FMAT pure1 rational functions and graphs

Question 1

What is a rational function?

Answer 1

A rational function is of the form f(x)/g(x), where f(x) and g(x) are polynomials, and g(x) has a degree of at least 1.

Question 2

What are the steps to sketch the graph of a rational function?

Answer 2

1. Find points where the graph cuts the axes.
2. Identify vertical asymptotes.
3. Determine horizontal or oblique asymptotes.
4. Complete the sketch considering the behaviour near asymptotes.

Question 3

How do you find where a rational function cuts the axes?

Answer 3

• For x-axis: Find x where the numerator is zero.
• For y-axis: Substitute x = 0, unless undefined.

Question 4

What determines the horizontal or oblique asymptotes of a rational function?

Answer 4

1. Degree of denominator > degree of numerator: y → 0 (horizontal asymptote at y = 0).
2. Degree of denominator = degree of numerator: y → leading coefficients ratio (horizontal asymptote).
3. Degree of numerator > degree of denominator: Oblique asymptote found by polynomial division.

Question 5

Example: Sketch y = (x-2)(x+3) / (x+2)(x-3)

Answer 5

1. Cuts axes at (2, 0) (-3, 0) and (0, 1)
2. Vertical asymptotes: x = -2, x = 3.
3. Horizontal asymptote: y = 1.
4. Use sign analysis to complete the graph.

Question 6

What is the range of a rational function?

Answer 6

The range is the set of y-values the function can take. Use calculus or quadratic theory to determine turning points and gaps in the range.

Question 7

Example: For y = (x² + 3x + 3) / (x + 1), find asymptotes and turning points.

Answer 7

1. Vertical asymptote: x = -1.
2. Divide to find oblique asymptote: y = x + 2.
3. Turning points at (0, 3) and (-2, -1).

Question 8

How can inequalities involving rational functions be solved?

Answer 8

1. Sketch the graph of the function.
2. Identify intervals where the graph is above or below the x-axis or another graph.

Question 9

Example: Solve (x-1)(x+2)/(x+1)(x-3) < 0

Answer 9

1. Find zeros: x = 1, x = -2.
2. Find vertical asymptotes: x = -1, x = 3.
3. Solution: -2 < x < -1 or 1 < x < 3.

Question 10

What is the behaviour of a rational function near a vertical asymptote?

Answer 10

The graph tends to ±∞ as x approaches the vertical asymptote from either side, depending on the sign of the function near the asymptote.

Question 11

Example: Solve (2x + 3)/(x + 1) ≥ 1

Answer 11

1. Rearrange: (2x + 3)/(x + 1) - 1 ≥ 0.
2. Simplify: (x + 2)/(x + 1) ≥ 0.
3. Solution: x ≤ -2 or x > -1.

Question 12

How do you find the equation of an oblique asymptote?

Answer 12

Divide the numerator by the denominator using polynomial division. The quotient is the oblique asymptote.

Question 13

What does the modulus transformation f(|x|) do to a graph?

Answer 13

1. Reflects the portion of the graph where x > 0 in the y-axis.
2. The graph is symmetrical about the y-axis.

Question 14

What does the modulus transformation |f(x)| do to a graph?

Answer 14

1. Retains the portion of the graph where f(x) ≥ 0.
2. Reflects the portion of the graph where f(x) < 0 in the x-axis.

Question 15

How do you sketch the graph of y = |f(x)|?

Answer 15

1. Sketch y = f(x).
2. Reflect any part of the graph below the x-axis above it.

Question 16

Example: Sketch y = |(x-2)(x+1)|

Answer 16

1. Sketch y = (x-2)(x+1).
2. Reflect the portion of the curve below the x-axis.

Question 17

What does squaring a function do to its graph?

Answer 17

1. Retains the positive part of the graph.
2. Reflects the negative part above the x-axis and squares the y-values.

Question 18

How do you sketch the graph of y = f(x)^2?

Answer 18

1. Sketch y = f(x).
2. Square the y-values and reflect negative values above the x-axis.

Question 19

Example: Sketch y = [(x-1)(x+3)]^2.

Answer 19

1. Sketch y = (x-1)(x+3).
2. Square the y-values and reflect negative parts above the x-axis.

Question 20

What is the effect of taking the reciprocal of a function on its graph?

Answer 20

1. Vertical asymptotes of f(x) become horizontal asymptotes of 1/f(x).
2. Points where f(x) = 1 or -1 are unchanged.
3. Points where f(x) = 0 become vertical asymptotes.

Question 21

How do you sketch the graph of y = 1/f(x)?

Answer 21

1. Identify zeros of f(x) (these become vertical asymptotes).
2. Identify points where f(x) = 1 or -1.
3. Sketch reciprocal behaviour near these points.

Question 22

Example: Sketch y = 1 / [(x-2)(x+3)]

Answer 22

1. Vertical asymptotes: x = 2, x = -3.
2. Points where f(x) = ±1 remain fixed.
3. Reciprocal behaviour: y → 0 as x → ±∞.

Question 23

What happens to asymptotes when y = 1/f(x)?

Answer 23

1. Horizontal asymptotes of f(x) at y = k become vertical asymptotes of y = 1/k.
2. Vertical asymptotes of f(x) become zeros of 1/f(x).

Question 24

How do transformations affect turning points on a graph?

Answer 24

1. If f(x) has a maximum at (a, b), then y = 1/f(x) has a minimum at (a, 1/b).
2. If f(x) has a minimum at (a, b), then y = 1/f(x) has a maximum at (a, 1/b).