12. Graph Transformations

Question 1

LEVEL 8

Given the following graph of f(x)=x^3+3x^2, find the graph of f(x)+1.

Select the correct plot from the list below:

A:

B:

C:

D:

CORRECT ANSWER:  A

WORKED SOLUTION:

We are trying to draw the equation f(x)+1.

Because the +1 is outside the bracket we know that it will be affecting the y values, and because it is positive it will be moving upward.

So, the points we are given will go from (-2,4)\rightarrow(-2,4+1)\rightarrow(-2,5) and (0,0)\rightarrow(0,0+1)\rightarrow(0,1).

And now we just need to connect these points in the same way as the original graph.

Question 2

LEVEL 8

Given the following graph of f(x)=x^2-9x+20, find the graph of f(x-1).

Select the correct answer from the list below:

A:

B:

C:

D:

CORRECT ANSWER:  C

WORKED SOLUTION:

We are trying to draw the equation f(x-1). Because the -1 is inside the bracket we know that it will be affecting the x values, and because it is negative, we do the opposite of what we expect and move right. So, the point we are given will go from (4.5,-0.25)\rightarrow(4.5+1,-0.25)\rightarrow(5.5,-0.25). In fact, we can see where this graph crosses the x-axis, so we can see how these points will change too: (5,0)\rightarrow(5+1,0)\rightarrow(6,0) and (6,0)\rightarrow(6+1,0)\rightarrow(7,0).

And now we just need to connect these in the same way as the original graph.

Question 3

LEVEL 8

Given the equation f(x)=x^2-16, find the graph of -f(x).

Select the correct plot from the list below:

A:

B:

C:

D:

CORRECT ANSWER:  D

WORKED SOLUTION:

To sketch -f(x) we have a couple of methods:

Method 1: Draw -f(x) from f(x)

To sketch f(x) we need to first factorise it (note that this is the difference of two squares).
f(x)=x^2-16=(x+4)(x-4)

If we now put this as y=(x+4)(x-4), it will cross the x-axis at -4 and 4.

And now we just need to find out where it crosses the y-axis, which will be when x=0

y=(x+4)(x-4)
y=(0+4)(0-4)
y=4\times(-4)
y=-16

And now we just need to plot the points and connect them

Now that we have f(x) we just have to remember that -f(x) is just a flip in the x-axis. In this case, the points on the x-axis will stay the same and the y co-ordinates will be multiplied by -1.

Method 2: Drawing -f(x) directly

We know that f(x)=x^2-16, so -f(x)=-(x^2-16)=16-x^2. Now, all we have to do is follow the same steps:

-f(x)=16-x^2
-f(x)=(4-x)(4+x)
y=(4-x)(4+x)

As we can see, graph will cross at (-4,0) and (4,0), which is the same as the graph above. And then, if we put x=0 to find y we get

y=(4-x)(4+x)
y=(4-0)(4+0)
y=4\times4=16

These points will therefore gives us the same graph as in Method 1.

Question 4

LEVEL 8

Given that the point E(1,5) lies on the graph f(x), what will be the new position of A under the transformation f(x+4)?

Select the correct answer from the list below:

A: (-3,5)

B: (5,5)

C: (1,1)

D: (1,9)

CORRECT ANSWER:  A

WORKED SOLUTION:

Because the +4 is inside the bracket we know it will be affecting the x value. However, because it is in the bracket, it will do the opposite of what we expect, so will move left 4.

Question 5

LEVEL 8

Given that the point E(-7,3) lies on the graph f(x), what will be the new position of A under the transformation f(-x).

Select the correct answer from the list below:

A: (-7,-3)

B: (7,-3)

C: (7,3)

D: (-7,3)

CORRECT ANSWER:  C

WORKED SOLUTION:

Because the minus is inside the brackets, we know it will be affecting the x value. However, we know that this means we flip everything in the y-axis, which we do my multiplying the x co-ordinate by -1.