Quadratic Functions
Definition of a Quadratic Function
A

quadratic function is defined as:
f(x) = ax² + bx + c where a, b and c are real
numbers and a ≠0.
 Graphs
of quadratic functions are
parabolas (u-shaped).
Vertex of a parabola

The vertex (x, y) of a parabola is the turning point of a
parabola.

The vertex (x, y) is at a minimum for a regular U-shaped
parabola. The y-value of the vertex is the absolute
minimum value for the function.

The vertex (x, y) is at a maximum for an “upside down” U
shaped parabola. The y-value of the vertex is the
absolute maximum value for the function.
Vertex/Maximum
Vertex/Minimum
Axis of Symmetry

The axis of symmetry (A.O.S.) divides the
parabola into 2 mirror image halves. The
equation for the AOS is the equation for a
vertical line, written as “x =___”.

The x-coordinate of the vertex is the value that
you use for the AOS.
Axis of Symmetry – Vertical Line ( x = # )
Reflection Points

Reflection points are found at equal distances
from the axis of symmetry, or vertex, and always
share the same y-values.

To graph a parabola by hand, find the vertex
and then choose x values at equal distances on
either side of the vertex to evaluate.
Reflection Points
Vertex/Minimum
Graphing parabolas by hand

To find the x-coordinate of the vertex: Use
the formula b
2a

Then, substitute the x-coordinate into the
equation and evaluate f(x) or y.
Graph: f(x) = 2x²-8x+1

1) Find the x-coordinate of the vertex. Note: a
= 2, b = -8 and c = 1
b (8) 8
x

 2
2a 2(2) 4

2) Evaluate the function at x = 2 (find f(2))
f (2)  2(2) 2  8(2)  1
 2(4)  16  1
 8  16  1
 7

The coordinates of the vertex are (2, -7)

Create a table of values with the
vertex in the middle of the table.
Then, choose x-values on either
side of the vertex to graph the
parabola.
x
y
0
1

Evaluate f(0), f(1), f(3) and f(4).

Then graph the parabola and
draw the axis of symmetry.




What is the equation for the axis
of symmetry?
Is the vertex at a maximum or
minimum?
What is the max/min y-value?
What is the domain/range?
2
3
4
-7
Task 1: Given two reflection points,
find the equation for the line of
symmetry
 Example:
Find the equation for the line of symmetry given
two reflection points: (3, -2) and (-8,-2)
Example:

Find the equation for the line of symmetry given
two reflection points:
(3, -2) and (-8,-2)
Reflection Points
will have same
Y-values.
(-8, -2)
5.5
5.5
(3, -2)
11
Find the mid point of the segment joining the two
reflection points.
3 + -8 = -5
A.O.S.: x = -2.5
2
2
Given two reflection points, find
equation for line of symmetry
 1)
(3, 7); (13, 7)
 2)
(5, 9); (12,9)
 3)
(-2, -3); (8, -3)
 4)
(6.4, 5.2); (8.6, 5.2)
Given two reflection points, find line
of symmetry-Answers
 1)
(3, 7); (13, 7)
x=8
 2)
(5, 9); (12,9)
x= 8.5
 3)
(-2, -3); (8, -3)
x=3
 4)
(6.4, 5.2); (8.6, 5.2) x=7.5
Task 2: Given the vertex (V) and a
point on parabola, find another point on
the parabola. (Find a reflection point)
Example:
V:( -2,4) P(1.5,-8) Find another point
on the parabola.
 Example:
V:( -2,4) P(1.5,-8)
(-2, 4)
The Reflection Point will
be at ( - 5.5, - 8 )
( ? , - 8)
(1.5, - 8)
3.5
? = - 5.5
3.5
Reflection Points
Will be at same height
Same y value
Given vertex and one point, find
another point on same parabola.
 5)
V: (1, 1); P (3, 4)
 6)
V: (-5, 6); P (0, 5)
 7)
V: (4, -6); P (-3.2, 11)
 8)
V: (-3, -4); P (5, 6)
Given vertex and one point, find
another point on parabola-Answers
 5)
V: (1, 1); P (3, 4)
P’ = (-1, 4)
 6)
V: (-5, 6); P (0, 5)
P’ = (-10, 5)
V: (4, -6); P (-3.2, 11)
11)
P’ = (11.2,
V: (-3, -4); P (5, 6)
P’=( -11, 6)
 7)
 8)
Algebraically:
V:( -2,4) P(1.5,-8)
 The distance between the given xcoordinates is:
d  2  1.5
d  3.5


