9.1 – graphing quadratic functions
DESCRIPTION
9.1 – Graphing Quadratic Functions. Ex. 1 Use a table of values to graph the following functions. a. y = 2 x 2 – 4 x – 5. Ex. 1 Use a table of values to graph the following functions. a. y = 2 x 2 – 4 x – 5. Ex. 1 Use a table of values to graph the following functions. - PowerPoint PPT PresentationTRANSCRIPT
9.1 – Graphing Quadratic Functions
Ex. 1 Use a table of values to graph the following functions.
a. y = 2x2 – 4x – 5
Ex. 1 Use a table of values to graph the following functions.
a. y = 2x2 – 4x – 5
x y
Ex. 1 Use a table of values to graph the following functions.
a. y = 2x2 – 4x – 5
x y
-2
-1
0
1
2
Ex. 1 Use a table of values to graph the following functions.
a. y = 2x2 – 4x – 5
x y
-2
-1
0
1
2
Ex. 1 Use a table of values to graph the following functions.
a. y = 2x2 – 4x – 5
y = 2(-2)2 – 4(-2) – 5 x y
-2
-1
0
1
2
Ex. 1 Use a table of values to graph the following functions.
a. y = 2x2 – 4x – 5
y = 2(-2)2 – 4(-2) – 5
y = 8 + 8 – 5 = 11x y
-2
-1
0
1
2
Ex. 1 Use a table of values to graph the following functions.
a. y = 2x2 – 4x – 5
y = 2(-2)2 – 4(-2) – 5
y = 8 + 8 – 5 = 11x y
-2 11
-1
0
1
2
Ex. 1 Use a table of values to graph the following functions.
a. y = 2x2 – 4x – 5
y = 2(-2)2 – 4(-2) – 5
y = 8 + 8 – 5 = 11x y
-2 11
-1 1
0 -5
1 -7
2 -5
Ex. 1 Use a table of values to graph the following functions.
a. y = 2x2 – 4x – 5
x y
-2 11
-1 1
0 -5
1 -7
2 -5
Ex. 1 Use a table of values to graph the following functions.
a. y = 2x2 – 4x – 5
x y
-2 11
-1 1
0 -5
1 -7
2 -5
Ex. 1 Use a table of values to graph the following functions.
a. y = 2x2 – 4x – 5
x y
-2 11
-1 1
0 -5
1 -7
2 -5
b. y = -x2 + 4x – 1
x y
-2
-1
0
1
2
b. y = -x2 + 4x – 1
x y
-2 -13
-1 -6
0 -1
1 2
2 3
b. y = -x2 + 4x – 1
x y
-2 -13
-1 -6
0 -1
1 2
2 3
b. y = -x2 + 4x – 1
x y
-2 -13
-1 -6
0 -1
1 2
2 3
3
4
b. y = -x2 + 4x – 1
x y
-2 -13
-1 -6
0 -1
1 2
2 3
3 2
4 -1
b. y = -x2 + 4x – 1
x y
-2 -13
-1 -6
0 -1
1 2
2 3
3 2
4 -1
b. y = -x2 + 4x – 1
x y
-2 -13
-1 -6
0 -1
1 2
2 3
3 2
4 -1
• Axis of symmetry:
• Axis of symmetry: x = - b 2a
• Axis of symmetry: x = - b 2a
• Vertex:
• Axis of symmetry: x = - b 2a
• Vertex: (x, y)
• Axis of symmetry: x = - b 2a
• Vertex: (x, y), where the x = axis of sym.
• Axis of symmetry: x = - b 2a
• Vertex: (x, y), where the x = axis of sym.• Maximum vs. Minimum:
• Axis of symmetry: x = - b 2a
• Vertex: (x, y), where the x = axis of sym.• Maximum vs. Minimum: For ax2 + bx + c,
• Axis of symmetry: x = - b 2a
• Vertex: (x, y), where the x = axis of sym.• Maximum vs. Minimum: For ax2 + bx + c,
–If a is positive, then the vertex is a Minimum.
