```Quadratic Equations
Function of a different shape
There are many uses of parabolas in real-world
applications.
Functions
If you remember a relation is a
correspondence between two sets of numbers
called the domain and range. If each member
of the domain is assigned exactly one member
of the range, then the relation is a function.
A function can be represented as a list or a
table of ordered pairs, a graph in the
coordinate plane, or an equation in two
variables.
If you notice, the right side of the equation
y = 3x + 2 is a polynomial. Can you classify the
polynomial by degree?
A function of this form (y = mx + b) is called a
linear function. Note the graph is a straight line.
Y = 3x + 2
Now consider the
equation
y = x2 + 6x – 1
Classify the polynomial
on the right.
A function defined by
an equation of this form
y = ax2 + bx + c is
function.
Now we are going to
investigate this form.
Let’s try a little experiment with your graphing
calculator.
Graph the equation y = x2 on the coordinate plane.
Now graph y = 3x2 on the same coordinate plane.
How are the graphs the same?
How are they different?
Can you predict how the graph of y = ¼x2 will be
similar or different?
HINT: Graph y = x2 first,
next graph y = 3x2 to see
what differences or
similarities are present.
Now graph y = ¼x2 to see how
the shape of the graph
changes.
Type in y = x2 on the
graphing calculator.
Graph of y = x2
Now graph y = 3x2
Type in y = 3x2 on the
graphing calculator.
Graph of y = x2 and
y = 3x2 on the same
graph.
How are the graphs the same?
How are they different?
Can you predict what y = ¼x2 looks?
Type in y = x2, y = 3x2,
and y = ¼x2.
What differences do
you notice in the new
graph y = ¼x2.
Graphs of y = x2, y = 3x2,
and y = ¼x2.
functions.
Standard Form of a Quadratic Function
A quadratic function - is a function that can be
written in the form y = ax2 + bx + c, where a ≠ 0.
This form is called the standard form of a
Ex: y = 5x2
y = x2 + 7 y = x2 –x -3
The variable in a
squared (x2), so the
graph forms a curved
line called a parabola.
have the same shape.
The graph of y = x2
forms this U-shaped
graph called a parabola.
Axis of symmetry
You can fold a
parabola so that the
two sides match
exactly. This property
is called symmetry.
The fold or line that
divides the parabola
into two matching
halves is called the
axis of symmetry.
Minimum
point
The highest or lowest point of
a parabola is its vertex, which
is on the axis of symmetry.
If a > 0 in (a positive number)
y = ax2 + bx + c
Maximum
point
the parabola opens upward.
The vertex is the minimum
point or lowest point of the
graph.
If a < 0 in (a negative number)
y = -ax2 + bx + c
the parabola opens downward.
The vertex is the maximum
point or highest point of the
graph
Identifying a Vertex
Identify the vertex of each graph. Tell
whether it is a minimum or maximum.
On the left is the graph
of a parabola. Below are
examples of equations of
parabolas.
y = x2
x = y2
y = x2 + 2x + 3
Graphing y = ax2
You can use the fact that a
parabola is symmetric to
graph it quickly.
Remember our original
experiment when we
graphed y = x2 and y = 3x2
First, find the coordinates
of the vertex and several
points on either side of the
vertex. Then reflect the
points across the axis of
symmetry. For functions of
the form of y = ax2, the
vertex is at the origin.
Make a table for the
function y = x2 using x = 0,
1, 2, and 4.
Try this
Make a table for y = x2 using x = 0, 1, 2, 4.
Graph the points on the graph then reflect the
x-values to the other side of the graph.
Try another
Make a table of values and graph the quadratic
function f(x) = -2x2 using x = 0, 1, 2, 4
Remember these are
functions so we also
use the function
notation f(x)
Graphing y = ax2 + c
Graphing y = ax2 + c (y = 2x + 4)
The value of c, the constant term in a quadratic
function, translates the graph up or down.
2
Make a table and graph
y = 2x2
Make a table and graph
Y = 2x2 + 3
Try it
Graph y = x2 and y = x2 - 4
This time let’s
use the same
graph for both.
Real world application
height (feet)
The graph shows at 0
seconds the object is at
50 feet, after one second
fallen to 34 feet, and at
1½ seconds the object has
hit the ground
seconds
Seagull drops a clam to
break the shell so it
can eat it. The gull
drops the clam from 50
feet in the air.
Real World Problem
You can model the height of
an object moving under the
influence of gravity using a
object falls, its speed
continues to increase. You
can find the height of a
falling object using the
function h = -16t2 + c.
The height h is in feet, the
time t is in seconds, and the
initial height of the object c
is in feet.
Try one
Suppose a squirrel is in a tree 60 feet off the
ground. She drops an acorn. The function
h = -16t2 + 60 gives the height h of the acorn
in feet after t seconds. Make a table and graph
this function.
Graph each function
• y = -x2
• y = 2x2
• y = 3x2 – 6
• y = -½x2 + 3
Match the graph
Can you match these graphs with their
functions?
f(x) = x2 + 4
f(x) = -x2 + 2
Function
f(x) = ax2 + bx + c
So far we have investigated the graphs of y = ax2
and y = ax2 + c. In these functions c has always
been 0, which means the axis of symmetry has
always been the y-axis.
In the quadratic function y = ax2 + bx + c, the
value of b affects the position of the axis of
symmetry, moving it left or right.
In the next slide we are going to consider
functions in the form of y = ax2 + bx + c
Notice that both graphs have the same
y-intercept. This is because in both equations c = 0
y = 2x2 + 2x
Y = 2x2 + 4x
The axis of symmetry changes
with each change in the b
value.
