Vertex Form of Quadratics November 10, 2014 Notes.
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Vertex Form of Quadratics November 10, 2014 Notes
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Transcript of Vertex Form of Quadratics November 10, 2014 Notes.
Vertex Formof
Quadratics
November 10, 2014Notes
Objective
relate representations of quadratic functions, such as algebraic, tabular, graphical, and verbal descriptions[6.B]
Essential Question
• What parts of a quadratic function can I determine from the vertex form and how can I graph it?
Vocabulary• parabola: the shape of a quadratic function• vertex: the highest (maximum) or lowest
(minimum) point on a parabola; the turning point
• y-intercept: the point where the graph crosses the y-axis; x = 0
• x-intercepts: the points where the graph crosses the x-axis; y = 0; may be none, 1, or 2
• axis of symmetry: the vertical line that divides a parabola into two equal parts; goes through the vertex
Vertex Form
• f(x) = a(x – h)2 + k
– ‘a’ reflection across the x-axis (PEMDAS order) and/or vertical stretch or compression
– ‘h’ horizontal translation– ‘k’: vertical translation
What can we determine from the Vertex Form?Vertex Form:y=a(x-h)2 + k
h
k
Vertex:(h, k)
Axis of symmetry:
x = h
y-intercept:(0, y)
What can we determine from the Vertex Form?Vertex Form:y=a(x-h)2 + k y=(x–2)2
h
k
Vertex:(h, k)
Axis of symmetry:
x = h
y-intercept:(0, y)
What can we determine from the Vertex Form?Vertex Form:y=a(x-h)2 + k y=(x–2)2
h 2
k
Vertex:(h, k)
Axis of symmetry:
x = h
y-intercept:(0, y)
What can we determine from the Vertex Form?Vertex Form:y=a(x-h)2 + k y=(x–2)2
h 2
k 0
Vertex:(h, k)
Axis of symmetry:
x = h
y-intercept:(0, y)
What can we determine from the Vertex Form?Vertex Form:y=a(x-h)2 + k y=(x–2)2
h 2
k 0
Vertex:(h, k) (2, 0)
Axis of symmetry:
x = h
y-intercept:(0, y)
What can we determine from the Vertex Form?Vertex Form:y=a(x-h)2 + k y=(x–2)2
h 2
k 0
Vertex:(h, k) (2, 0)
Axis of symmetry:
x = hx = 2
y-intercept:(0, y)
What can we determine from the Vertex Form?Vertex Form:y=a(x-h)2 + k y=(x–2)2
h 2
k 0
Vertex:(h, k) (2, 0)
Axis of symmetry:
x = hx = 2
y-intercept:(0, y) (0, 4)
What can we determine from the Vertex Form?Vertex Form:y=a(x-h)2 + k y=(x–2)2 y = (x+3)2 – 1
h 2
k 0
Vertex:(h, k) (2, 0)
Axis of symmetry:
x = hx = 2
y-intercept:(0, y) (0, 4)
What can we determine from the Vertex Form?Vertex Form:y=a(x-h)2 + k y=(x–2)2 y = (x+3)2 – 1
h 2 -3
k 0
Vertex:(h, k) (2, 0)
Axis of symmetry:
x = hx = 2
y-intercept:(0, y) (0, 4)
What can we determine from the Vertex Form?Vertex Form:y=a(x-h)2 + k y=(x–2)2 y = (x+3)2 – 1
h 2 -3
k 0 -1
Vertex:(h, k) (2, 0)
Axis of symmetry:
x = hx = 2
y-intercept:(0, y) (0, 4)
What can we determine from the Vertex Form?Vertex Form:y=a(x-h)2 + k y=(x–2)2 y = (x+3)2 – 1
h 2 -3
k 0 -1
Vertex:(h, k) (2, 0) (-3, -1)
Axis of symmetry:
x = hx = 2
y-intercept:(0, y) (0, 4)
What can we determine from the Vertex Form?Vertex Form:y=a(x-h)2 + k y=(x–2)2 y = (x+3)2 – 1
h 2 -3
k 0 -1
Vertex:(h, k) (2, 0) (-3, -1)
Axis of symmetry:
x = hx = 2 x = -3
y-intercept:(0, y) (0, 4)
What can we determine from the Vertex Form?Vertex Form:y=a(x-h)2 + k y=(x–2)2 y = (x+3)2 – 1
h 2 -3
k 0 -1
Vertex:(h, k) (2, 0) (-3, -1)
Axis of symmetry:
x = hx = 2 x = -3
y-intercept:(0, y) (0, 4) (0, 8)
What can we determine from the Vertex Form?Vertex Form:y=a(x-h)2 + k y=(x–2)2 y = (x+3)2 – 1 y= -3(x+2)2+4
h 2 -3
k 0 -1
Vertex:(h, k) (2, 0) (-3, -1)
Axis of symmetry:
x = hx = 2 x = -3
y-intercept:(0, y) (0, 4) (0, 8)
What can we determine from the Vertex Form?Vertex Form:y=a(x-h)2 + k y=(x–2)2 y = (x+3)2 – 1 y= -3(x+2)2+4
h 2 -3 -2
k 0 -1
Vertex:(h, k) (2, 0) (-3, -1)
