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Transcript of Evaluating and Graphing Quadratic Functions 1. Graphing Quadratic Functions The quadratic function...
Evaluating and Graphing Quadratic Functions
1. Graphing Quadratic Functions
• The quadratic function is a second-order polynomial function
• It is always written in this format, with the coefficient parameters: a, b, c
f(x) = ax2 + bx + c
OR
ax2 + bx + c = f(x)
Evaluating and Graphing Quadratic Functions
1. Graphing Quadratic Functions
• Example 1: Identify the coefficient parameters (a, b, c) in the following quadratic functions:a. f(x) = 5x2 + 10x + 15
b. f(n) = n2 – 10n + 22
c. h(r) = 7r2 + 14r – 7
d. g(x) = x2 – 25
e. g(x) = 7x2
f. h(n) = 5x – 24
a = 5, b = 10, c = 15
a = 1, b = -10, c = 22
a = 7, b = 14, c = -7
a = 1, b = 0, c = -25
a = 7, b = 0, c = 0
a = 0, b = 5, c = -24 (linear)
Evaluating and Graphing Quadratic Functions
1. Graphing Quadratic Functions
• This is a graph of a quadratic function. We call it a parabola
• What are some observations we can make about the graph?
x
y
Evaluating and Graphing Quadratic Functions
1. Graphing Quadratic Functions
• The axis of symmetry is the x-coordinate that forms the middle and splits the parabola in two halves• Axis of Symmetry: x = - b
Formula 2a
• The vertex is the (x,y) ordered pair on the axis of symmetry• Plug in the axis of symmetry for x to
find the y coordinate
Evaluating and Graphing Quadratic Functions
1. Graphing Quadratic Functions
1. Find axis of symmetry• X = - b
2a
2. Find the vertex• plug in the axis of
symmetry for x to find the y-coordinate of the vertex
Ex 2: How to Graph a Quadratic Function?
1. f(x) = x2 – 10x + 24a=1, b=-10, c=24
x = -b = -(-10) = 10 = 5 2a 2(1) 2
2. f(x) = x2 – 10x + 24 f(5) = (5)2 – 10(5) + 24f(5) = 25 – 50 + 24f(5) = -1
vertex: (5, -1)
1. Identify a, b, c
2. Find the axis of symmetry
3. Find the vertex (x,y)
4. Fill in data table values
5. Graph the function
Evaluating and Graphing Quadratic Functions
1. Graphing Quadratic Functions
3. Make a data table (x/y or in/out table)
• Put the vertex in the middle value
• Plug-in two x-values lower and two x-values higher than the axis of symmetry
• You should notice some symmetry in your output values
X Y
3
4
5 -1
6
7
f(x) = x2 – 10x + 24a=1, b=-10, c=24vertex: (5, -1)
Ex 2: How to Graph a Quadratic Function?
1. Identify a, b, c
2. Find the axis of symmetry
3. Find the vertex (x,y)
4. Fill in data table values
5. Graph the function
Evaluating and Graphing Quadratic Functions
1. Graphing Quadratic Functions
4. First plot the vertex. Then plot all the other points on your graph. Draw your parabola. Done!
X Y
3 3
4 0
5 -1
6 0
7 3
f(x) = x2 – 10x + 24a=1, b=-10, c=24vertex: (5, -1)
Ex 2: How to Graph a Quadratic Function?
1. Identify a, b, c
2. Find the axis of symmetry
3. Find the vertex (x,y)
4. Fill in data table values
5. Graph the function
Evaluating and Graphing Quadratic Functions
1. Graphing Quadratic Functions
X Y
f(x) = x2 – 6x + 8a= , b= , c= vertex: ( __ , __ )
Ex 3: How to Graph a Quadratic Function?
1. Identify a, b, c
2. Find the axis of symmetry
3. Find the vertex (x,y)
4. Fill in data table values
5. Graph the function
x = -b = = ___ 2a
f(x) = x2 – 6x + 8f( ) = ( )2 – 6( ) + 8
f( ) = ___
Vertex goes here
Evaluating and Graphing Quadratic Functions
1. Graphing Quadratic Functions
f(x) = (x – 4)2 + 1
f(x) = (x – 4)(x – 4) + 1
f(x) = x2 – 8x + 16 + 1
f(x) = x2 – 8x + 17
Ex 4: How to Graph a Quadratic Function?
1. Identify a, b, c
2. Find the axis of symmetry
3. Find the vertex (x,y)
4. Fill in data table values
5. Graph the function
Use the box method to multiply the binomials
x2 -4x
-4x 16
x -4
x
-4
f(x) = x2 – 8x + 17
a= , b= , c=
vertex: ( __ , __ )
x = -b = = ___ 2a
f(x) = x2 – 8x + 17f( ) = ( )2 – 8( ) + 17
f( ) = ___
Evaluating and Graphing Quadratic Functions
1. Graphing Quadratic Functions
X Y
Ex 4: How to Graph a Quadratic Function?