• Axis of symmetry: x = - b 2a
• Vertex: (x, y), where the x = axis of sym.• Maximum vs. Minimum: For ax2 + bx + c,
–If a is positive, then the vertex is a Minimum.
–If a is negative, then the vertex is a Maximum.
• Axis of symmetry: x = - b 2a
• Vertex: (x, y), where the x-value = axis of sym.• Maximum vs. Minimum: For the form ax2 + bx + c,
– If a is positive, then the vertex is a Minimum.– If a is negative, then the vertex is a Maximum.
Ex. 2 Write the equation of the axis of symmetry, and find the coordinates of the vertex fo the graph of each function. Identify the vertex as a max or min. Then graph the function.
• Axis of symmetry: x = - b 2a
• Vertex: (x, y), where the x-value = axis of sym.• Maximum vs. Minimum: For the form ax2 + bx + c,
– If a is positive, then the vertex is a Minimum.– If a is negative, then the vertex is a Maximum.
Ex. 2 Write the equation of the axis of symmetry, and find the coordinates of the vertex fo the graph of each function. Identify the vertex as a max or min. Then graph the function.
a. -x2 + 2x + 3
• Axis of symmetry: x = - b 2a
• Vertex: (x, y), where the x-value = axis of sym.• Maximum vs. Minimum: For the form ax2 + bx + c,
– If a is positive, then the vertex is a Minimum.– If a is negative, then the vertex is a Maximum.
Ex. 2 Write the equation of the axis of symmetry, and find the coordinates of the vertex fo the graph of each function. Identify the vertex as a max or min. Then graph the function.
a. -x2 + 2x + 3
1) axis of sym.:
• Axis of symmetry: x = - b 2a
• Vertex: (x, y), where the x-value = axis of sym.• Maximum vs. Minimum: For the form ax2 + bx + c,
– If a is positive, then the vertex is a Minimum.– If a is negative, then the vertex is a Maximum.
Ex. 2 Write the equation of the axis of symmetry, and find the coordinates of the vertex fo the graph of each function. Identify the vertex as a max or min. Then graph the function.
a. -x2 + 2x + 31) axis of sym.: x = - b
2a
• Axis of symmetry: x = - b 2a
• Vertex: (x, y), where the x-value = axis of sym.• Maximum vs. Minimum: For the form ax2 + bx + c,
– If a is positive, then the vertex is a Minimum.– If a is negative, then the vertex is a Maximum.
Ex. 2 Write the equation of the axis of symmetry, and find the coordinates of the vertex fo the graph of each function. Identify the vertex as a max or min. Then graph the function.
a. -x2 + 2x + 31) axis of sym.: x = - b = - 2
2a 2(-1)
• Axis of symmetry: x = - b 2a
• Vertex: (x, y), where the x-value = axis of sym.• Maximum vs. Minimum: For the form ax2 + bx + c,
– If a is positive, then the vertex is a Minimum.– If a is negative, then the vertex is a Maximum.
Ex. 2 Write the equation of the axis of symmetry, and find the coordinates of the vertex fo the graph of each function. Identify the vertex as a max or min. Then graph the function.
a. -x2 + 2x + 31) axis of sym.: x = - b = - 2 = -2 = 1
2a 2(-1) -2
• Axis of symmetry: x = - b 2a
• Vertex: (x, y), where the x-value = axis of sym.• Maximum vs. Minimum: For the form ax2 + bx + c,
– If a is positive, then the vertex is a Minimum.– If a is negative, then the vertex is a Maximum.
Ex. 2 Write the equation of the axis of symmetry, and find the coordinates of the vertex fo the graph of each function. Identify the vertex as a max or min. Then graph the function.
a. -x2 + 2x + 31) axis of sym.: x = - b = - 2 = -2 = 1
2a 2(-1) -22) vertex:
• Axis of symmetry: x = - b 2a
• Vertex: (x, y), where the x-value = axis of sym.• Maximum vs. Minimum: For the form ax2 + bx + c,
– If a is positive, then the vertex is a Minimum.– If a is negative, then the vertex is a Maximum.
