4. Graphs and Transformations

Question 1

Order of an equation

Answer 1

The highest power of x

Question 2

Shape of a positive cubic

Answer 2

Starts low, goes up, down again and all the way up

Question 3

Shape of a negative cubic

Answer 3

Starts high, goes down, back up and then all the way down

Question 4

Roots when axis is touched/crossed

Answer 4

When an axis is crossed, there is one root there, 2+ when it only touches at a point

Question 5

How to sketch a cubic

Answer 5

Shape (+/- x^3)
Roots (Factorise)
y-intercept (Substitute x = 0)

Question 6

When there are 3 equal roots in a cubic

Answer 6

The curve meets the x-axis once at the point of inflection, carries along the x-axis before rising/falling

Question 7

If some of the roots are imaginary

Answer 7

Draw the same graph shape even if you don’t know where the curve is just only touching the x-axis at the real roots

Question 8

To go from roots to equation

Answer 8

Multiply each (x - root) together
If the y-intercepts aren't equal, multiply all by a factor so you have the correct y-intercept

Question 9

Positive/negative quartic shape

Answer 9

Positive is like a w, negative like an m

Can have other shapes due to touching at repeated roots

Question 10

How to sketch a quartic?

Answer 10

Shape (+/- x^4)
Roots (Factorise)
y-intercept (Substitute x = 0)

Question 11

If there are two equal roots

Answer 11

The graph emerges on the same side of the axis as it arrived from

Question 12

What to add on reciprocal graphs?

Answer 12

Dotted lines on the axes as they are asymptotes (lines the graph approaches but never touches)

Question 13

Positive reciprocal

Answer 13

Curve on top-right and bottom left of graph

Question 14

Negative reciprocal

Answer 14

Curves on top left and bottom right of graph

Question 15

How to draw a reciprocal graph?

Answer 15

Use sign of a to locate curves and size to gauge steepness
Sketch the graph
Label points at x=1 and y=1 as guides
Draw dotted lines for asymptotes

Question 16

Reciprocal over x^2

Answer 16

Positive: curve on top half of graph
Negative: curve on bottom half of graph
Steeper gradient than just x
Plot x = 1 and -1
Draw in asymptotes

Question 17

f(x+a)

Answer 17

Translates along the x-axis by -a

Question 18

f(x) + a

Answer 18

Translates along the y-axis by a

Question 19

f(ax)

Answer 19

Enlargement of x by 1/a

Question 20

af(x)

Answer 20

Enlargement of y by a

Question 21

f(-x)

Answer 21

Reflection in the y-axis