Quadratic Functions. Definition of a Quadratic Function A quadratic function is defined as: f(x) =...

Quadratic FunctionsQuadratic Functions

Definition of a Quadratic FunctionDefinition of a Quadratic Function

A quadratic function is defined as:A quadratic function is defined as:

f(x) = axf(x) = ax² + bx + c where a, b and c are real ² + bx + c where a, b and c are real numbers and a ≠0.numbers and a ≠0.

Graphs of quadratic functions are Graphs of quadratic functions are parabolas (u-shaped).parabolas (u-shaped).

Vertex of a parabolaVertex of a parabola The vertex (x, y) of a parabola is the turning point of a The vertex (x, y) of a parabola is the turning point of a

parabola. parabola.

The vertex (x, y) is at a The vertex (x, y) is at a minimumminimum for a regular U-shaped for a regular U-shaped parabola. The y-value of the vertex is the parabola. The y-value of the vertex is the absoluteabsolute minimum value for the function.minimum value for the function.

The vertex (x, y) is at a The vertex (x, y) is at a maximummaximum for an “upside down” U for an “upside down” U shaped parabola. The y-value of the vertex is the shaped parabola. The y-value of the vertex is the absoluteabsolute maximum value for the function. maximum value for the function.

Vertex/Minimum

Vertex/Maximum

Axis of SymmetryAxis of Symmetry The axis of symmetry (A.O.S.) divides the The axis of symmetry (A.O.S.) divides the

parabola into 2 mirror image halves. The parabola into 2 mirror image halves. The equation for the AOS is the equation for a equation for the AOS is the equation for a vertical line, written as “x =___”.vertical line, written as “x =___”.

The x-coordinate of the vertex is the value that The x-coordinate of the vertex is the value that you use for the AOS.you use for the AOS.

Axis of Symmetry – Vertical Line ( x = # )

Reflection PointsReflection Points Reflection points are found at equal distances Reflection points are found at equal distances

from the axis of symmetry, or vertex, and always from the axis of symmetry, or vertex, and always share the same y-values.share the same y-values.

To graph a parabola by hand, find the vertex To graph a parabola by hand, find the vertex and then choose x values at equal distances on and then choose x values at equal distances on either side of the vertex to evaluate.either side of the vertex to evaluate.

Vertex/Minimum

Reflection Points

Graphing parabolas by handGraphing parabolas by hand

To find the x-coordinate of the vertex: Use To find the x-coordinate of the vertex: Use the formula the formula

Then, substitute the x-coordinate into the Then, substitute the x-coordinate into the equation and evaluate f(x) or y.equation and evaluate f(x) or y.

2

b

a

Graph: f(x) = 2xGraph: f(x) = 2x²-8x+1²-8x+1 1) Find the x-coordinate of the vertex. Note: a 1) Find the x-coordinate of the vertex. Note: a

= 2, b = -8 and c = 1= 2, b = -8 and c = 1

2) Evaluate the function at x = 2 (find f(2))2) Evaluate the function at x = 2 (find f(2))

The coordinates of the vertex are (2, -7)The coordinates of the vertex are (2, -7)

( 8) 82

2 2(2) 4

bx

a

2(2) 2(2) 8(2) 1

2(4) 16 1

8 16 1

7

f

Create a table of values with the Create a table of values with the vertex in the middle of the table. vertex in the middle of the table. Then, choose x-values on either Then, choose x-values on either side of the vertex to graph the side of the vertex to graph the parabola.parabola.

Evaluate f(0), f(1), f(3) and f(4). Evaluate f(0), f(1), f(3) and f(4).

Then graph the parabola and Then graph the parabola and draw the axis of symmetry. draw the axis of symmetry.

What is the equation for the axis What is the equation for the axis of symmetry?of symmetry?

Is the vertex at a maximum or Is the vertex at a maximum or minimum? minimum?

What is the max/min y-value?What is the max/min y-value? What is the domain/range?What is the domain/range?

xx yy

00

11

22 -7-7

33

44

Task 1: Given two reflection points, Task 1: Given two reflection points, find the equation for the line of find the equation for the line of

symmetrysymmetry

Example:Example: Find the equation for the line of symmetry given two reflection points: (3, -2) and (-8,-2)

Example:Example: Find the equation for the line of symmetry given Find the equation for the line of symmetry given

two reflection points:two reflection points:

(3, -2) and (-8,-2)(3, -2) and (-8,-2)

Find the mid point of the segment joining the two Find the mid point of the segment joining the two reflection points. reflection points.

