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Example Questions
Example Question #1 : Graphs Of Polynomial Functions
For what values of will the given polynomial pass through the x-axis if plotted in Cartesian coordinates?
None of the other answers
One can remember that if the multiplicity of a zero is odd then it passes through the x-axis and if it's even then it 'bounces' off the x-axis. You can think about this analytically as well. What happens when we plug a number into our function just slightly above or below a zero with an even multiplicity? You find that the sign is always positive. Whereas a zero with an odd multiplicity will yield a positive on one side and a negative on the other. For zeros with odd multiplicity this alters the sign of our output and the function passes through the x-axis. Whereas the zero with even multiplicity will output a number with the same sign just above and below its zero, thus it 'bounces' off the x-axis.
Example Question #1 : Graphs Of Polynomial Functions
For this particular question we are restricting the domain of both to nonnegative values, or the interval
.
Let and
.
For what values of is
?
The cubic function will increase more quickly than the quadratic, so the quadratic function must have a head start. At , both functions evaluate to 8. After than point, the cubic function will increase more quickly.
The domain was restricted to nonnegative values, so this interval is our only answer.
Example Question #1 : Graph A Polynomial Function
Which of the following is an accurate graph of ?
is a parabola, because of the general
structure. The parabola opens downward because
.
Solving tells the x-value of the x-axis intercept;
The resulting x-axis intercept is: .
Setting tells the y-value of the y-axis intercept;
The resulting y-axis intercept is:
Example Question #2 : Graph A Polynomial Function
Give the -intercept of the graph of the function
Round to the nearest tenth, if applicable.
The graph has no -interceptx
The -intercept is
, where
:
The -intercept is
.
Example Question #1 : Graph Polynomial Functions, Identify Zeros, Factor, And Identify End Behavior.: Css.Math.Content.Hsf If.C.7c
Graph the following function and identify the zeros.
This question tests one's ability to graph a polynomial function.
For the purpose of Common Core Standards, "graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior" falls within the Cluster C of "Analyze Functions Using Different Representations" concept (CCSS.MATH.CONTENT.HSF-IF.C.7).
Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.
Step 1: Use algebraic technique to factor the function.
Separating the function into two parts...
Factoring a negative one from the second set results in...
Factoring out from the first set results in...
The new factored form of the function is,
.
Now, recognize that the first binomial is a perfect square for which the following formula can be used
since
thus the simplified, factored form is,
.
Step 2: Identify the roots of the function.
To find the roots of a function set its factored form equal to zero and solve for the possible x values.
Step 3: Create a table of pairs.
The values in the table are found by substituting in the x values into the function as follows.
Step 4: Plot the points on a coordinate grid and connect them with a smooth curve.
Example Question #4 : Graph A Polynomial Function
Graph the function and identify the roots.
This question tests one's ability to graph a polynomial function.
For the purpose of Common Core Standards, "graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior" falls within the Cluster C of "Analyze Functions Using Different Representations" concept (CCSS.MATH.CONTENT.HSF-IF.C.7).
Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.
Step 1: Use algebraic technique to factor the function.
Recognize that the binomial is a perfect square for which the following formula can be used
since
thus the simplified, factored form is,
.
Step 2: Identify the roots of the function.
To find the roots of a function set its factored form equal to zero and solve for the possible x values.
Step 3: Create a table of pairs.
The values in the table are found by substituting in the x values into the function as follows.
Step 4: Plot the points on a coordinate grid and connect them with a smooth curve.
Example Question #5 : Graph A Polynomial Function
Graph the function and identify its roots.
This question tests one's ability to graph a polynomial function.
For the purpose of Common Core Standards, "graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior" falls within the Cluster C of "Analyze Functions Using Different Representations" concept (CCSS.MATH.CONTENT.HSF-IF.C.7).
Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.
Step 1: Use algebraic technique to factor the function.
Recognize that the binomial is a perfect square for which the following formula can be used
since
thus the simplified, factored form is,
.
Step 2: Identify the roots of the function.
To find the roots of a function set its factored form equal to zero and solve for the possible x values.
Step 3: Create a table of pairs.
The values in the table are found by substituting in the x values into the function as follows.
Step 4: Plot the points on a coordinate grid and connect them with a smooth curve.
Example Question #6 : Graph A Polynomial Function
Graph the function and identify its roots.
This question tests one's ability to graph a polynomial function.
For the purpose of Common Core Standards, "graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior" falls within the Cluster C of "Analyze Functions Using Different Representations" concept (CCSS.MATH.CONTENT.HSF-IF.C.7).
Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.
Step 1: Use algebraic technique to factor the function.
Recognize that the binomial is a perfect square for which the following formula can be used
since
thus the simplified, factored form is,
.
Step 2: Identify the roots of the function.
To find the roots of a function set its factored form equal to zero and solve for the possible x values.
Step 3: Create a table of pairs.
The values in the table are found by substituting in the x values into the function as follows.
Step 4: Plot the points on a coordinate grid and connect them with a smooth curve.
Example Question #1 : Write The Equation Of A Polynomial Function Based On Its Graph
Which could be the equation for this graph?
This graph has zeros at 3, -2, and -4.5. This means that ,
, and
. That last root is easier to work with if we consider it as
and simplify it to
. Also, this is a negative polynomial, because it is decreasing, increasing, decreasing and not the other way around.
Our equation results from multiplying , which results in
.
Example Question #2 : Write The Equation Of A Polynomial Function Based On Its Graph
Write the quadratic function for the graph:
Because there are no x-intercepts, use the form , where vertex
is
, so
,
, which gives
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