x - 2)2 + (-4)
h = 2; k = -4 Vertex: (2, -4)
3. The line x = h is the axis of symmetry. Draw the axis of symmetry.

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x = 2 is the axis of symmetry
4. Find two or three points on one side of the axis of symmetry, by substituting your chosen x-values into the equation.

For this problem, we chose (to the left of the axis of symmetry):

x = 1; y = 3(1 - 2)2 - 4 = -1

x = 0; y = 3(0 - 2)2 - 4 = 8

Plot (1, -1) & (0,8)


5. Plot the mirror images of these points across the axis of symmetry, or plot new points on the right side. Draw the parabola. Remember, when drawing the parabola to avoid "connecting the dots" with straight line segments. A parabola is curved, not straight, as its slope is not constant.

Topical Outline | Algebra 1 Outline | orsini-gotha.com | MathBits" Teacher Resources Terms of Use tương tác Person: Donna Roberts