1. Identify a, b, c
2. Find the axis of symmetry
3. Find the vertex (x,y)
4. Fill in data table values
5. Graph the function
Vertex goes here
f(x) = x2 – 8x + 17
a= , b= , c=
vertex: ( 4 , 1 )
Evaluating and Graphing Quadratic Functions
2. Solving/
Finding Roots/
Finding Zeroes
Of Quadratic
Functions
What is true about where the curve intercepts the x-axis? How many times does it intercept the x-axis?
x
y
• Y = f(x) = 0 at the x-intercepts (curve crosses x-axis)
• The x-coordinates where y=0 are called solutions, or the roots, or the zeroes of the quadratic function
Evaluating and Graphing Quadratic Functions
• I DO: Find the solution of the following function:
f(x) = (x – 3)(x + 2)
2. Solving/
Finding Roots/
Finding Zeroes
Of Quadratic
Functions
(x – 3)(x + 2) = 0
2. Solve for variable.
You will have two solutions/roots/zeroes
(in most cases)
(x – 3)(x + 2) = 0
x – 3 = 0
+3 +3
x = 3
x + 2 = 0
-2 -2
x = -2
3. Write the solutionSolutions: (3,0) and (-2,0)
The parabola crosses the x-axis at x = -2 and x = 3
1. Factor the polynomial and set the function = 0
Evaluating and Graphing Quadratic Functions
• WE DO: Find the solution of the following function:
f(n) = (2n + 5)(n – 4)
2. Solving/
Finding Roots/
Finding Zeroes
Of Quadratic
Functions
(2n + 5)(n – 4) = 0
2. Solve for variable.
You will have two solutions/roots/zeroes
(in most cases)
(2n + 5)(n – 4) = 0
2n + 5 = 0
-5 -5
2n = -5
2 2
n = -2.5
n – 4 = 0
+4 +4
x = 4
3. Write the solutionSolutions: (-2.5, 0) and (4, 0)
The parabola crosses the x-axis at x = -2.5 and x = 4
1. Factor the polynomial and set the function = 0
Evaluating and Graphing Quadratic Functions
• I DO: Find the solution of the following function:
f(x) = x2 – 4x + 3
2. Solving/
Finding Roots/
Finding Zeroes
Of Quadratic
Functions
1. Factor the polynomial and set the function = 0
x2 – 4x + 3 = 0
(x – 1)(x – 3) = 0
2. Solve for variable.
You will have two solutions/roots/zeroes
(in most cases)
(x – 1)(x – 3) = 0
x – 1 = 0
+1 +1
x = 1
x – 3 = 0
+3 +3
x = 3
3. Write the solutionSolutions: (1,0) and (3,0)
The parabola crosses the x-axis at x = 1 and x = 3
Evaluating and Graphing Quadratic Functions
• I DO: Find the solution of the following function:
x2 – 4x + 3 = 15
2. Solving/
Finding Roots/
Finding Zeroes
Of Quadratic
Functions
1. Factor the polynomial and set the function = 0
x2 – 4x + 3 = 15
-15 -15
x2 – 4x – 12 = 0
(x – 6)(x + 2) = 0
2. Solve for variable.
You will have two solutions/roots/zeroes
(in most cases)
(x – 6)(x + 2) = 0
3. Write the solutionSolutions: (-2,0) and (6,0)
The parabola crosses the x-axis at x = -2 and x = 6
x – 6 = 0
+6 +6
x = 6
x + 2 = 0
-2 -2
x = -2
Evaluating and Graphing Quadratic Functions
• WE DO: Find the solution of the following function:
x2 – 11x + 15 = -9
2. Solving/
Finding Roots/
Finding Zeroes
Of Quadratic
Functions
1. Factor the polynomial and set the function = 0
x2 – 11x + 15 = -9
+9 +9
x2 – 11x + 24 = 0
(x – 8)(x – 3) = 0
2. Solve for variable.
You will have two solutions/roots/zeroes
(in most cases)
(x – 8)(x – 3) = 0
3. Write the solutionSolutions: (3,0) and (8,0)
The parabola crosses the x-axis at x = 3 and x = 8
x – 8 = 0
+8 +8
x = 8
x – 3 = 0
+3 +3
x = 3
Evaluating and Graphing Quadratic Functions
3. Find the Max/Min of a Quadratic Function
What is Concavity?
f(x) = x2 – 4x – 5a= 1, b=-4, c=-5
f(x) = x2
a= 1, b=0, c=0
a= -1, b=0, c=0
f(x) = -x2 – 2x + 3a= -1, b=-2, c=3
Evaluating and Graphing Quadratic Functions
3. Find the Max/Min of a Quadratic Function
What is Concavity?
Coefficient Parameter “a”
Concavity Shape of Parabola
Vertex is Max or Min?
a > 0 Concave UpVertex is Minimum
a < 0Concave Down
Vertex is Maximum
A = 0Linear
No Concavity
No max/min
No vertex
Evaluating and Graphing Quadratic Functions
f(x) = x2 – 6x + 8a= , b= , c=
Ex 1 (I DO): What is the vertex of the quadratic function? Is it a relative max or min?
x = -b = = ___ 2a
f(x) = x2 – 6x + 8f( ) = ( )2 – 6( ) + 8
f( ) = ___
3. Find the Max/Min of a Quadratic Function
What is Concavity?
Is it a max or min? Is a>0 or a<0?
a=1, greater than 0, concave up, vertex is minimum
The minimum of -1 is where x = 3
vertex: ( __ , __ )
Evaluating and Graphing Quadratic Functions
f(x) = -x2 + 8x – 20a= , b= , c=
Ex 2 (WE DO): What is the vertex of the quadratic function? Is it a relative max or min?
x = -b = = ___ 2a
f(x) = -x2 + 8x – 20 f( ) = -( )2 + 8( ) – 20
f( ) = ___
3. Find the Max/Min of a Quadratic Function
What is Concavity?
Is it a max or min? Is a>0 or a<0?
a=-1, less than 0, concave down, vertex is maximum
The maximum of -4 is where x = 4
vertex: ( __ , __ )