Axis of symmetry:
x = hx = 2 x = -3
y-intercept:(0, y) (0, 4) (0, 8)
What can we determine from the Vertex Form?Vertex Form:y=a(x-h)2 + k y=(x–2)2 y = (x+3)2 – 1 y= -3(x+2)2+4
h 2 -3 -2
k 0 -1 4
Vertex:(h, k) (2, 0) (-3, -1)
Axis of symmetry:
x = hx = 2 x = -3
y-intercept:(0, y) (0, 4) (0, 8)
What can we determine from the Vertex Form?Vertex Form:y=a(x-h)2 + k y=(x–2)2 y = (x+3)2 – 1 y= -3(x+2)2+4
h 2 -3 -2
k 0 -1 4
Vertex:(h, k) (2, 0) (-3, -1) (-2, 4)
Axis of symmetry:
x = hx = 2 x = -3
y-intercept:(0, y) (0, 4) (0, 8)
What can we determine from the Vertex Form?Vertex Form:y=a(x-h)2 + k y=(x–2)2 y = (x+3)2 – 1 y= -3(x+2)2+4
h 2 -3 -2
k 0 -1 4
Vertex:(h, k) (2, 0) (-3, -1) (-2, 4)
Axis of symmetry:
x = hx = 2 x = -3 x = -2
y-intercept:(0, y) (0, 4) (0, 8)
What can we determine from the Vertex Form?Vertex Form:y=a(x-h)2 + k y=(x–2)2 y = (x+3)2 – 1 y= -3(x+2)2+4
h 2 -3 -2
k 0 -1 4
Vertex:(h, k) (2, 0) (-3, -1) (-2, 4)
Axis of symmetry:
x = hx = 2 x = -3 x = -2
y-intercept:(0, y) (0, 4) (0, 8) (0, -8)
What can we determine from the Vertex Form?Vertex Form:y=a(x-h)2 + k y=(x–2)2 y = (x+3)2 – 1 y= -3(x+2)2+4 y= 2(x+3)2+1
h 2 -3 -2
k 0 -1 4
Vertex:(h, k) (2, 0) (-3, -1) (-2, 4)
Axis of symmetry:
x = hx = 2 x = -3 x = -2
y-intercept:(0, y) (0, 4) (0, 8) (0, -8)
What can we determine from the Vertex Form?Vertex Form:y=a(x-h)2 + k y=(x–2)2 y = (x+3)2 – 1 y= -3(x+2)2+4 y= 2(x+3)2+1
h 2 -3 -2 -3
k 0 -1 4
Vertex:(h, k) (2, 0) (-3, -1) (-2, 4)
Axis of symmetry:
x = hx = 2 x = -3 x = -2
y-intercept:(0, y) (0, 4) (0, 8) (0, -8)
What can we determine from the Vertex Form?Vertex Form:y=a(x-h)2 + k y=(x–2)2 y = (x+3)2 – 1 y= -3(x+2)2+4 y= 2(x+3)2+1
h 2 -3 -2 -3
k 0 -1 4 1
Vertex:(h, k) (2, 0) (-3, -1) (-2, 4)
Axis of symmetry:
x = hx = 2 x = -3 x = -2
y-intercept:(0, y) (0, 4) (0, 8) (0, -8)
What can we determine from the Vertex Form?Vertex Form:y=a(x-h)2 + k y=(x–2)2 y = (x+3)2 – 1 y= -3(x+2)2+4 y= 2(x+3)2+1
h 2 -3 -2 -3
k 0 -1 4 1
Vertex:(h, k) (2, 0) (-3, -1) (-2, 4) (-3, 1)
Axis of symmetry:
x = hx = 2 x = -3 x = -2
y-intercept:(0, y) (0, 4) (0, 8) (0, -8)
What can we determine from the Vertex Form?Vertex Form:y=a(x-h)2 + k y=(x–2)2 y = (x+3)2 – 1 y= -3(x+2)2+4 y= 2(x+3)2+1
h 2 -3 -2 -3
k 0 -1 4 1
Vertex:(h, k) (2, 0) (-3, -1) (-2, 4) (-3, 1)
Axis of symmetry:
x = hx = 2 x = -3 x = -2 x = -3
y-intercept:(0, y) (0, 4) (0, 8) (0, -8)
What can we determine from the Vertex Form?Vertex Form:y=a(x-h)2 + k y=(x–2)2 y = (x+3)2 – 1 y= -3(x+2)2+4 y= 2(x+3)2+1
h 2 -3 -2 -3
k 0 -1 4 1
Vertex:(h, k) (2, 0) (-3, -1) (-2, 4) (-3, 1)
Axis of symmetry:
x = hx = 2 x = -3 x = -2 x = -3
y-intercept:(0, y) (0, 4) (0, 8) (0, -8) (0, 19)
To Graph from Vertex Form:
1. Identify the vertex and axis of symmetry and graph.
2. Find the y-intercept and graph along with its reflection across the axis of symmetry.
3. Make a table (with the vertex in the middle) to calculate at least 5 points on the parabola.
Example 1 (Left Side): y = (x + 3)2 - 1
h:_______ k:_______ vertex:__________axis of symmetry: ___________ y-int: ________
x y
-3 -1 (-3, -1)x = -3
y = (x + 3)2 – 1y = (0 + 3)2 – 1y = (3)2 – 1y = 9 – 1y = 8
(0, 8)
-3 -1-4 0-5 3
-2 0-1 3
y = -3(x – 2)2 + 4y = -3(0 – 2)2 + 4y = -3(-2)2 + 4y = -3 • 4 + 4y = -8
Example 2 (Left Side): y = -3(x – 2)2 + 4
h:_______ k:_______ vertex:__________axis of symmetry: ___________ y-int: ________
2 4 (2, 4)x = 2 (0, -8)
2 4 1 1 0 -8
3 1 4 -8
x y
Assignment
1. f(x) = x2 – 2
2. g(x) = -(x – 4)2
3. h(x) = (x + 1)2 – 3
4. j(x) = (x + 2)2 + 2
Reflection
1. How do you know if a parabola will open upward or downward?
2. When does the parabola have a maximum point?
3. When does the parabola have a minimum point?
