Download Vertex Form of a Quadratic Equation

Warm Up
Expand the following pairs of binomials:
1.(x-4)(2x+3)
2.(3x-1)(x-11)
3.(x+8)(x-8)
Quadratics Equations
Standard Form, Vertex Form and
Graphing
Vertex Form of a Quadratic Equation
• Recall that the standard form of a quadratic equation is
y = a·x2 + b·x + c
where a, b, and c are numbers and a does not equal 0.
• The vertex form of a quadratic equation is
y = a·(x – h)2 + k
where (h, k) are the coordinates of the vertex of the
parabola and a is a number that does not equal 0.
Vertex Form of a Quadratic Equation
• Vertex form y = a·(x – h)2 + k allows us to find vertex
of the parabola without graphing or creating a x-y table.
y = (x – 2)2 + 5
a=1
vertex at (2, 5)
y = 4(x – 6)2 – 3
y = 4(x – 6)2 + –3
a=4
vertex at (6, –3)
y = –0.5(x + 1)2 + 9
y = –0.5(x – –1)2 + 9
a = –0.5
vertex at (–1, 9)
Vertex Form of a Quadratic Equation
• Check your understanding…
1. What are the vertex coordinates of the parabolas with
the following equations?
vertex at (4, 1)
a. y = (x – 4)2 + 1
b. y = 2(x + 7)2 + 3
vertex at (–7, 3)
c. y = –3(x – 5)2 – 12
vertex at (5, –12)
2. Create a quadratic equation in vertex form for a
"wide"
parabola with vertex at (–1, 8).
y = 0.2(x + 1)2 + 8
Vertex Form of a Quadratic Equation
• Finding the a value.
• Recall that the vertex form of a quadratic equation is
y = a·(x – h)2 + k
where (h, k) are the coordinates of the vertex of the
parabola and a is a number that does not equal 0.
Also, the values of x and y represent the coordinates of
any point (x, y) that is on the parabola.
• We can see that (2, 9) is a point on y = (x – 4)2 + 5
9 = (2 – 4)2 + 5
9=4+5
…because the equation is true
9=9
Vertex Form of a Quadratic Equation
• Finding the a value (cont'd)
• If we know the coordinates of the vertex and some
other point on the parabola, then we can find the a value.
• For example,
What is the a value in the equation for a parabola that
has a vertex at (3, 4) and an x-intercept at (7, 0)?
substitute
y = a·(x – h)2 + k
simplify
0 = a·(7 – 3)2 + 4
simplify
0 = a·(4)2 + 4
subtract 4
0 = a·16 + 4
divide by 16
-4 = a·16
y = -0.25·(x – 3)2 + 4
-0.25 = a
Vertex Form of a Quadratic Equation
• Finding the a value (cont'd)
What is the a value in the equation for a parabola that
has a vertex at (2, -10) and other point at (3, -15)?
Vertex Form of a Quadratic Equation
• Classwork assignment
A particular parabola has its vertex at (-3, 8) and an
xintercept at (1, 0). Your task is to determine which of
the following are other points on that same
parabola.
1. (-1, 6)
2. (0, 3)
3. (4, -16)
4. (5, -24)
Homework:
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