Graphs and transformations

Question 1

If p is a root of the function f(x), what happens to the graph of y = f(x)?

Answer 1

The graph touches or crosses the x-axis at the point (p, 0).

A root of a function is a value for which the function equals zero.

Question 2

What are the asymptotes of the graphs of y = k/x and y = k/x² where k is a real constant?

Answer 2

Asymptotes at x = 0 and y = 0.

Asymptotes are lines that the graph approaches but never touches.

Question 3

What do the x-coordinates at the points of intersection of the curves y = f(x) and y = g(x) represent?

Answer 3

They are the solution(s) to the equation f(x) = g(x).

Intersection points indicate where two functions have the same value.

Question 4

What does the graph of y = f(x) + a represent?

Answer 4

A translation of the graph y = f(x) by the vector (0, a).

This shifts the graph vertically by ‘a’ units.

Question 5

What does the graph of y = f(x + a) represent?

Answer 5

A translation of the graph y = f(x) by the vector (-a, 0).

This shifts the graph horizontally by ‘-a’ units.

Question 6

What happens to asymptotes when you translate a function?

Answer 6

Any asymptotes are also translated.

This means their positions change according to the translation applied.

Question 7

What does the graph of y = af(x) represent?

Answer 7

A stretch of the graph y = f(x) by a scale factor of ‘a’ in the vertical direction.

If ‘a’ > 1, the graph stretches; if 0 < ‘a’ < 1, it compresses.

Question 8

What does the graph of y = f(ax) represent?

Answer 8

A stretch of the graph y = f(x) by a scale factor of 1/a in the horizontal direction.

This affects the width of the graph.

Question 9

What does the graph of y = -f(x) represent?

Answer 9

A reflection of the graph of y = f(x) in the x-axis.

This flips the graph over the x-axis.

Question 10

What does the graph of y = f(-x) represent?

Answer 10

A reflection of the graph of y = f(x) in the y-axis.

This flips the graph over the y-axis.