3 + -8 3 + -8 = = -5 -5 A.O.S.: x = -2.5 A.O.S.: x = -2.5

22 2 2

Reflection Points

will have same

Y-values.(3, -2)(-8, -2) 5.5 5.5

11

Given two reflection points, find Given two reflection points, find equation for line of symmetryequation for line of symmetry

1) (3, 7); (13, 7) 1) (3, 7); (13, 7)

2) (5, 9); (12,9)2) (5, 9); (12,9)

3) (-2, -3); (8, -3)3) (-2, -3); (8, -3)

4) (6.4, 5.2); (8.6, 5.2)4) (6.4, 5.2); (8.6, 5.2)

Given two reflection points, find line Given two reflection points, find line of symmetry-Answersof symmetry-Answers

1) (3, 7); (13, 7) 1) (3, 7); (13, 7) x = 8x = 8

2) (5, 9); (12,9) 2) (5, 9); (12,9) x= 8.5x= 8.5

3) (-2, -3); (8, -3) 3) (-2, -3); (8, -3) x=3x=3

4) (6.4, 5.2); (8.6, 5.2) x=7.54) (6.4, 5.2); (8.6, 5.2) x=7.5

Task 2: Given the vertex (V) and a Task 2: Given the vertex (V) and a point on parabola, find another point on point on parabola, find another point on the parabola. (Find a reflection point)the parabola. (Find a reflection point)

Example:

V:( -2,4) P(1.5,-8) Find another point on the parabola.

Example:Example:

V:( -2,4) P(1.5,-8)V:( -2,4) P(1.5,-8)

Reflection Points

Will be at same height

Same y value

(-2, 4)

(1.5, - 8)( ? , - 8)

3.53.5

? = - 5.5

The Reflection Point will be at ( - 5.5, - 8 )

Given vertex and one point, find Given vertex and one point, find another point on same parabola.another point on same parabola.

5) V: (1, 1); P (3, 4)5) V: (1, 1); P (3, 4)

6) V: (-5, 6); P (0, 5)6) V: (-5, 6); P (0, 5)

7) V: (4, -6); P (-3.2, 11)7) V: (4, -6); P (-3.2, 11)

8) V: (-3, -4); P (5, 6)8) V: (-3, -4); P (5, 6)

Given vertex and one point, find Given vertex and one point, find another point on parabola-Answersanother point on parabola-Answers 5) V: (1, 1); P (3, 4) P’ = (-1, 4)5) V: (1, 1); P (3, 4) P’ = (-1, 4)

6) V: (-5, 6); P (0, 5) P’ = (-10, 5)6) V: (-5, 6); P (0, 5) P’ = (-10, 5)

7) V: (4, -6); P (-3.2, 11) P’ = (11.2, 7) V: (4, -6); P (-3.2, 11) P’ = (11.2, 11)11)

8) V: (-3, -4); P (5, 6) P’=( -11, 6)8) V: (-3, -4); P (5, 6) P’=( -11, 6)

Algebraically:Algebraically:

V:( -2,4) P(1.5,-8)V:( -2,4) P(1.5,-8) The distance between the given x-The distance between the given x-

coordinates is:coordinates is:

The point we are looking for is to the left The point we are looking for is to the left of the vertex, so of the vertex, so subtractsubtract from -2. from -2.

(-2 - 3.5 = -5.5)(-2 - 3.5 = -5.5)

The reflection point is (-5.5, -8)The reflection point is (-5.5, -8)

2 1.5

3.5

d

d

Task: Given the vertex (V) and a point Task: Given the vertex (V) and a point on parabola, find another pointon parabola, find another point

Basically, find the reflection point. To do this:Basically, find the reflection point. To do this: 1) Draw a quick sketch. The y-coordinate of the 1) Draw a quick sketch. The y-coordinate of the

point you are looking for is the point you are looking for is the samesame as the given as the given point.point.

To find the x-coordinate of the reflection point: To find the x-coordinate of the reflection point: 2) Determine the distance between the x-2) Determine the distance between the x-

coordinates of V and Pcoordinates of V and P 3) If reflection point is to left of vertex, subtract 3) If reflection point is to left of vertex, subtract

from the x-coordinate of V. If reflection point is to from the x-coordinate of V. If reflection point is to right of vertex, add from the x-coordinate of V. right of vertex, add from the x-coordinate of V.

d a b

Task 3: Write a quadratic equation for Task 3: Write a quadratic equation for a parabola with vertex at the origin a parabola with vertex at the origin

passing through a given point.passing through a given point.Example:

(-2, 8)

Example:Example:

( -2,8)( -2,8)

(0, 0)

(-2, 8)

2

2

2

8 ( 2)

8 4

2

2

y ax

a

a

a

y x

What is the equation for this parabola if it were reflected over the x-axis?

Task: Write a quadratic equation for a parabola Task: Write a quadratic equation for a parabola with vertex at the origin passing through a given with vertex at the origin passing through a given point, then find equation for the reflection of the point, then find equation for the reflection of the

parabola.parabola. Try!Try!

