### All Precalculus Resources

## 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:**

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 ?

**Possible Answers:**

**Correct answer:**

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^{ }?

**Possible Answers:**

**Correct answer:**

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 #1 : How To Graph An Exponential Function

Give the -intercept of the graph of the function

Round to the nearest tenth, if applicable.

**Possible Answers:**

The graph has no -interceptx

**Correct answer:**

The -intercept is , where :

The -intercept is .

### Example Question #91 : High School: Functions

Graph the following function and identify the zeros.

**Possible Answers:**

**Correct answer:**

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 #1 : Graph A Polynomial Function

Graph the function and identify the roots.

**Possible Answers:**

**Correct answer:**

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 : Graph A Polynomial Function

Graph the function and identify its roots.

**Possible Answers:**

**Correct answer:**

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 #92 : High School: Functions

Graph the function and identify its roots.

**Possible Answers:**

**Correct answer:**

This question tests one's ability to graph a polynomial function.

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.

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?

**Possible Answers:**

**Correct answer:**

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 #1 : Write The Equation Of A Polynomial Function Based On Its Graph

Write the quadratic function for the graph:

**Possible Answers:**

**Correct answer:**

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

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