Ex. 2 Write the equation of the axis of symmetry, and find the coordinates of the vertex fo the graph of each function. Identify the vertex as a max or min. Then graph the function.
a. -x2 + 2x + 31) axis of sym.: x = - b = - 2 = -2 = 1
2a 2(-1) -22) vertex: (x, y)
• Axis of symmetry: x = - b 2a
• Vertex: (x, y), where the x-value = axis of sym.• Maximum vs. Minimum: For the form ax2 + bx + c,
– If a is positive, then the vertex is a Minimum.– If a is negative, then the vertex is a Maximum.
Ex. 2 Write the equation of the axis of symmetry, and find the coordinates of the vertex fo the graph of each function. Identify the vertex as a max or min. Then graph the function.
a. -x2 + 2x + 31) axis of sym.: x = - b = - 2 = -2 = 1
2a 2(-1) -22) vertex: (x, y) = (1,
• Axis of symmetry: x = - b 2a
• Vertex: (x, y), where the x-value = axis of sym.• Maximum vs. Minimum: For the form ax2 + bx + c,
– If a is positive, then the vertex is a Minimum.– If a is negative, then the vertex is a Maximum.
Ex. 2 Write the equation of the axis of symmetry, and find the coordinates of the vertex fo the graph of each function. Identify the vertex as a max or min. Then graph the function.
a. -x2 + 2x + 31) axis of sym.: x = - b = - 2 = -2 = 1
2a 2(-1) -22) vertex: (x, y) = (1, ?)
• Axis of symmetry: x = - b 2a
• Vertex: (x, y), where the x-value = axis of sym.• Maximum vs. Minimum: For the form ax2 + bx + c,
– If a is positive, then the vertex is a Minimum.– If a is negative, then the vertex is a Maximum.
Ex. 2 Write the equation of the axis of symmetry, and find the coordinates of the vertex fo the graph of each function. Identify the vertex as a max or min. Then graph the function.
a. -x2 + 2x + 31) axis of sym.: x = - b = - 2 = -2 = 1
2a 2(-1) -22) vertex: (x, y) = (1, ?)
-x2 + 2x + 3
• Axis of symmetry: x = - b 2a
• Vertex: (x, y), where the x-value = axis of sym.• Maximum vs. Minimum: For the form ax2 + bx + c,
– If a is positive, then the vertex is a Minimum.– If a is negative, then the vertex is a Maximum.
Ex. 2 Write the equation of the axis of symmetry, and find the coordinates of the vertex fo the graph of each function. Identify the vertex as a max or min. Then graph the function.
a. -x2 + 2x + 31) axis of sym.: x = - b = - 2 = -2 = 1
2a 2(-1) -22) vertex: (x, y) = (1, ?)
-x2 + 2x + 3 -(1)2 + 2(1) + 3
• Axis of symmetry: x = - b 2a
• Vertex: (x, y), where the x-value = axis of sym.• Maximum vs. Minimum: For the form ax2 + bx + c,
– If a is positive, then the vertex is a Minimum.– If a is negative, then the vertex is a Maximum.
Ex. 2 Write the equation of the axis of symmetry, and find the coordinates of the vertex fo the graph of each function. Identify the vertex as a max or min. Then graph the function.
a. -x2 + 2x + 31) axis of sym.: x = - b = - 2 = -2 = 1
2a 2(-1) -22) vertex: (x, y) = (1, ?)
-x2 + 2x + 3 -(1)2 + 2(1) + 3-1 + 2 + 3
• Axis of symmetry: x = - b 2a
• Vertex: (x, y), where the x-value = axis of sym.• Maximum vs. Minimum: For the form ax2 + bx + c,
– If a is positive, then the vertex is a Minimum.– If a is negative, then the vertex is a Maximum.
Ex. 2 Write the equation of the axis of symmetry, and find the coordinates of the vertex fo the graph of each function. Identify the vertex as a max or min. Then graph the function.
a. -x2 + 2x + 31) axis of sym.: x = - b = - 2 = -2 = 1
2a 2(-1) -22) vertex: (x, y) = (1, ?)
