plz help!! functions graphing

Question

Graph the transformations that are applied to 𝑦 = 𝑥2 to obtain the quadratic: 𝑓(𝑥) = 2(𝑥 + 1)2 + 5

1. Include a table of values for your base graph (𝑦 = 𝑥2)
2. Be sure to label each graph
3. List the transformations applied to 𝑦 = 𝑥2 to obtain the quadratic assigned to you

2. Given 𝑓(𝑥) = 2𝑥2 − 3𝑥 + 4 and 𝑔(𝑥) = −3𝑥 + 2, determine 𝑓(2𝑦) − 𝑔(−4𝑦)

Answer 1

Starting with the parent function y = x², notice in going to f(x) = 2(x + 1)² + 5 that grouped with the "x" is a "+1". This translates the graph in the x-direction (side-to-side) but in the opposite direction of the way it reads so, in this case, it moves the graph of y = x² to the left by 1 unit, or, in other words, you subtract 1 from every x-value.

Then, notice that the multiplier "2" is outside the squaring function which means that it stretches the graph in the y-direction by a factor of 2, or, in other words, you multiply every y-value by 2.

Then notice that you have a "+5" at the end. This means that the graph is moved up 5 units, or, in other words, you add 5 to every y-value.

Table of values:

x y(for parent function) new x (left 1) first y-transformation (x 2) 2nd y-transformation (up 5)

-2 4 -3 8 13

-1 1 -2 2 7

0 0 -1 0 5

1 1 0 2 7

2 4 1 8 13

In other words:

1. What used to be (-2, 4) is now (-3, 13)
2. What used to be (-1, 1) is now (-2, 7)
3. What used to be (0, 0) is now (-1, 5)
4. What used to be (1, 1) is now (0, 7)
5. What used to be (2, 4) is now (1, 13)

For problem 2:

If f(x) = 2𝑥² − 3𝑥 + 4 and g(x) = −3𝑥 + 2:

f(2y) = 2(2y)² − 3(2y) + 4 = 2(4y²) - 6y + 4 = 8y² - 6y + 4

g(-4y) = -3(-4y) + 2 = 12y + 2

So f(2y) - g(-4y) = (8y² - 6y + 4) - (12y + 2) = 8y² - 6y + 4 - 12y - 2 = 8y² - 18y + 2