In other words, the Intermediate Value Theorem tells us that when a polynomial function changes from a negative value to a positive value, the function must cross the
axis.
[link] shows that there is a zero between
and
Using the Intermediate Value Theorem to show there exists a zero.
Intermediate value theorem
Let
be a polynomial function. The
Intermediate Value Theorem states that if
and
have opposite signs, then there exists at least one value
between
and
for which
Using the intermediate value theorem
Show that the function
has at least two real zeros between
and
As a start, evaluate
at the integer values
See
[link] .
1
2
3
4
5
0
–3
2
We see that one zero occurs at
Also, since
is negative and
is positive, by the Intermediate Value Theorem, there must be at least one real zero between 3 and 4.
We have shown that there are at least two real zeros between
and
Now that we know how to find zeros of polynomial functions, we can use them to write formulas based on graphs. Because a
polynomial function written in factored form will have an
intercept where each factor is equal to zero, we can form a function that will pass through a set of
intercepts by introducing a corresponding set of factors.
Factored form of polynomials
If a polynomial of lowest degree
has horizontal intercepts at
then the polynomial can be written in the factored form:
where the powers
on each factor can be determined by the behavior of the graph at the corresponding intercept, and the stretch factor
can be determined given a value of the function other than the
x -intercept.
Given a graph of a polynomial function, write a formula for the function.
Identify the
x -intercepts of the graph to find the factors of the polynomial.
Examine the behavior of the graph at the
x -intercepts to determine the multiplicity of each factor.
Find the polynomial of least degree containing all the factors found in the previous step.
Use any other point on the graph (the
y -intercept may be easiest) to determine the stretch factor.
Writing a formula for a polynomial function from the graph
Write a formula for the polynomial function shown in
[link] .
This graph has three
intercepts:
and
The
intercept is located at
At
and
the graph passes through the axis linearly, suggesting the corresponding factors of the polynomial will be linear. At
the graph bounces at the intercept, suggesting the corresponding factor of the polynomial will be second degree (quadratic). Together, this gives us
To determine the stretch factor, we utilize another point on the graph. We will use the
intercept
to solve for
The graphed polynomial appears to represent the function
A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance
Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes
Adjei
please, I'm a physics student and I need help in physics
Adjanou
chemistry could also be understood like the sexual attraction/repulsion of the male and female elements. the reaction varies depending on the energy differences of each given gender. + masculine -female.
Pedro
A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity
2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s
can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
Joseph
"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?