-x2 + 2x + 3 -(1)2 + 2(1) + 3-1 + 2 + 3 = 4
• Axis of symmetry: x = - b 2a
• Vertex: (x, y), where the x-value = axis of sym.• Maximum vs. Minimum: For the form ax2 + bx + c,
– If a is positive, then the vertex is a Minimum.– If a is negative, then the vertex is a Maximum.
Ex. 2 Write the equation of the axis of symmetry, and find the coordinates of the vertex fo the graph of each function. Identify the vertex as a max or min. Then graph the function.
a. -x2 + 2x + 31) axis of sym.: x = - b = - 2 = -2 = 1
2a 2(-1) -22) vertex: (x, y) = (1, ?)
-x2 + 2x + 3 -(1)2 + 2(1) + 3-1 + 2 + 3 = 4, so (1, 4)
• Axis of symmetry: x = - b 2a
• Vertex: (x, y), where the x-value = axis of sym.• Maximum vs. Minimum: For the form ax2 + bx + c,
– If a is positive, then the vertex is a Minimum.– If a is negative, then the vertex is a Maximum.
Ex. 2 Write the equation of the axis of symmetry, and find the coordinates of the vertex fo the graph of each function. Identify the vertex as a max or min. Then graph the function.
a. -x2 + 2x + 31) axis of sym.: x = - b = - 2 = -2 = 1
2a 2(-1) -22) vertex: (x, y) = (1, ?)
-x2 + 2x + 3 -(1)2 + 2(1) + 3-1 + 2 + 3 = 4, so (1, 4)
3) Max OR Min.?
• Axis of symmetry: x = - b 2a
• Vertex: (x, y), where the x-value = axis of sym.• Maximum vs. Minimum: For the form ax2 + bx + c,
– If a is positive, then the vertex is a Minimum.– If a is negative, then the vertex is a Maximum.
Ex. 2 Write the equation of the axis of symmetry, and find the coordinates of the vertex fo the graph of each function. Identify the vertex as a max or min. Then graph the function.
a. -x2 + 2x + 31) axis of sym.: x = - b = - 2 = -2 = 1
2a 2(-1) -22) vertex: (x, y) = (1, ?)
-x2 + 2x + 3 -(1)2 + 2(1) + 3-1 + 2 + 3 = 4, so (1, 4)
3) Max OR Min.? (1, 4) is a max b/c a is neg.
4) Graph:
4) Graph:
*Plot vertex:
4) Graph:
*Plot vertex: (1, 4)
4) Graph:
*Plot vertex: (1, 4)
*Make a table
based on vertex
4) Graph:
*Plot vertex: (1, 4)
* Make a table
based on vertexx y
1 4
4) Graph:
*Plot vertex: (1, 4)
* Make a table
based on vertexx y
0
1 4
4) Graph:
*Plot vertex: (1, 4)
* Make a table
based on vertexx y
-1
0
1 4
4) Graph:
*Plot vertex: (1, 4)
* Make a table
based on vertexx y
-1
0
1 4
2
4) Graph:
*Plot vertex: (1, 4)
* Make a table
based on vertexx y
-1
0
1 4
2
3
4) Graph:
*Plot vertex: (1, 4)
* Make a table
based on vertexx y
-1 0
0 3
1 4
2 3
3 0
4) Graph:
*Plot vertex: (1, 4)
* Make a table
based on vertexx y
-1 0
0 3
1 4
2 3
3 0
4) Graph:
*Plot vertex: (1, 4)
* Make a table
based on vertexx y
-1 0
0 3
1 4
2 3
3 0
b. 2x2 – 4x – 5
b. 2x2 – 4x – 5
1) axis of sym.:
2) vertex: (x, y) =
3) Max OR Min.?
4) Graph:
b. 2x2 – 4x – 5
1) axis of sym.: x = - b = -(-4) = 4 = 1
2a 2(2) 4
2) vertex: (x, y) = (1, ?)
2x2 – 4x – 5
2(1)2 – 4(1) – 5
2 – 4 – 5 = -7, so (1, -7)
3) Max OR Min.? (1, -7) is a min b/c a is neg.
4) Graph:
*Plot vertex: (1, -7)
* Make a table
based on vertex
x y
-1 1
0 -5
1 -7
2 -5
3 1