1) P:(1,1)1) P:(1,1)

2) P:(1,-4)2) P:(1,-4)

3) P: (2, -4)3) P: (2, -4)

4) P: (-3, -45)4) P: (-3, -45)

Quadratic Function-a function in the form 2( )f x ax bx c where a, b, and c are real numbers and 0a

Parabola-the U-shaped (or upside down U) curve that EVERY quadratic makes when graphed

Vertex-The lowest or highest point on a parabola (always given as an ordered pair)

Minimum/Maximum-The lowest or highest y-value (always the y-value of the vertex) given in the form y =

Axis of Symmetry-the vertical line through the vertex that cuts the parabola into two mirror images (always the x-value of the vertex) given in

the form x =

Image/Reflection Points-points on the parabola that are equidistant from the line of symmetry (given as a coordinate and always have the same y-value)

Quadratic Terms

Graphing Quadratic FunctionsGraphing Quadratic Functionson the Calculatoron the Calculator

Graph in y1 = screen and find a good Graph in y1 = screen and find a good window (you must be able to see the window (you must be able to see the vertex)vertex)

To find the coordinates of the vertex:To find the coordinates of the vertex: 22ndnd Calc Maximum or Minimum Calc Maximum or Minimum Move cursor to left side of vertex, Move cursor to left side of vertex, EnterEnter Move cursor to right side of vertex, Move cursor to right side of vertex, EnterEnter EnterEnter

Graphing Quadratic FunctionsGraphing Quadratic Functionson the Calculator cont.on the Calculator cont.

To find reflection points:To find reflection points: Go to Table of Values and find vertexGo to Table of Values and find vertex Look in table on either side of vertex for sets of Look in table on either side of vertex for sets of

reflection pointsreflection points To find the x-intercepts:To find the x-intercepts:

Graph the x-axis in Y2 (y = 0)Graph the x-axis in Y2 (y = 0) Note how many times it crosses x-axisNote how many times it crosses x-axis

• If it does not touch the x-axis, there are no x-intercepts.If it does not touch the x-axis, there are no x-intercepts.• If the vertex (x,y) is on the x-axis, there is one x-intercept.If the vertex (x,y) is on the x-axis, there is one x-intercept.• Otherwise, there are two x-intercepts.Otherwise, there are two x-intercepts.

22ndnd Calc Intersect to find the places where Y1 and Y2 Calc Intersect to find the places where Y1 and Y2 intersectintersect

• Do not move cursor, just push Enter 3 times.Do not move cursor, just push Enter 3 times.

Problem Solving Problem Solving with Quadratic Functionswith Quadratic Functions

Quadratic Functions model the path of a falling object.Quadratic Functions model the path of a falling object.

After After tt seconds, the height of an object with an initial upward velocity seconds, the height of an object with an initial upward velocity of of vv00 meters per second and an initial height of meters per second and an initial height of hh00 meters is: meters is:

If If hh0 0 is measure in feet and is measure in feet and vv00 in feet per second, then the height is: in feet per second, then the height is:

20 04.9h t t v t h meters

20 016h t t v t h feet

In each equation, the force of gravity is represented by a squared term in In each equation, the force of gravity is represented by a squared term in the negative direction. As the time increases, the the negative direction. As the time increases, the tt22 term overpowers the term overpowers the tt term, and the object falls.term, and the object falls.

Example:Example:


0 6v

A flea jumps straight up from the ground with an initial upward velocity of 6 feet per second. What will the height of the flea be after 0.2 seconds?

Because this problem uses feet as it units, we must use the equation that is written for this…

0 0h 216 6h t t t

20.2 16 0.2 6 0.2

16 .04 1.2

.64 1.2

.56

h

feet

Another Example:Another Example:


0 96v

A ball is thrown directly upward from an initial height of 200 feet with and initial velocity of 96 feet per second. After how many seconds will the ball

reach its maximum height? And, what is the maximum height?

Because this problem uses feet as it units, we must use the equation that is written for this…

0 200h 216 96 200h t t t

23 16 3 96 3 200

16 9 288 200

144 488

344

h

feet

96 963

2 2( 16) 32

bt

a

After 3 seconds, the ball reaches its maximum height.

Calculator Example:Calculator Example:Rob’s FootballRob’s Football

Rob is playing football and he throws the ball from an initial height of 5.5 feet at an initial velocity of 80 feet per second.

Write an equation that represents the height of the rocket at any time.

Now write the equation as you would enter it on the calculator.

What do x and y represent?

What will the height of the football be after 1.625 seconds?


After how many seconds will the ball reach its maximum height?

What will this maximum height be?

How long does it take the ball to hit the ground?

How long does it take the ball to initially rise to 60 feet?

How long does it take the ball to get to 60 feet on the way down?

Write an equation that represents the height of the rocket at any time.

Now write the equation as you would enter it on the calculator.

What do x and y represent?


h(t) = -16t2 + 80t + 5.5

y = -16x2 + 80x + 5.5

x = time (seconds)

y = height (feet)

Graph the function. Note your window size.



The height of the football will be 93.25 feet.


The height of the football will be 24.5 feet.





After 2.5 seconds…

105.5 feet

How long does it take the ball to hit the ground?


It takes 5.07 seconds to hit the ground

How long does it take the ball to initially rise to 60 feet?

How long does it take the ball to get to 60 feet on the way down?


It takes .81 seconds to initially

rise to 60 feet.

It takes 4.19 seconds to get to 60 feet on the way down.

Another Calculator Example:Another Calculator Example:

Because this problem uses meters as it units, we must use the equation that is written for this…

A ball is thrown directly upward from an initial height of 50 meters with an initial velocity of 30 meters per second.



After how many seconds will the ball hit the ground?

How high is the ball after 2 seconds?

20 04.9h t t v t h meters

0 30v

0 50h 24.9 30 50h t t t

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