Since the axis of symmetry is related to the
change in the b value, the equation of the axis
of symmetry is related to the ratio b/a
x = -b/2a
Let’s try one!
To find the y-value, first substitute a and b
into the equation x = -b/2a and solve to
find x. Then substitute x back into the
original equation to determine y.
Find the coordinates of the vertex and an
equation for the axis of symmetry. Then
graph the function.
y = x2 – 4x + 3
a = 1, b = -4, and c = 3
x = -b/2a
x = -(-4)/2(1) = 4/2 = 2
axis of symmetry: x = 2
If x = 2,
then y = x2 -4x + 3 =
y = 22 - 4(2) + 3 = -1
Use the equation for the
axis of symmetry.
x = -b/2a
Substitute the x-value into
the original equation and
solve for y. the vertex is
(2, -1)
Now make a table.
Since the vertex of the axis of symmetry is (2, -1)
and we know the parabola turns upward (a > 0), we
can use values on both sides of (2, -1).
X
Y
-1
8
0
3
1
0
2
-1
3
0
4
3
5
8
Now graph your
points and draw a
curved line.
Try this one.
y=
-x2
+4
a =
,b =
c =
Find x = -b/2a
x =
x
y
Substitute the xvalue into the
equation
y = -x2 + 4
Now use your
vertex as the
middle of your
table.
Solve problems
Graph each function. Label the axis of
symmetry and the vertex.
1) y = x2 + 4x + 3
2) y = 2x2 – 6x
3) y =x2 + 4x – 4
4) y = 2x2 + 3x + 1
Real world problem
In professional fireworks displays, aerial fireworks carry “stars”
upward, ignite them, and project them into the air.
The equation h = -16t2 + 72t + 520 gives the star’s height h in
feet at time t in seconds. Since the coefficient of t2 is negative,
the curve opens downward, and the vertex is the maximum point.
Find the t-coordinate of the vertex
x = -b/2a = -72/2(-16) = 2.25
After 2.25 seconds, the star will be at its greatest
height.
Find the h-coordinate of the vertex.
h = -16(2.25)2 + 72(2.25) + 520 = 601
The maximum height of the star will be 601 feet.
TRY THIS
The shape of the Gateway Arch in St. Louis,
Missouri, is a catenary curve that resembles a
parabola. The equation
h = -0.00635x2 + 4.0005x – 0.07875
represents the parabola, where h is the height in
feet and x is the distance from one base in feet.
What is the equation of the axis of symmetry?
What is the maximum height of the arch?
Formula
In our earlier lesson, you
equations by factoring.
Another method, which
equation, is to use the
seen left.
Here values of a, b, and c
are substituted into the
formula to determine x.
Be sure to write a
standard form before
formula.
Solve:
x2 + 6 = 5x
x2 -5x + 6 = 0
Solve:
x2 + 2 = -3x
VERTICAL MOTION
FORMULA
h = -16t2 + vt + c
The initial upward
velocity is v, and the
starting height is c
formula to solve real-world
problems.
Suppose a football player
kicks a ball and gives it an
initial velocity of 47ft/s.
The starting height of the
football is 3 ft. If no one
catches the football how
long will it be in the air?
Using the vertical motion
formula and the information
given, the formula
h = -16t2 + vt + c
represents this illustration.
You must decide whether a solution makes sense in the real-world situation.
For example, a negative value for time is not a reasonable solution.
Use the vertical motion formula h = -16t2 + vt + c
1) A child tosses a ball upward with a starting
velocity of 10 ft/s from a height of 3 ft.
a. Substitute the values into the vertical motion
formula. Let h = 0
b. Solve. If it is not caught, how long will the ball be
in the air? Round to the nearest tenth of a second.
2) A soccer ball is kicked with a starting velocity of
50 ft/s from a starting height of 3.5 ft.
a. Substitute the values into the vertical motion
formula. Let h = 0
b. Solve. If no one touches the ball, how long will
the ball be in the air?
TRY THIS
The function below models the United States
population P in millions since 1900, where t is
the number of years after 1900.
P = 0.0089t2 + 1.1149t + 78.4491
a. Use the function to estimate the US
population the year I graduated from high
school.
b. Estimate the US population in 2025.
c. Estimate the US population in 2050.
Try Another
A carnival game involves striking a lever that
forces a weight up a tube to strike a bell
which will win you a prize. If the weight
reaches 20 feet and strikes the bell, you win.
The equation
h = -16t2 + 32t + 3
gives the height h of the weight if the initial
velocity v is 32 ft/s.
Find the maximum height of the weight.
Will the contestant win a prize?
One More
The Sky Concert in Peoria, Illinois, is a 4th of
July fireworks display set to music. If a
rocket (firework) is launched with an initial
velocity of 39.2 m/s at a height of 1.6 m above
the ground, the equation,
h = -4.9t2 + 39.2t + 1.6
represents the rockets height h in meters
after t seconds. The rocket will explode at
approximately the highest point.
At what height will the rocket explode?
Review
If a quadratic equation is written in the
form ax2 + bx + c = 0, the solutions can be
In the quadratic equation, the expression
under the radical sign, b2 – 4ac, is called
the discriminant.
1) If b2 – 4ac is a negative number, the
square root cannot be found as a real
number. There are no real-number
solutions.
2) If b2 – 4ac equals 0, there is only one
solution of the equation.
3) If b2 – 4ac is a positive number, there
are two solutions of the equation. The
graph of the quadratic intersects the x-axis
twice.
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