Precalculus : Graphs of Polynomial Functions

Study concepts, example questions & explanations for Precalculus

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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? 

Possible Answers:

None of the other answers

Correct answer:

Explanation:

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 #2 : 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 ?

Possible Answers:

Correct answer:

Explanation:

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 #3 : Graphs Of Polynomial Functions

Graph .

Possible Answers:

Varsity11

Varsity2

Varsity1

Varsity10

Varsity12

Correct answer:

Varsity1

Explanation:

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 #4 : Graphs Of Polynomial Functions

Which could be the equation for this graph?

Polynomial

Possible Answers:

Correct answer:

Explanation:

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 #5 : Graphs Of Polynomial Functions

Write the quadratic function for the graph:

Varsity8

Possible Answers:

Correct answer:

Explanation:

Because there are no x-intercepts, use the form , where vertex  is , so , , which gives

          

          

          

          

      

Example Question #6 : Graphs Of Polynomial Functions

Write the quadratic function for the graph:

Varsity9

Possible Answers:

Correct answer:

Explanation:

Method 1:

The x-intercepts are .  These values would be obtained if the original quadratic were factored, or reverse-FOILed and the factors were set equal to zero.

For , .  For , .  These equations determine the resulting factors and the resulting function; .

Multiplying the factors and simplifying, 

.

Answer: .

 

Method 2:

Use the form , where is the vertex. 

 is , so , .

Answer:

                       

                       

                       

 

Example Question #7 : Graphs Of Polynomial Functions

Write the equation for the polynomial in this graph:

Graph 1 write funct

Possible Answers:

Correct answer:

Explanation:

The zeros for this polynomial are .

This means that the factors are equal to zero when these values are plugged in for x.

multiply both sides by 2

so one factor is

 

multiply both sides by 3

so one factor is

 

so one factor is

Multiply these three factors:

Example Question #8 : Graphs Of Polynomial Functions

Write the equation for the polynomial shown in this graph:

Graph 2 write funct

Possible Answers:

Correct answer:

Explanation:

The zeros of this polynomial are . This means that the factors equal zero when these values are plugged in.

One factor is

One factor is

The third factor is equivalent to . Set equal to 0 and multiply by 2:

Multiply these three factors:

The graph is negative since it goes down then up then down, so we have to switch all of the signs:

Example Question #9 : Graphs Of Polynomial Functions

Write the equation for the polynomial in the graph:

Graph 4 write funct

Possible Answers:

Correct answer:

Explanation:

The zeros of the polynomial are . That means that the factors equal zero when these values are plugged in.

The first factor is or equivalently multiply both sides by 5: 

The second and third factors are and

Multiply:

Because the graph goes down-up-down instead of the standard up-down-up, the graph is negative, so change all of the signs:

Example Question #10 : Graphs Of Polynomial Functions

Write the equation for the polynomial in this graph:

Graph 3 write funct

Possible Answers:

Correct answer:

Explanation:

The zeros for this polynomial are . That means that the factors are equal to zero when these values are plugged in. 

or equivalently multiply both sides by 4

the first factor is 

 

multiply both sides by 3

the second factor is

 

the third factor is

 

Multiply the three factors:

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