Graph Question (Please Help!)

Question

A graph with two linear functions; f of x passes through 1, 3 and 3, 13, and g of x passes through negative 1, 3 and 1, 13

Part A: Describe two types of transformations that can be used to transform f(x) to g(x). (2 points)

Part B: Solve for k in each type of transformation. (4 points)

Part C: Write an equation for each type of transformation that can be used to transform f(x) to g(x). (4 points)

Answer 1

Before you can answer this, you must first define the function f(x) by find the slope of the line (using the slope formula) and the point-slope equation. I'll use the points in order as given.

(x₁,y₁)=(1,3) & (x₂,y₂)=(3,13)

First, use the slope formula to find the slope [Slope formula: m=(y₂-y₁)/(x₂-x₁)]

(y2-y1) = (13-3) = (10) = (5)

---------- -------- ---- --- = 5 so the slope is a whole number, m=5 (rises 5 units as it runs 1 unit)

(x2-x1) = (3-1) = (2) = (1)

Next, use the point-slope formula to find the equation of the line (Note: either point can be used as they will be on the same line, but the slope must be used corrected) [Point-slope formula: y-y₁=m*(x-x₁)]

Using m=5 & (x₁,y₁)=(1,3) we will get:

y-y₁=m*(x-x₁) written as y-(3)=(5)(x-(1)), y-3=5(x-1)

Now, we will solve for y to get the liner equation form (y=mx+b), so:

y-3=5(x-1) [distribute multiplication]

y-3=5x-5 [eliminate number next to variable by canceling]

y-3+3=5x-5+3 [this cancelation involves adding 3 to both sides]

y=5x-2

[Check to make sure this works before advancing further!

If x=1, then y=5(1)-2; y=5-2, y=3 So when x=1, y=3 (1,3) This works!

If x=3, then y=5(3)-2; y=15-2; y=13, So when x=3, y=13 (3,13) This works so our line is correct!

Now, is this a function? Absolutely! Remember a function is an equation where one independent variable equals one dependent variable (meaning, x will not equal 2 or more y numbers. It exhibits 1 to 1, 1 variable for one answer). Linear equations will generally be functions (with the exception of x=a). So, this linear equation is a function! f(x)=y so

f(x)=mx+b; meaning m=5 & b=(-2) (this is the slope-intercept). Substituting this into the formula gives us: f(x)=5x-2

Now for g(x) (this will be the same process for f(x)):

(x₁,y₁)=(-1,3) & (x₂,y₂)=(1,13)

(y2-y1) = (13-(3)) = (10) = (5)

---------- -------- ---- --- = 5 so m=5 again

(x2-x1) = (1-(-1)) = (2) = (1)

Using m=5 & (x₁,y₁)=(-1,3) we will get:

y-y₁=m*(x-x₁) written as y-(3)=(5)(x-(-1)), y-3=5(x+1)

y-3=5(x+1)

y-3=5x+5

y+3-3=5x+5+3

y=5x+8

g(x)=5x+8, where m=5 & b=8

So our two functions, f(x)=5x-2 & g(x)=-5x+8, can now be transformed

Part A), there are two types of transformation we are looking for and two that are easily done. First is the vertical transformation (which we’ll be solving for in Part B) and another is horizontal transform. We could also do a reflection or vertical/horizontal stretches as well, but not needed in this problem as the slopes are the same. A vertical transformation shifts the y intercept b up or down a number of units. This is usually represented as f(x)+k and f(x)-k. A horizontal shift is similar, it shifts the equation at the x variable a number of units to the left or the right. This is usually represented as f(x+h) and f(x-h). For these two functions, we will use the vertical and horizontal shifts, because we are trying to transform f(x) into g(x) (meaning, we want to turn the first equation into the second!) and their slopes are the same (m=5).

(This may answer Part C)

1) f(x)=5x-2+k

2) f(x)=5(x+h)-2

In the next part, we will solve the constant k to find out the value needed to make f(x)=g(x) in both equations.

Part B) We will be putting in the variables given to us from the points in g(x) [(x₁,y₁)=(-1,3) or (x₂,y₂)=(1,13)] to give us the same answers. We will put x1 & y1 (remember, f(x)=y) into f(x) to solve for k

1) f(x)=5x-2+k & (x₁,y₁)=(-1,3) so (3)=5(-1)-2+k

3=-5-2+k

3(+7)=-7+k(+7)

10=k

So if k=10, then f(x)=5x-2+10

f(x)=5x+8 (which is g(x)=5x+8)

f(x)=5x-2-k using the same point, so (3)=5(-1)-2-k

3=-5-2-k

3=-7-k

3(+7)=-7(+7)-k

10=-k (Note: you can divide by -1 and end up with k=-10, however this solution will not work as it will produce the line y=5x-12)

The horizontal shift

f(x+h) with f(x)=5x-2 gives us f(x)=5(x+h)-2. So using the same point (x₁,y₁)=(-1,3), we get

(3)=5((-1)+h)-2, which simplifies to 3=5(h-1)-2

3=5h-5-2

3=5h-7

3(+7)=5h-7(+7)

10=5h

2=h

So, when h=2, then f(x)=5(x+h)-2 is

f(x)=5(x+2)-2

f(x)=5x+10-2

f(x)=5x+8 which is g(x)=5x+8

Try solving these transformations with the other point (3,13) on your own for practice.