Further Graphs

Question 1

On a number line, what is the relative position of a if a < b?

Answer 1

a is to the left of b.

Question 2

Algebraically, how is the inequality a < b defined in terms of b - a?

Answer 2

b - a > 0.

Question 3

What operation requires you to flip the sign of an inequality?

Answer 3

Multiplying or dividing by a negative number.

Question 4

What is the first step when solving a quadratic inequality?

Answer 4

Move everything to one side of the inequality.

Question 5

When reading a graph to solve an inequality, what do the regions above the x-axis represent?

Answer 5

Positive values.

Question 6

When reading a graph to solve an inequality, what do the regions below the x-axis represent?

Answer 6

Negative values.

Question 7

What is the core geometric idea represented by the absolute value|x|?

Answer 7

The distance from 0.

Question 8

What are the two solutions for the equation |x| = a?

Answer 8

x = pm a.

Question 9

What region of values satisfies the absolute value inequality |x| < a?

Answer 9

Values between -a and a.

Question 10

What region of values satisfies the absolute value inequality |x| > a?

Answer 10

Values outside -a and a.

Question 11

What is the primary algebraic method used to solve absolute value expressions?

Answer 11

Split the expression into two cases (positive and negative).

Question 12

When a variable is in the denominator of an inequality, what should you multiply by to solve it?

Answer 12

The square of the denominator.

Question 13

Why must you exclude values where the denominator equals zero when solving fractional inequalities?

Answer 13

The function is undefined at those points.

Question 14

At what two types of points can the sign of a function change?

Answer 14

Zeroes and discontinuities.

Question 15

What tool is used to determine the sign of a function between critical points?

Answer 15

A sign table using test values.

Question 16

What are the two requirements for a vertical asymptote at x = c in a rational function?

Answer 16

The denominator equals zero and the numerator does not equal zero.

Question 17

How do you determine the direction of a graph as it approaches a vertical asymptote?

Answer 17

Use a sign table to check values near the asymptote.

Question 18

How does the sign of a reciprocal graph 1/f(x) compare to the original function f(x)?

Answer 18

It has the same sign as the original function.

Question 19

What do the zeros of f(x) become in the reciprocal graph 1/f(x)?

Answer 19

Vertical asymptotes.

Question 20

How does the domain of 1/f(x) relate to the zeros of f(x)?

Answer 20

The domain excludes all values where f(x) = 0

Question 21

In a reciprocal transformation, what happens to the intervals where the original graph was increasing?

Answer 21

They become decreasing.

Question 22

In a reciprocal transformation, what happens to the maximum points of the original graph?

Answer 22

They become minimum points.

Question 23

How are the y-values calculated for the sum graph f(x) + g(x)?

Answer 23

By adding the y-values of f(x) and g(x) at each point.

Question 24

If f(x) = 0, what is the result of the sum graph f(x) + g(x) at that point?

Answer 24

The result is equal to the other graph g(x).

Question 25

If f(x) = g(x), what happens to the $y$-value of the sum graph f(x) + g(x)?

Answer 25

The y-value is doubled.

Question 26

If f(x) and g(x) are opposites at a point, what is the result of their sum?

Answer 26

Zero.

Question 27

What is the resulting sign of the product graph f(x) x g(x) if both functions have the same sign?

Answer 27

Positive.

Question 28

What is the resulting sign of the product graph f(x) x g(x) if the functions have different signs?

Answer 28

Negative.

Question 29

What is the 'Golden Rule' for applying graph transformations?

Answer 29

Start with the original graph and apply changes step-by-step.

Question 30

Which values are affected by a transformation inside the function, such as f(|x|)?

Answer 30

x-values.

Question 31

Which values are affected by a transformation outside the function, such as |f(x)|?

Answer 31

y-values.

Question 32

How do you graphically construct y = f(|x|) from y = f(x)?

Answer 32

Keep the right side and reflect it to the left side.

Question 33

What type of symmetry is created by the transformation y = f(|x|)?

Answer 33

Symmetry about the y-axis.

Question 34

How do you graphically construct y = |f(x)| from y = f(x)?

Answer 34

Keep the part above the x-axis and reflect the negative parts upward.

Question 35

What symmetry is associated with the transformation y = |f(x)|?

Answer 35

Symmetry about the x-axis.

Question 36

How do you graphically construct |y| = f(x) from y = f(x)?

Answer 36

Keep the top half and reflect it downward.

Question 37

How many branches are typically formed in the graph of |y| = f(x)?

Answer 37

Two branches (top and bottom).

Question 38

What symmetry is found in the graph of |y| = f(|x|)?

Answer 38

Symmetry in both the x and y axes.

Question 39

What part of the original graph must be removed when transforming to y = sqrtf(x)?

Answer 39

Any part where f(x) < 0.

Question 40

How does taking the square root affect the shape of large y-values on a graph?

Answer 40

The values shrink.

Question 41

How does taking the square root affect y-values between 0 and 1?

Answer 41

The values rise.

Question 42

In the transformation y = sqrtf(x), what happens to the overall steepness of the graph?

Answer 42

The graph becomes less steep.

Question 43

What is the algebraic rewrite of the relation y^2 = f(x)?

Answer 43

y = pm sqrt f(x).

Question 44

Why is the graph of y^2 = f(x) generally not considered a function?

Answer 44

It has two branches (top and bottom) for each x value.

Question 45

What are the two primary 'Domain Awareness' checks for functions?

Answer 45

Square roots cannot have negatives and denominators cannot be zero.

Question 46

Which specific transformation results in symmetry across both axes, saving time in exams?

Answer 46

|y| = f(|x|).

Question 47

What is the procedural step for finding the inverse of a function algebraically?

Answer 47

Swap the x and y variables.

Question 48

Across which line is a graph reflected to find its inverse?

Answer 48

The line y = x

Question 49

How do the domain and range of a function relate to its inverse?

Answer 49

They swap (domain becomes range, and range becomes domain).

Question 50

What test must a graph pass for its inverse to be a function?

Answer 50

The horizontal line test.

Question 51

What is the standard notation for an inverse function?

Answer 51

f^{-1}(x)

Question 52

In parametric equations, x and y are both expressed as functions of what variable?

Answer 52

The parameter t

Question 53

How do you find the Cartesian equation from a set of parametric equations?

Answer 53

Eliminate the parameter t.