Chapter 2: Rational Functions & Inequalities

Question 1

Describe how to graph the line y=3x-1.

Answer 1

1. Plot the y-intercept, -1, first.
2. The slope is 3 (rise = 3, run = 1). From the y-intercept, count up 3 and right 1.

Question 2

Describe how to graph the parabola y=(x-2)(x+4).

Answer 2

1. Plot the zeros, 2 and -4.
2. Since -1 is halfway between the zeros, the vertex has x-value = -1 and y-value = (-1-2)(-1+4) = (-3)(3) = -9.
3. Plot the vertex (-1,-9) and connect the three points to complete the graph.

*If the equation is given in expanded form, factor to begin.

Question 3

Where are the vertical asymptotes on the graph of y=1/f(x)?

Answer 3

The reciprocal function has vertical asymptotes where f(x)=0.

Question 4

If the graph of f(x) includes the point (2, 4), what is the corresponding point on the graph of y=1/f(x)?

Answer 4

(2, 1/4)

Take the reciprocal of the y-value.

Question 5

What is the reciprocal of 1?

What is the reciprocal of -1?

Answer 5

The reciprocal of 1 is 1.

The reciprocal of -1 is -1.

This is why we identify points with y-values of 1 and -1 when graphing reciprocal functions. These points appear on both graphs.

Question 6

When might the graph of a rational function have an oblique asymptote?

Answer 6

When the degree of the numerator is greater than the degree of the denominator.

Proceed using synthetic division or long division to rewrite the rational funciton in quotient form.

Question 7

When does the graph of rational function have a horizontal asymptote at y=0?

Answer 7

When the degree of the numerator is less than the degree of the denominator.

Question 8

What can we conclude if the degree of the numerator and denominator of a rational function is the same?

Answer 8

The graph of the rational function has a horizontal asymptote at y=a, where a is the ratio of the leading coefficients.

Question 9

How do you identify the vertical asymptote(s) of a rational function?

Answer 9

1. Factor the denominator if possible.
2. Identify values of x that make the denominator equal to zero.
3. If these values of x give a non-zero value in the numerator, then you have found a VA.

Question 10

How do you identify a hole in a rational function?

Answer 10

A hole occurs when the numerator and denominator have a common factor.

For example, a common factor of (x-2) implies a hole at x=2. Subbing in 2 into the function gives zero over zero.

Question 11

How do you solve a polynomial equation algebraically?

Answer 11

1. If necessary manipulate the equation so that one side equals zero.
2. Factor the polynomial to easily identify values of x for which f(x)=0.

Question 12

How do you solve the polynomial equation f(x)=0 using graphing technology?

Answer 12

1. Graph y=f(x).
2. Indentify the zeros (x-intercepts)

​On a TI-83/TI-84, use 2:zero from the CALC menu.

Question 13

What are the solutions of (x²-x+6)/(x-4)=0?

Answer 13

1. Since the right side equals zero, we only need to identify values for which the numerator equals zero.
2. In this case, x²-x+6=(x-3)(x+2), so the solutions are x=3 and x=-2.

Note that the only restriction is that x cannot equal 4, so both solutions are valid.

Question 14

When is cross-multiplying a valid strategy for solving a rational equation?

Answer 14

It can be used when both the left side and right side are single fractions.

If you have more than one fraction on either side, they can first be combined by finding a common denominator.

Question 15

What is the formula for problems that involve comparing the time is takes for two people to complete a task alone versus together?

Answer 15

1/(t_together) = 1/t₁ + 1/t₂

Question 16

When solving a linear inequality, when does the direction of the inequality change.

Answer 16

The direction of the inequality changes when both side are multiplied or divided by a negative number.

Question 17

What is the first step in solving the polynomial inequality f(x)>0?

Answer 17

Factor the polynomial to determine when f(x)=0.

We do this because a sign change will always occur at an x-intercept.

Question 18

How is a sign chart used to solve an inequality?

Answer 18

Values for which f(x)=0 allow the domain to be divided into intervals. For a rational function, also include values for which f(x) is undefined.

Include a row for each factor of f(x) in the table and then consider their product/quotient in the bottom row.

A test value from each interval indicates whether values within the interval are positive or negative.

Question 19

Can you cross-multiply/multiply by the LCD to solve a rational inequality?

Answer 19

NO!

Assuming the denominator contains at least one variable, that means multiplying by a factor that could be either positive or negative, depending on the value of x.

Since an inequality maintained when we multiply by a positive number, but reversed when we multiple by a negative number, we cannot proceed this way.