The point we are looking for is to the left
of the vertex, so subtract from -2.
(-2 - 3.5 = -5.5)
The reflection point is (-5.5, -8)
Task: Given the vertex (V) and a point
on parabola, find another point

Basically, find the reflection point. To do this:
 1) Draw a quick sketch. The y-coordinate of the
point you are looking for is the same as the given
point.
To find the x-coordinate of the reflection point:
 2) Determine the distance between the xcoordinates of V and P d  a  b

3) If reflection point is to left of vertex, subtract
from the x-coordinate of V. If reflection point is to
right of vertex, add from the x-coordinate of V.
Task 3: Write a quadratic equation for
a parabola with vertex at the origin
passing through a given point.
Example:
(-2, 8)
 Example:
( -2,8)
y  ax
2
8  a ( 2)
8  4a
2a
y  2x
2
(-2, 8)
(0, 0)
2
What is the equation for this parabola if it were reflected over the x-axis?
Task: Write a quadratic equation for a parabola
with vertex at the origin passing through a given
point, then find equation for the reflection of the
parabola.

Try!
1) P:(1,1)
2) P:(1,-4)
3) P: (2, -4)
4) P: (-3, -45)
Quadratic Terms
2
Quadratic Function-a function in the form f ( x)  ax  bx  c
where a, b, and c are real numbers and a  0
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)
Graphing Quadratic Functions
on the Calculator
 Graph
in y1 = screen and find a good
window (you must be able to see the
vertex)
 To find the coordinates of the vertex:




2nd Calc Maximum or Minimum
Move cursor to left side of vertex, Enter
Move cursor to right side of vertex, Enter
Enter
Graphing Quadratic Functions
on the Calculator cont.

To find reflection points:



Go to Table of Values and find vertex
Look in table on either side of vertex for sets of
reflection points
To find the x-intercepts:


Graph the x-axis in Y2 (y = 0)
Note how many times it crosses x-axis
• 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.
• Otherwise, there are two x-intercepts.

2nd Calc Intersect to find the places where Y1 and Y2
intersect
• Do not move cursor, just push Enter 3 times.
Problem Solving
with Quadratic Functions

Quadratic Functions model the path of a falling object.

After t seconds, the height of an object with an initial upward velocity
of v0 meters per second and an initial height of h0 meters is:
h t   4.9t  v0t  h0 meters
2

If h0 is measure in feet and v0 in feet per second, then the height is:
h t   16t  v0t  h0 feet
2

In each equation, the force of gravity is represented by a squared term in
the negative direction. As the time increases, the t2 term overpowers the t
term, and the object falls.
Example:
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…
h t   16t 2  v0t  h0 feet
v0  6
h0  0
h t   16t 2  6t
h  0.2   16  0.2   6  0.2 
2
 16 .04   1.2
 .64  1.2
 .56 feet
Another Example:
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…
h t   16t 2  v0t  h0 feet
v0  96
h0  200
b
96
96
t


3
2a 2(16) 32
After 3 seconds, the ball reaches its
maximum height.
h t   16t  96t  200
2
h  3  16  3  96  3  200
2
 16  9   288  200
 144  488
 344 feet
Calculator Example:
Rob’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?
What will the height of the football be after 4.75 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?
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.
h(t) = -16t2 + 80t + 5.5
Now write the equation as you would enter it on the calculator.
y = -16x2 + 80x + 5.5
What do x and y represent?
x = time (seconds)
y = height (feet)
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.
Graph the function. Note your window size.
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.

What will the height of the football be after 1.625 seconds?
The height of the football will be 93.25 feet.

What will the height of the football be after 4.75 seconds?
The height of the football will be 24.5 feet.
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.

After how many seconds will the ball reach its maximum height?
After 2.5 seconds…

What will this maximum height be? 105.5 feet
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.
How long does it take the ball to hit the ground?
It takes 5.07 seconds to
hit the ground
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.

How long does it take the ball to initially rise to 60 feet?
It takes .81
seconds to initially
rise to 60 feet.

How long does it take the ball to get to 60 feet on the way down?
It takes 4.19 seconds to get to
60 feet on the way down.
Another Calculator Example:
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 reach its maximum height?
What will this maximum height be?
After how many seconds will the ball hit the ground?
How high is the ball after 2 seconds?
Because this problem uses meters as it units, we must use the equation that is
written for this…
2
h t   4.9t  v0t  h0 meters
v0  30
h0  50
h t   4.9t 2  30t  50