Chapter 4 - Graphs and Transformations

Question 1

If p is a root of the function y= f(x), where does the graph cross the x-axis?

Answer 1

It crosses at the point: (p,0).

Question 2

Where are the asymptotes of the graph

y = k

x

Answer 2

Asymptotes at x= 0 and at y= 0. The lines are in top right and bottom left section of the graph.

Question 3

Where are the asymptotes for the graph

y = - k

x

Answer 3

Asymptotes at x= 0 and at y= 0. The lines are in the top left and bottom right section of the graph.

Question 4

Where are the asymptotes for the graph

y = k

x²

Answer 4

Asymptotes at x= 0 and at y= 0. The lines are in the top right and top left sections of the graph.

Question 5

Where are the asymptotes for the graph

y = - k

x²

Answer 5

Asymptotes at x= 0 and at y= 0. The lines are in the bottom right and bottom left sections of the graph.

Question 6

How do you find the x-coordinates at the point of intersections of two curves y= f(x) and y= g(x).

Answer 6

You find the solutions of the equation: f(x) = g(x).

Question 7

what happens to the graph of y= f(x) when it is translated by y= f(x) + a?

Answer 7

It is translated by the column vector:

( 0 )

( a )

It moves vertically.

Question 8

what happens to the graph of y= f(x) when it is translated by y= f(x + a)?

Answer 8

It is translated by the vector:

( -a )

( 0 )

It moves horizontally.

Question 9

When you translate a function, what happens to the asymptotes?

Answer 9

They get translated too by the same amount.

Question 10

what happens to the graph of y= f(x) when it is stretched by y= af(x)?

Answer 10

It is stretched by a scale factor of a in the vertical direction.

Question 11

what happens to the graph of y= f(x) when it is stretched by y= f(ax)?

Answer 11

It is stretched by a scale factor of: 1/a in the horizontal direction.

Question 12

what happens to the graph of y= f(x) when it is reflected by y= -f(x)?

Answer 12

It is reflected in the x-axis ie all the y values change sign

Question 13

what happens to the graph of y= f(x) when it is reflected by y= f(-x)?

Answer 13

It is reflected in the y-axis ie. all the x values change sign

Question 14

What kind of shape does a repeated root in y= f(x) cause? e.g. (x + 1)²

Answer 14

It causes a parabola shape (down-stationary-up) Or (Up-stationary-down)

Question 15

What kind of point does a triple repeated root in Y= f(x) cause? E.g. (x + 1)³

Answer 15

It causes a point of inflection. (Down-stationary-down) Or (Up-stationary-up)

Question 16

How do you sketch a curve?

Answer 16

• Identify what kind of curve it is eg is it a straight line, quadratic, cubic
• work out which way it slopes ie is it negative gradient (straight line), + or - coeff of x² or x³
• work out where it crosses y axis (when x = 0)
• factorise to work out where it crosses x axis (when y = 0)
• mark on graph every point where curve cuts axes